Modelling Crowd Dynamics: a Multiscale, Measure-theoretical Approach

Mo delling Cro wd Dynamics: a Multiscale, Measure-theoretical Approac h Master’s Thesis Industrial and Applied Mathematics Eindho ven Univ ersity of T ec hnology , The Netherlands Author: Jo ep Ev ers Sup ervisor: Dr. Adrian Mun tean Ma y 30, 2011 Abstract W e presen t a strategy capable of describing basic features of the dynamics of cro wds. The b eha viour of the cro wd is considered from a tw ofold p erspective: b oth macroscopically and microscopically . W e examine the large scale b eha viour of the cro wd (considering it as a con tinuum), sim ultaneously b eing able to capture phenomena happ ening at the individual p edestrian’s lev el. W e unify the micro and macro approac hes in a single mo del, b y working with general mass measures and their transp ort. W e impro ve existing mo delling by coupling a measure-theoretical framework with basic ideas of mixture theory formulated in terms of measures. This strategy allo ws us to define sev eral constituen ts (subp opulations) of the large crowd, each having its own partial velocity . W e th us ha ve the p ossibilit y to examine the in teractive b eha viour b et ween subp opulations that ha ve distinct characteristics. W e giv e sp ecial features to those p edestrians that are repre- sen ted b y the microscopic (discrete) part. In real life situations they w ould play the role of firemen, tourist guides, leaders, terrorists, predators etc. Since we are in terested in the global b eha viour of the rest of the cro wd, w e mo del this part as a con tin uum. By iden tifying a suitable concept of entrop y , we derive an entrop y inequalit y and sho w that our mo del agrees with a Clausius-Duhem-lik e inequalit y . F rom this inequalit y natural re- strictions on the proposed v elo cit y fields follo w; ob eying these restrictions mak es our mo del compatible with thermo dynamics. W e prov e existence and uniqueness of a solution to a time-disc rete transp ort problem for general mass measures. Moreo ver, we show prop erties lik e p ositivity of the solution and con- serv ation of mass. Although afterw ards w e opt for a particular form, our results are v alid for mass measures in their most general app earance. W e giv e a robust scheme to appro ximate the solution and illustrate n umerically tw o-scale micro-macro b ehaviour. W e exp eriment with a num b er of scenarios, in order to capture the emergen t qualitative behaviour. Finally , we formulate op en problems and basic researc h questions, inspired by our mo delling, analysis and sim ulation results. Keyw ords: Cro wd dynamics; Social and b ehavioral sciences; Conserv ation la ws; Micro- macro mo dels; Mass measures; Thermo dynamics; Mixture theory; So cial net works; Initial v alue problems; Sim ulation MSC 2010: 35Q91; 35L65; 35Q80; 28A25; 91D30; 65L05 P ACS 2010: 45.50.-j; 47.10.ab; 02.30.Cj; 02.60.Cb; 05.70.-a; 47.51.+a I w ant to suggest that ev en with our woeful ignorance of why h umans b ehav e the w ay they do, it is p ossible to mak e some predictions ab out ho w they b ehav e collectiv ely . Philip Ball, Critic al Mass 1 1 Quotation taken from [7], p. 6. Con ten ts 1 In tro duction 4 1.1 Pro cess leading to this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 So cietal back ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Mo delling approac hes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 A few p eople in the field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Con tent and structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Measure theory 13 2.1 Measures and their Leb esgue decomp osition . . . . . . . . . . . . . . . . . . . 13 2.2 Radon-Nik o dym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Prop erties of Radon-Nikodym deriv ativ es . . . . . . . . . . . . . . . . . . . . 17 2.4 Mass measure µ m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.1 Microscopic mass measure m m . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Macroscopic mass measure M m . . . . . . . . . . . . . . . . . . . . . . 18 3 Mixture theory and thermo dynamics 20 3.1 Description of a mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Balance of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 En tropy inequalit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Application to crowd dynamics 32 4.1 W eak form ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 F urther sp ecification of the velocity fields . . . . . . . . . . . . . . . . . . . . 35 4.3 Deriv ation of an en tropy inequality . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.1 One p opulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.2 Multi-comp onen t cro wd . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.3 Generalization to discrete measures . . . . . . . . . . . . . . . . . . . . 46 4.4 Deriv ation of the time-discrete mo del . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Solv abilit y of Problem ( P ) and prop erties of the solution . . . . . . . . . . . . 50 4.5.1 Solv abilit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.2 Basic prop erties of the time-discrete solution . . . . . . . . . . . . . . 54 4.5.3 Relaxing the conditions on the motion mapping and v elo city fields . . 57 4.6 Reform ulation of the prop osed v e lo cit y fields in the time-discrete setting . . . 60 4.7 En tropy inequalit y for the time-discrete problem . . . . . . . . . . . . . . . . 60 2 4.7.1 One p opulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.7.2 Multi-comp onen t cro wd . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7.3 Generalization to discrete measures . . . . . . . . . . . . . . . . . . . . 67 4.8 Tw o-scale phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Numerical sc heme 72 5.1 T yp es of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.1 Discrete measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.2 Absolutely con tinuous measure: spatial appro ximation of density . . . 72 5.2 Calculating v elo cities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.1 Discrete µ α n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.2 Absolutely con tinuous µ α n . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.3 Summary: The appro ximation ˜ v α n . . . . . . . . . . . . . . . . . . . . . 77 5.3 Push forw ard of the m ass measures . . . . . . . . . . . . . . . . . . . . . . . . 78 6 Numerical illustration: sim ulation results 81 6.1 Reference setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2 Tw o basic critical exam ples: one micro, one macro . . . . . . . . . . . . . . . 83 6.3 Tw o-scale interactions of repulsiv e nature . . . . . . . . . . . . . . . . . . . . 84 6.4 In teraction with tw o individuals . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5 Mo delling leadership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7 Discussion 93 Ac knowledgemen ts 96 A Pro of of Theorem 2.1.4 97 B Pro of of Theorem 2.1.9 99 C Proof of Lemma 2.3.1 101 C.1 Pro of of Part 1 of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 101 C.2 Pro of of Part 2 of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.3 Pro of of Part 3 of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 104 C.4 Pro of of Part 4 of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 105 D Deriv ation of the en tropy densit y for an ideal gas 108 E Mo difications in the pro ofs of Theorem 4.5.1 and Corollary 4.5.4 111 E.1 The pro of of Theorem 4.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 E.2 The pro of of Corollary 4.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 F P ap er ‘Mo deling micro-macro p edestrian coun terflow in heterogeneous domains’ 114 Bibliograph y 123 Chapter 1 In tro duction I w ould like to use this first section to introduce the central challenge treated in this thesis: the mo delling of the b ehaviour of crowds. This section is also used to place this sub ject in a broader con text. 1.1 Pro cess leading to this thesis In 2010 I w as offered the opp ortunity to take part in the Honors Program Industrial and Ap- plied Mathematics: a new initiativ e of the Departmen t of Mathematics and Computer Science (Eindho ven Univ ersity of T echnology) to c hallenge ”the b est students of IAM” by w orking in an academic, scientific setting. During these months I analyzed and extended a n umber of curren tly existing approaches to the mo delling of crowd dynamics, under sup ervision of Dr. Adrian Muntean and Prof.dr. Mark Peletier. Since this research pro ject to ok place in a relativ ely short perio d of time, w e ultimately had to conclude that there were many more things to b e explored in this field. As a natural consequence, we th us decided to dedicate my master’s thesis to the same topic. The next part of this in tro duction is intended to emphasize that we hav e not b een dealing with an irrelev ant problem. Indeed, a series of everyda y-life even ts hav e shown the imp or- tance of b eing able to predict the dynamics of a crowd. Sev eral attempts to capture human b eha viour in mathematical models hav e found their w ay into useful applications. In the follo wing sections these statements are discussed in more detail. 1.2 So cietal bac kground In his 1895 b o ok L a psycholo gie des foules , the F renc h so cial psyc hologist Gustav e Le Bon wrote: ”L’ˆ age o` u nous entrons sera v ´ eritablement l’` ere des foules ”. 1 Although Le Bon could probably at the time (the fin de si ` ecle ) not quite foresee what the t wen tieth cen tury w ould lo ok like, his words ha ve turned out to b e prophetic. In their most literal sense - disregarding an y p olitical or metaphorical meaning that Le Bon might ha ve in tended to attach to them - they describe what our world has b ecome. Indeed, during the t wen tieth cen tury the global 1 ”The age we are ab out to enter, will truly b e the era of cro wds”. The F rench quotation is taken from [39], p. 12, a republication of the original work published in 1895. 4 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology p opulation has b een ev er-increasing. Occasions inv olving large n umbers of p eople in crowded areas ha v e become familiar sights. W e exp erience such situations both under everyda y urban conditions, and during large-scale manifestations with h uge audiences (whic h tak e place oc- casionally). Most of the ”ev eryday situations” mentioned ab ov e happ en in a ‘normal’ setting, that is, without the p eople b eing in a state of panic. Although this tends to sound reassuring, it do es b y no means imply that no s p ecial atten tion is to b e paid. In order to design safe and comfortable public space, the presence of p edestrians cannot b e disregarded. The b ehaviour of individual p edestrians, p ossibly clustering together to form larger crowds, is an imp ortan t factor in urban design and traffic managemen t. Indeed, un safe and un comfortable situations are often related to congestion and high densities. T o assess the quality of the p edestrian en vironment, the follo wing questions may b e taken as a guideline (see [40]): • Are routes direct, leading the p eople where they actually wan t to go? • Is it easy to find and follo w a certain route? • Are crossings easy to use; ho w long do p edestrians hav e to wait b efore they can cross a road? • Are fo ot wa ys well-lit, and sufficiently wide; what obstructions are there? • If p edestrians, cyclists and v ehicles are mixed, do es the t ypical sp eed of road users allo w for this? These questions also indicate which measures can preven t the aforemen tioned cases of un- safet y/discomfort. The necessit y of these measures dep ends on the typical scale of p edestrian traffic (i.e. the ‘cro wdedness’) that the area has to deal with. The effectiv eness of each sp ecific measure is also determined b y its in terplay with the others. The commen ts made here, should b e extended to the semi-public domain. T o b e more con- crete, we do not only need to consider the behaviour of p eople walking in the street, on sidew alks and in parks, but also in railwa y stations (see Figure 1.1), sports stadiums and shopping malls. In fact, the inv estigation of crowd dynamics is essen tial in any setting in whic h a p erson’s desired motion is imp eded by the presence of other p eople, to preven t that incon venience escalates in to danger. If prop er understanding of, or resp onse to a crowd’s b eha viour is lac king, even ts in the (recen t) past ha v e shown us what the consequences are. These are dramatic situations during whic h people’s liv es are at stak e. An example is the pilgrimage ( Hajj ) to Mecca and other holy places in Saudi Arabia, that Muslims p erform every y ear during the Dhu al-Hijjah month. Since all adult Muslims are required to p erform such Ha jj at least once during their lifetimes (unless ph ysical or financial circumstances mak e this imp ossible), h uge n um b ers of p eople (sev eral millions) gather ev ery y ear during a relatively short p erio d of time. Near the town of Mina, pilgrims as a ritual throw p ebbles at three stone pillars ( jamar ahs ), represen ting the devil. 2 A sp ecial bridge (the Jamarat Bridge) was built is the 1960s to allow pilgrims to 2 Go d/Allah had commanded Abraham/Ibrahim to sacrifice his son Ishmael, but Satan urged Abraham to disob ey Go d. The ‘stoning of the devil’ ritual is to commemorate Abraham rejecting the temptation of the devil.[2] Mo delling Cro wd Dynamics 5 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Figure 1.1: Passengers rushing to wards the exit of Eindhov en Railwa y Station, after alighting from tw o trains that ha ve just arrived. At the transition from the platform to the staircase a b ottlenec k o ccurs. (Photo: Jo ep Julicher) tak e part in the ritual at tw o flo or levels. How ever, during the 1424 AH (2004) and 1426 AH (2006) Ha jjs several hundreds of p eople were trampled to death due to o v ercro wding at the bridge. As this urged authorities to tak e action, cro wd exp erts were asked to in vestigate the situation, and giv e recommendations for impro vemen ts; see [6, 30]. The site was completely reconstructed afterwards. The three pillars w ere replaced by wider and taller oblong concrete structures, to allo w more pilgrims at the same time, and pre- v ent them from accidentally thro wing p ebbles at each other. Crowds flow around these new ‘pillars’ more easily , by which congestion is reduced. A new multi-storey bridge w as built with b etter entrance facilities and more emergency exits. Bottlenecks were remov ed. More- o ver, the proto cols for cro wd managemen t by stew ards and other officials were reconsidered.[1] Unfortunately , crowd disasters do not alw ays happ en far a w a y from us. Recently , in July 2010, 21 p eople died at the Lo veparade in the German cit y of Duisburg, at a distance of less than 100 kilometers from Eindho ven. This festiv al to ok place in a closed-off area, which could b e en tered via a small n um b er of tunnels. Each of these ended at a common ramp/staircase, ev entually leading to the festiv al ground. Due to o vercro wding at the b ottom end of the staircase, a stamp ede o ccurred. All lethal victims died b ecause of suffo cation. After these disastrous ev en ts, there was a lot of criticism on the safet y precautions that had, or had not, b een taken. The organization of the Lov eparade w as blamed for ha ving provided to o few emergency exits to the Lov eparade area. Moreov er, the n um b er of security agen ts w as smaller than promised. On the other hand, lo cal authorities are said to ha v e ignored the p olice’s ob jections, b ecause they were so keen on ha ving this prestigious ev ent in their to wn. F or more information, see e.g. [4, 5, 3]. The ab ov e emphasizes the imp ortance and urgency of b eing able to prop erly describ e the dynamics of a crowd. This is a prerequisite for predicting cro wd b ehaviour, which in its turn is needed to an ticipate life-threatening situations. 6 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 1.3 Mo delling approac hes Ev er since the Renaissance scientists hav e w ondered how h uman b eha viour can b e captured in mathematical formalism (cf. e.g. [7]). W e fo cus now in particular on the developmen ts of the last decades, during which the description and analysis of a cro wd’s motion has gained the attention of the scientific w orld. This is mainly due to the gradual increase of the num b er of large-audience ev en ts b eing organized, and the accidents happ ening at such even ts. An illustration of these developmen ts has b een given ab ov e. Sev eral mo dels ha ve b een developed and explored to catch the crowd-related phenomena we exp erience in real life, in a scientific (mathematical) framework. These mo dels w ere treated b oth analytically and numerically , and w ere based on a n umber of distinct p ersp ectiv es. F o cussing on what happens at the level of an individual p edestrian, one obtains discrete, agen t-based or microscopic models. In this approac h the dynamics of eac h single p erson in the cro wd are mo delled and traced. This p ersp ective is adopted by Dirk Helbing et al. within the framework of a so-called so cial for c e mo del , see e.g. [31]. Roughly sp eaking, each p edes- trian is driven via Newton’s Second Law, by means of a so cial force that dep ends on the presence and so cial b eha viour of other p eople. Sim ulation is their main to ol for obtaining qualitativ e and quantitativ e information. On the other hand, one can also ‘zo om out’, considering a crowd on a more global scale. W orking on this macroscopic lev el, one uses densities, rather than individual pedestrians. Via this approac h one ends up in what is essentially a fluid-dynamic setting. A sp ecific prop ert y of these mo dels is that w e cannot capture lo cal interactions any more. Macroscopic mo delling is thus only useful if w e are in terested in av erage characteristics of the crowd and its motion. This wa y of mo delling is for instance adopted by Bertrand Maury et al. in [41], where initially a gradien t flow structure is prop osed. Their results are deriv ed analytically . Up to the initial conditions (which might b e generated by means of a random sampling), the mo dels describ ed ab ov e are fully deterministic. In this thesis w e fo cus an approac h that differs in nature from the aforemen tioned ones. Our motiv ation is that we do not wan t to b e forced to choose either of the tw o p ersp ectiv es; w e aim at ‘marrying’ the tw o p ersp ectives in one single mo del. This strategy was describ ed b y Benedetto Piccoli, Andrea T osin et al. in [45, 44, 17]. In most of the fluid-dynamic mo d- els of p edestrians, non-linear hyperb olic conserv ation laws app ear. Certain analytical and n umerical problems inheren tly arise when treating these PDEs, especially in more than one space dimension. F or instance the solution might not b e unique, sho ck wa v es can o ccur, it is imp ortant whether the corresp onding flux is con vex or not, b oundary conditions are diffi- cult to imp ose, etc. Numerically , p ositivit y of the densit y might not alwa ys b e guaran teed. In order to circum v ent these problems, Piccoli et al.’s work takes place in a (time-discrete) measure-theoretical framework. This thesis is mainly built up on the fundamen ts Piccoli cum suis pro vides. Ho wev er, we wan t to b e able to cov er a m uc h wider range of situations, for instance a crowd consisting of several distinct subp opulations (rather than only one p opulation), each of which is willing to mov e with its own desired velocity . Moreov er, we draw parallels with mixture theory and thermo dynamics, and extract inspiration from those fields. Mo delling Cro wd Dynamics 7 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology The microscopic, macroscopic and micro-macro approaches describ ed ab o v e, are not the only existing strategies. See e.g. [48] (pp. 212–214) for a systematic classification of mo dels. Par- tially based on [48], w e describ e a n um b er of differen t p ersp ectives. A t an intermediate level b etw een micro and macro are so-called mesoscopic mo dels. These mo dels do not distinguish b etw een individuals and hence cannot trace their tra jectories. Ho wev er, individual b ehaviour can b e sp ecified. The description is in terms of probabil- it y densities: mostly , the probabilit y to find an individual with sp ecified sp eed in a certain lo cation at a certain time. T o give an example, Dirk Helbing describ es vehicular traffic in suc h framework (also indicated by ‘Boltzmann-like’ or ‘gas-kinetic’ mo dels) in [29]. Moreo v er, in [28] he deriv es h ydro dynamic equations for pedestrians, based on the Boltzmann-approac h. Up to now, all mo dels hav e mainly b een based on considerations related to physics. This mostly matc hes our p oin t of view. Other views are (of course) also p ossible. The fact that we do not fully understand the underlying mechanisms of h uman b ehaviour, is most naturally incorp orated in a mo del by adding sto chasticit y . A small amount of external noise can b e added to av oid undesired situations. One should think here for example of a situation in whic h t wo individuals are p ositioned ‘head-on’ and b oth wan t to mo v e straigh t ahead. In a mo del it might happ en that a deadlo ck o ccurs, although in reality the t wo most lik ely just mov e aside slightly , and get passed one another. A bit of sto chastic fluctuation forces a breakthrough if the describ ed deadlo ck happ ens. Aw a y from suc h configurations, the influence of the noise is only small. F ul ly sto chastic mo dels are different in the sense that they do not just add random p erturba- tion to intrinsically deterministic dynamics. Here, the underlying decision-making pro cesses are influenced directly b y random effects. If you take the deterministic limit in these mo dels (i.e. v anishing sto chastic effects) the o verall phenomena are completely different from those in the sto chastic ‘normal’ case. This makes fully sto chastic models in trinsically differen t from mo dels that contain external noise only . In a c el lular automata mo del all v ariables are discrete. The crowd is describ ed at the in- dividual’s level. The spatial domain is sub divided into a n umber of cells, where each cell p ossesses a state v ariable (e.g. 0 = ‘empt y’, 1 = ‘o ccupied’). A t sp ecified (discrete) points in time, the mo del decides whic h cells are o ccupied in the subsequen t generation; the state v ariables are then up dated accordingly . F or realit y-mimicking mo dels the total num b er of o ccupied cells is constant. T ypically , the up date is rule-based, i.e. each o ccupied cell (read: particle/p edestrian) makes a decision where to go, based on the current situation and its goals. These rules are often supp orted b y psyc hological argumen ts. Possibly , some sto c hastic effects are included. The update can either b e executed in parallel (for all cells simultane- ously), or for randomly selected individuals only . Serge Ho ogendo orn and Piet Bovy mo del individuals as play ers in a differential game. This approac h w as first applied to driving b eha viour in traffic flo ws in [34]. The ideas presented therein w ere subsequently extended to p edestrian dynamics; see [33]. V ehicles and/or p edes- trians are assumed to maximize their exp ected success or profit, or follow a Zipfian principle of le ast effort . In suc h a game-theoretical approach, individuals are allow ed to mo dify their 8 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology con trol decisions based on their observ ations, and on predictions of the b ehaviour of other pla yers in their neigh b ourho o d. 1.4 A few p eople in the field Throughout 2010 and 2011 Adrian and I ha v e b een in con tact with many p eople, whose area of exp ertise was related to our research. When meeting them, w e w ere mainly in terested in what kind of questions these exp erts would lik e to b e answered by crowd models. Moreo ver, w e obtained insight in the wide range of p ossible applications. The p erson who provided us with the actual idea for studying the topic of crowd dynamics is Prof. Chris Budd (Universit y of Bath, UK). Mark Peletier in vited me to meet Prof. Budd during his stay in Eindhov en, Spring 2010. At that time, I needed to decide what I w anted to do in m y Honors Program, and these t wo p eople suggested the idea (that has ev entually led to this thesis). In January 2011, Adrian Mun tean and I visited Mark Peletier during his sabbatical stay in Bath, and we met Prof. Budd again. He was the one drawing our attention to t w o-scale phenomena in nature (birds and fish), b y sho wing us a couple of fascinating BBC mo vies. 3 Prof.dr.ir. Serge Ho ogendo orn, Dr.ir. Winnie Daamen and Mario Campanella MSc (Delft Univ ersity of T echnology) w ork on mo delling/simulation of pedestrian/traffic dynamics, and on the calibration of their models b y real-life exp eriments. 4 Winnie Daamen ga v e a talk in Eindho ven (during the CASA collo quium) in Septem b er 2010. Adrian Mun tean and I paid a visit to their T ransp ort & Planning Department in January 2011. They hav e dev elop ed a softw are pac k age called Nomad , which is a microscopic simulation to ol that can e.g. be used to assess the geometry of infrastructure. The underlying mo del is the one describ ed in [33]. The b ehaviour of the individuals is comprised in their acceleration, whic h consists of an uncon trollable and a controllable part. The uncontrollable part is due to physical interactions with their surroundings, that is, with other individuals and obstacles (part of the geometry of the domain). The controllable part contains the tendency to mini- mize the w alking cost (cf. Section 1.3). Calibration of the sim ulation is done by observing real-life traffic and p edestrian flows (as far as priv acy regulations p ermit to do so) and lab oratory exp eriments; see e.g. [14, 35]. These lab oratory exp eriments tak e place in standardized environmen ts, and participants’ p ositions are obtained by video image analysis. An example of such standard setting (which is v ery w ell-known in the field) is the c ounterflow or bidir e ctional flow : a narro w corridor in whic h t wo groups of people w ant to mo ve in opp osite directions. In this exp eriment, one expe cts to observe a sp ecific phenomenon of self-or ganization : lane-formation. That is, p eople tend to align in such a wa y that they just follo w someone going in the same direction. 5 In Figure 1.2 a schematic impression of a bidirectional flo w is given. Another standard scenario is a b ottlenec k. It o ccurs for instance if a corridor suddenly gets narrow er. At this transition, in- 3 These phenomena will b e addressed in Section 4.8 of this thesis. 4 See www.pedestrians.tudelft.nl for more information on either of these tw o areas of exp ertise. 5 A video of a bidirectional flo w exp eriment is av ailable at www.youtube.com/watch?v=J4J lOOV2E . Lanes are formed without the participan ts b eing instructed to do so (note that these lanes are unsteady). The exp erimen t w as p erformed in connection with the German Hermes pro ject; in Delft similar exp eriments are done. Mo delling Cro wd Dynamics 9 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology dividuals get clogged up in circular structures ( ar ches ) which are hard to break; this scenario is indicated in Figure 1.3. See also [48], pp. 418–419, for a description of b oth settings and of the emerging phenomena. Figure 1.2: Sc hematic dra wing of a counterflo w scenario with lane-formation. At b oth ends of the corridor individuals are supplied; they intend to w alk to the other side. Halfw ay , four lanes emerge. Figure 1.3: Sc hematic dra wing of arc hes at a b ottlenec k. the flow is obstructed there. Ing. Gilian Brouw er works for Adviesburo Nieman: a company sp ecialized in consultancy on quality , safety and building physics. 6 He is sp ecialized in fire safety engineering. A t Nie- man, sev eral softw are to ols are applied to simulate the ev acuation of p eople in case of fire in a building. In case a building do es not comply with Dutch regulations (‘Bouwbesluit’), simu- lations are needed to make feasible that nevertheless an equiv alent level of safety is provided. In [38] a comparison is made b et ween tw o of the softw are pac k ages that are used by Nieman ( FDS + Evac and Simulex ). Both pack ages are based on a so cial force-like mo del, where FDS + Ev ac also includes the effect of smoke. An individual’s desired route is based on the geometry of the domain and the configuration of other p edestrians. The route is c hosen suc h that it is exp ected to take a minimal amoun t of time to reac h the destination. In Simulex ho wev er, in eac h spatial p osition the shortest route to the exit is precalculated (considering the distance). The ev acuation is assessed b y recording the times at whic h pedestrians lea ve the building, and consequen tly calculating exit flo w rates or av erage exit times. Prof.dr.ir. Bauke de V ries is the chairman of the Urban Management & Design Systems group at the Department of Arc hitecture, Building and Planning (Eindhov en Universit y of T ec hnology). W e met him in F ebruary 2011 and sp oke ab out his working in terests and area of exp ertise. He mainly fo cuses on in terior design of buildings, where for instance routing and the p ositions of obstacles are taken into accoun t. In the past, he also performed real-life exp erimen ts in his department, to obtain data ab out the walking b ehaviour of his co work ers 6 www.nieman.nl 10 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology within the office space. During our conv ersation, he also explained the w ork of Prof.dr. Harry Timmermans, who is a member of the same group. He is more sp ecialized in the design of exterior urban space, suc h as shopping malls. An example can b e found in [19], in whic h the t wo exp erts co op erated. Their simulat ion mo dels work with agents (read: p edestrians) that ha ve an agenda and an up dating mechanism for their ‘to do’-list; the individuals b ehav e and mo ve accordingly . W e also talked with p eople ab out cro wds in a broader context. An example is Dr. Petru Curseu who is an exp ert in group pro cesses and decision-making at Tilburg Univ ersity . T o get an impression of his area of in terest, see [18]. When he visited us in Eindhov en (No vem b er 2010), he show ed his in terest in microscopic p edestrian mo dels. He prop osed to use them as a metaphor for more abstract so cietal phenomena. F or instance, if there is strongly repul- siv e interaction b etw een a large group and a small group, the question arises whether this enables the large group to leav e a ro om relatively faster than the small group. According to Dr. Curseu, a parallel can b e dra wn b et ween this p edestrian simulation and a ma jority in a coun try that discriminates a minority . The ma jority group turns out to hav e b etter access to resources. I to ok part in the Study Group Mathematics with Industry 2011 (organized at the V rije Uni- v ersiteit Amsterdam, 24–28 Jan uary 2011). The problem w as pro vided b y Chess . 7 An ad ho c wireless netw ork is considered, in which eac h no de can broadcast and receive messages. How- ev er, it is uncertain whether sent messages reach a receiv er, and if y es, how many/whic h no des are reached. The aim w as to let each no de send, receive and pro cess ‘in telligent’ messages such that it can estimate the num b er of no des in its neighbourho o d and in the total netw ork. The application p ossibilities of suc h unreliable wireless net works are closely linked to the sub ject of this thesis. This was also pointed out to us at an earlier stage b y Prof.dr. F abian Wirth (Univ ersity of W ¨ urzburg, Germany), an exp ert in logistic net works. Typically , each no de is a simple micropro cessor, with limited computational p o wer, that measures a prop erty of its en vironment, like temp erature or p osition. A large num b er of such sensors can b e used to detect forest fires, but also to obtain information about e.g. bird flo c king, so cial behaviour and cro wd dynamics. The pro ceedings of the study group are to b e exp ected. In March 2011 Adrian Mun tean and I had a conv ersation with Dr. David F edson, a medical exp ert in the area of epidemics. W e talked ab out Malcolm Gladwell’s The Tipping Point , whic h treats phase transitions that tak e place everywhere around us. Certain even ts suddenly tak e off in so cial b ehaviour (fashion), epidemics of infectious diseases, and in animal groups. F or the latter see [36]. Another topic he brough t to our attention is synchronization. A connection b etw een synchronous b ehaviour in nature and in the financial world is made in [47]. Prof. Stev en Strogatz also explains this phenomenon in a fascinating w a y . 8 1.5 Con ten t and structure of this thesis In this thesis we fo cus on the mathematical background of crowd dynamics mo dels. This means that the emphasis will not b e on direct applications. W e are mainly in terested in iden- tifying underlying structure and mechanisms, and in understanding the coupling b et w een the micro- and macro-scale. W e w ork in a fully deterministic setting. F or more information in 7 www.chess.nl 8 www.youtube.com/watch?v=aSNrKS-sCE0 Mo delling Cro wd Dynamics 11 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology the direction of sto chastic mo delling approac hes (such as sto chastically interacting particles), the reader is referred e.g. to [48]. W e first describ e the theoretical fundamen ts on which we build our mo del. In Section 2 basic measure-theoretical concepts are introduced. The most imp ortant parts are the (re- fined) Leb esgue decomp osition and the Radon-Nikodym Theorem. Section 3 is dedicated to mixture theory , a branch of contin uum mechanics that turns out to b e suitable for describing a crowd consisting of m ultiple subp opulations. The corresp ond- ing theory is presented in measure-theoretical language. It provides us with conserv ation of mass equations, that are the basis for our mo del. In Section 3 we also co v er asp ects of ther- mo dynamics that are used later to derive an entrop y inequality for a particular cro wds setting. In Section 4, w e apply the mo delling ideas of the preceding sections to obtain a mo del for the dynamical b ehaviour of a crowd. After a n umber of (mainly formal) calculations w e first presen t a contin uous-in-time mo del and the accompanying entrop y inequality . W e also pro- p ose a velocity field, go v erning the dynamics. F rom this mo del, we deriv e a discrete-in-time v ersion. All our mathematical analysis concerns this time-discrete mo del. The main result is a pro of of global existence and uniqueness of a time-discrete solution. W e also derive a discrete-in-time equiv alen t of the entrop y inequality . Section 5 contains the full description of the n umerical scheme and parameter setting, which w e use to simulate our tw o-scale cro wd setting. The aim of the simulation part is to in ves- tigate the basic patterns pro duced by the in terpla y b et ween one or t wo individuals and a macroscopic cro wd. Section 6 contains our simulation results. In Section 7 we review the w ork done and the obtained results. Moreov er, w e giv e sug- gestions for future w ork and p ose a few basic questions that are still op en. Our first attempt [23] of mo delling pedestrians using the approach describ ed in this the- sis, was published in Nonline ar Phenomena in Complex Systems . Its conten t can b e found in App endix F. 12 Mo delling Cro wd Dynamics Chapter 2 Measure theory 2.1 Measures and their Leb esgue decomp osition Let  Ω , B (Ω)  b e a measurable space, where ∅ 6 = Ω ⊂ R d . Here, B (Ω) denotes the σ -algebra of Borel subsets of Ω. Supp ose that µ and λ are p ositive, finite measures defined on B (Ω). Definition 2.1.1 (Absolutely contin uous measures) . The me asur e µ is said to b e absolutely c ontinuous with r esp e ct to the me asur e λ if for any Ω 0 ∈ B (Ω) , λ (Ω 0 ) = 0 implies µ (Ω 0 ) = 0 . Notation: µ  λ. Definition 2.1.2 (Singular measures) . The me asur es µ and λ ar e said to b e mutual ly singular if ther e exists a B ∈ B (Ω) such that µ (Ω \ B ) = λ ( B ) = 0 . Notation: µ ⊥ λ. A lthough the r elation µ ⊥ λ is symmetric, it is also often said that µ is singular with r esp e ct to λ . Definition 2.1.2 is taken from [22], p. 40. An alternative definition is given by [46], p. 120, whic h we will no w show to b e equiv alen t. Lemma 2.1.3. The fol lowing two statements ar e e quivalent: (i) Ther e exists a B ∈ B (Ω) such that µ (Ω \ B ) = λ ( B ) = 0 . (ii) Ther e ar e disjoint A 1 , A 2 ∈ B (Ω) such that µ (Ω 0 ) = µ (Ω 0 ∩ A 1 ) and λ (Ω 0 ) = λ (Ω 0 ∩ A 2 ) for al l Ω 0 ∈ B (Ω) . Pr o of. 1. Assume that (i) holds. Define A 1 := B and A 2 := Ω \ B (these sets are ob viously disjoin t). F or all Ω 0 ∈ B (Ω) w e hav e (using prop erty (i)) µ (Ω 0 \ A 1 ) 6 µ (Ω \ A 1 ) = µ (Ω \ B ) = 0 , and λ (Ω 0 \ A 2 ) 6 λ (Ω \ A 2 ) = λ  Ω \ (Ω \ B )  = λ ( B ) = 0 . 13 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology It follo ws that µ (Ω 0 ) = µ (Ω 0 ∩ A 1 ) + µ (Ω 0 \ A 1 ) 6 µ (Ω 0 ∩ A 1 ) , and λ (Ω 0 ) = λ (Ω 0 ∩ A 2 ) + λ (Ω 0 \ A 2 ) 6 λ (Ω 0 ∩ A 2 ) . Since Ω 0 ∩ A 1 and Ω 0 ∩ A 2 are subsets of Ω 0 , it is clear that µ (Ω 0 ) > µ (Ω 0 ∩ A 1 ) , and λ (Ω 0 ) > λ (Ω 0 ∩ A 2 ) , th us we ha ve that µ (Ω 0 ) = µ (Ω 0 ∩ A 1 ) , and λ (Ω 0 ) = λ (Ω 0 ∩ A 2 ) . 2. Assume that (ii) holds. Define B := A 1 , then µ (Ω \ B ) = µ  (Ω \ B ) ∩ A 1  = µ  (Ω \ A 1 ) ∩ A 1  = µ ( ∅ ) = 0 , and λ ( B ) = λ ( B ∩ A 2 ) = λ ( A 1 ∩ A 2 ) = λ ( ∅ ) = 0 . In the following theorem w e relate an y p ositiv e, finite measure to absolutely con tinuous and singular measures. Theorem 2.1.4 (Leb esgue decomposition) . If µ and λ ar e p ositive, finite me asur es define d on B (Ω) , then ther e exists a unique p air of p ositive, finite me asur es µ ac and µ s define d on B (Ω) , such that: (i) µ = µ ac + µ s , (ii) µ ac  λ , (iii) µ s ⊥ λ . We c al l µ ac the absolutely c ontinuous p art and µ s the singular p art of µ w.r.t. λ . The p air ( µ ac , µ s ) is c al le d the L eb esgue de c omp osition of µ w.r.t. λ . Pr o of. The pro of of Theorem 2.1.4 is giv en in App endix A. Definition 2.1.5 (Discrete measures) . The me asur e µ is said to b e a discr ete me asur e with r esp e ct to the me asur e λ if ther e exists a c ountable set A := { x 1 , x 2 , . . . } ⊂ Ω such that µ (Ω \ A ) = λ ( A ) = 0 . Lemma 2.1.6. L et µ b e a p ositive, finite me asur e on B (Ω) . Then the fol lowing two statements ar e e quivalent: (i) µ is a discr ete me asur e with r esp e ct to the L eb esgue me asur e λ d . 14 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology (ii) Ther e is a c ountable set { x 1 , x 2 , . . . } ⊂ Ω and a set of c orr esp onding nonne gative c o- efficients { α 1 , α 2 , . . . } ⊂ R , such that µ = P ∞ i =1 α i δ x i . Her e δ x i is the Dir ac me asur e c enter e d at x i . Pr o of. 1. Assume that (i) holds. Let A := { y 1 , y 2 , . . . } ⊂ Ω b e the collection of p oin ts such that µ (Ω \ A ) = λ ( A ) = 0. Without loss of generality , assume that all elements of A are distinct. (In case not all elements of A are distinct, just delete y j from A if there is a y i ∈ A satisfying i < j and y i = y j .) Since µ (Ω \ A ) = 0, w e hav e for any Ω 0 ∈ B (Ω): µ (Ω 0 ∩ A ) 6 µ (Ω 0 ) = µ (Ω 0 ∩ A ) + µ (Ω 0 \ A ) 6 µ (Ω 0 ∩ A ) + µ (Ω \ A ) = µ (Ω 0 ∩ A ) , th us µ (Ω 0 ) = µ (Ω 0 ∩ A ). F or fixed Ω 0 , let J b e the index set, such that i ∈ J implies y i ∈ Ω 0 ∩ A . Since Ω 0 ∩ A = S i ∈J { y i } is a disjoin t union, it follo ws that µ (Ω 0 ) = µ (Ω 0 ∩ A ) = X i ∈J µ ( y i ) . If we define α i := µ ( y i ) > 0, and write 1 for the indicator function, then the ab o ve is equiv alen t to µ (Ω 0 ) = ∞ X i =1 α i 1 y i ∈ Ω 0 . This is exactly the definition of the Dirac measure, so w e can also write µ (Ω 0 ) = ∞ X i =1 α i δ y i . 2. Assume that (ii) holds, and let the set { x 1 , x 2 , . . . } b e called B . It follows by definition of the Dirac measure that µ (Ω \ B ) = 0. B is a countable collection of p oin ts, whic h (ob viously) all hav e Leb esgue measure zero. Thus λ d ( B ) = P ∞ i =1 λ d ( x i ) = 0. W e conclude that µ is discrete with resp ect to λ d . Definition 2.1.7 (Singular contin uous measures) . The me asur e µ is said to b e singular c on- tinuous with r esp e ct to the me asur e λ if µ ( x ) = 0 for al l x ∈ Ω , and ther e is a B ∈ B (Ω) such that µ (Ω \ B ) = λ ( B ) = 0 . Remark 2.1.8. W e only consider measures defined on B (Ω), i.e. we can only apply them to sets. If w e write µ ( x ), w e therefore actually mean µ ( { x } ). Theorem 2.1.9 (Decomp osition of singular measures) . If µ s is a p ositive, finite me asur e define d on B (Ω) , which is singular w.r.t. λ , then ther e exists a unique p air of p ositive, finite me asur es µ d and µ sc define d on B (Ω) , such that: (i) µ s = µ d + µ sc , (ii) µ d is a discr ete me asur e w.r.t. λ , (iii) µ sc is singular c ontinuous w.r.t. λ . Mo delling Cro wd Dynamics 15 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology We c al l µ d the discr ete p art and µ sc the singular c ontinuous p art of µ s w.r.t. λ . The singular c ontinuous p art is also c al le d the Cantor p art of the me asur e. Pr o of. The pro of of Theorem 2.1.9 is giv en in App endix B. Corollary 2.1.10 (Refined Leb esgue decomp osition) . Assume the hyp otheses of The or ems 2.1.4 and 2.1.9. Then for any p ositive, finite me asur e µ , ther e ar e unique me asur es µ ac , µ d and µ sc such that µ = µ ac + µ d + µ sc , wher e µ ac  µ , µ d is a discr ete me asur e, and µ sc is singular c ontinuous w.r.t. λ . Remark 2.1.11. The statement µ (Ω \ B ) = λ ( B ) = 0 (cf. Definitions 2.1.2, 2.1.5 and 2.1.7), is equiv alen t to: λ ( B ) = 0 and µ ( B ) = µ (Ω). This is due to the identit y µ (Ω) = µ (Ω ∩ B ) + µ (Ω \ B ) = µ ( B ) + µ (Ω \ B ). If B is suc h that µ ( B ) = µ (Ω), w e call B a set of ful l me asur e . 2.2 Radon-Nik o dym Theorem The Radon-Nikodym Theorem is of vital imp ortance in this context, esp ecially for the appli- cabilit y of measure theory to mixture theory as arising in real-life situations mimic king the dynamics of cro wds. The following theorem giv es sufficient conditions for a measure to b e expressed in terms of a densit y , with resp ect to another measure: Theorem 2.2.1 (Radon-Nikodym for finite measures) . Supp ose µ and λ ar e p ositive me asur es on a me asur able sp ac e  Ω , B (Ω)  such that 0 < µ (Ω) < ∞ , 0 < λ (Ω) < ∞ , and let µ b e absolutely c ontinuous with r esp e ct to λ . Then ther e exists a r e al, nonne gative, B (Ω) - me asur able function h on Ω such that µ (Ω 0 ) = Z Ω 0 hdλ, for al l Ω 0 ∈ B (Ω) . Pr o of. A comprehensible, well-structured pro of of Theorem 2.2.1 can b e found in [13]. Remark 2.2.2. The density h is often called R adon-Niko dym derivative and is denoted b y h := dµ dλ . (2.2.1) Lemma 2.2.3. Assume the hyp othesis of The or em 2.2.1. Then the R adon-Niko dym derivative is unique in the fol lowing sense: if b oth h 1 and h 2 satisfy µ (Ω 0 ) = R Ω 0 h k dλ , for al l Ω 0 ∈ B (Ω) , wher e k ∈ { 1 , 2 } , then h 1 = h 2 almost everywher e (w.r.t. λ ) in Ω . Pr o of. This statement can easily b e v erified. F or any Ω 0 ∈ B (Ω) w e hav e that Z Ω 0 ( h 1 − h 2 ) dλ = Z Ω 0 h 1 dλ − Z Ω 0 h 2 dλ = µ (Ω 0 ) − µ (Ω 0 ) = 0 . Th us, h 1 − h 2 = 0 almost ev erywhere w.r.t. λ in Ω, i.e. h 1 = h 2 λ -a.e. in Ω. Remark 2.2.4. Theorem 2.2.1 also holds for more general situations, for instance, including the case of signed σ -finite 1 measures (see [21], e.g.). 1 The measure λ is called σ -finite if there exists a sequence of sets { E k } ∞ k =1 ⊂ B (Ω) with Ω = S ∞ k =1 E k and | λ ( E k ) | < ∞ (for all k ∈ N ); see [21], p. 269. 16 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 2.3 Prop erties of Radon-Nik o dym deriv ativ es In the following lemma w e state a num b er of useful prop erties of Radon-Nikodym deriv atives: Lemma 2.3.1 (Basic prop erties) . Assume the hyp othesis of The or em 2.2.1 on the me asur es ν , µ , λ , ν 1 , ν 2 , µ 1 and µ 2 . The R adon-Niko dym derivatives satisfy the fol lowing gener al pr op erties: 1. If ν  µ  λ , then dν dλ = dν dµ dµ dλ . 2. If µ  λ and g ∈ L 1 µ (Ω) , then R Ω 0 g dµ = R Ω 0 g dµ dλ dλ for al l Ω 0 ∈ B (Ω) . 3. If µ  λ and λ  µ , then dµ dλ =  dλ dµ  − 1 . 4. If (Ω i , B (Ω i ) , µ i ) , i ∈ { 1 , 2 } , ar e two me asur e sp ac es with ν i  µ i , i ∈ { 1 , 2 } , then ν 1 ⊗ ν 2  µ 1 ⊗ µ 2 and d ( ν 1 ⊗ ν 2 ) d ( µ 1 ⊗ µ 2 ) = dν 1 dµ 1 dν 2 dµ 2 . Pr o of. The pro of of Lemma 2.3.1 is giv en in App endix C. 2.4 Mass measure µ m Here, we already give an indication of the particular measures we in tend to use in the rest of this thesis. Let Ω ⊂ R d b e a domain (read: ob ject, b o dy) with mass. F or physically relev an t situations, we consider d ∈ { 1 , 2 , 3 } . Let µ m (Ω 0 ) b e defined as the mass contained in Ω 0 ⊂ Ω. Note that w e use the concept of a mass measure v ery muc h in the spirit of [9]. Remark 2.4.1. As a rule, whenever we write Ω 0 ⊂ Ω, w e actually mean that Ω 0 is such that Ω 0 ∈ B (Ω). W e assume µ m to b e defined on all elements of B (Ω). In Sections 2.4.1 and 2.4.2 w e consider t wo sp ecific interpretations of this mass measure dep ending on the lo calization of the information we wish to capture. 2.4.1 Microscopic mass measure m m Supp ose that Ω contains a collection of N p oint masses (eac h of them of mass 1), and denote their p ositions by { p k } N k =1 ⊂ Ω, for N ∈ N . W e wan t the mass measure m m to b e a counting measure with resp ect to these p oint masses, i.e. for all Ω 0 ∈ B (Ω) m m (Ω 0 ) = # { p k ∈ Ω 0 } . (2.4.1) By (2.4.1) we mean that m m coun ts the n umber of individuals located in Ω 0 (with correspond- ing, known p ositions p k ∈ Ω 0 ). This can be ac hiev ed by represen ting m m as the sum of Dirac masses, with their singularities lo cated at the p oints p k , k ∈ { 1 , 2 , . . . , N } : m m = N X k =1 δ p k . (2.4.2) Mo delling Cro wd Dynamics 17 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology If m m is defined as in (2.4.1) or (2.4.2), we call it a micr osc opic mass measure. Note that suc h measure satisfies the conditions in Definition 2.1.5 and is thus a discrete measure. The c haracteristics of such measure are illustrated b y Figure 2.1. Ω Ω 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 Figure 2.1: A microscopic mass measure counts the num b er of p oints p k that are contained in Ω 0 , a subset of Ω. In this case m m (Ω 0 ) = 2 since p 2 and p 8 lie in Ω 0 . 2.4.2 Macroscopic mass measure M m Let us now regard a differen t mass measure M m . Assume that the following p ostulate applies to M m : P ostulate 2.4.2 (Prop erties of M m ) . 1. M m > 0 . 2. M m is σ -additive. 3. M m is finite. 4. M m  λ d (with λ d the L eb esgue me asur e in R d ). By Parts 1, 2 and 3 of Postulate 2.4.2, we hav e that M m is a p ositive, finite measure on Ω, whereas P art 4 implies that there is no mass present in a set that has no v olume (w.r.t. λ d ). W e refer to a mass measure satisfying Postulate 2.4.2 as a macr osc opic mass measure. No w Theorem 2.2.1 guarantees the existence of a real, nonnegative densit y ρ ∈ L 1 λ d (Ω) such that M m (Ω 0 ) = Z Ω 0 ρ ( x ) dλ d ( x ) , for all Ω 0 ∈ B (Ω) . An illustration of suc h measure is giv en in Figure 2.2. 18 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology ρ Ω Ω 0 Figure 2.2: A macroscopic mass measure is used to obtain the mass in Ω 0 ⊂ Ω. M m (Ω 0 ) dep ends on the density ρ of the area in Ω 0 . The densit y is here indicated b y grey shading. Mo delling Cro wd Dynamics 19 Chapter 3 Mixture theory and thermo dynamics In this section, we present some ideas from [11, 42, 50] regarding the theory of mixtures. W e cast their description in a measure-theoretical framework. Unlike [11, 42], here we mainly consider the Eulerian p oint of view. In Sections 3.1 – 3.5 we describ e the mixture as a contin- uum; this is the classical and most common wa y of doing so (cf. [11, 42] e.g.). In Section 3.4, w e extend the ideas of the preceding sections to more general mass measures. 3.1 Description of a mixture A t this stage, a mixture is defined as a contin uum consisting of a certain n umber of con- stituen ts (also called: comp onen ts). Constitutive relations typically differ from comp onent to comp onen t. Let Ω ⊂ R d b e the domain in which the mixture is lo cated, and consider the measurable space  Ω , B (Ω)  , where B (Ω) is the σ -algebra of Borel subsets of Ω. Supp ose the mixture consists of ν constituen ts, with index α ∈ { 1 , 2 , . . . , ν } . F or each constituent α we define its v olume and mass present in Ω 0 ∈ B (Ω) at time t ∈ (0 , T ) by τ α ( t, Ω 0 ) and µ α ( t, Ω 0 ), resp ec- tiv ely . Note that in fact t is the time v ariable and can b e understoo d as a parameter. The ob jects τ α ( t, · ) and µ α ( t, · ) on B (Ω) are for each t ∈ (0 , T ) fixed measures. W e assume that the follo wing p ostulate applies: P ostulate 3.1.1 (Prop erties of τ α and µ α ) . F or al l α ∈ { 1 , 2 , . . . , ν } and al l t ∈ (0 , T ) we p ostulate: 1. τ α > 0 . 2. τ α is σ -additive. 3. τ α is finite. 4. τ α  λ d (with λ d the L eb esgue me asur e in R d ). 20 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 5. µ α > 0 . 6. µ α is σ -additive. 7. µ α is finite. 8. µ α  τ α . Note that the concepts of Section 2.4.2 are incorp orated in this p ostulate. Remark 3.1.2. It follows in a straightforw ard wa y from Parts 4 and 8 of Postulate 3.1.1 that µ α  λ d for all α ∈ { 1 , 2 , . . . , ν } and all t ∈ (0 , T ). The absolute contin uit y of b oth τ α and µ α complies with the fact that w e define mixtures at the con tin uum level. Remark 3.1.3. W e c haracterize the presence of constituent α at a certain x by the presence of mass and volume here. Mathematically , this means that if at time t ∈ (0 , T ) a subset Ω 0 ∈ B (Ω) con tains some fraction of constituent α , then b oth µ α ( t, Ω 0 ) > 0 and τ α ( t, Ω 0 ) > 0 m ust hold. On the other hand, if a constituent α is not present in Ω 0 , then µ α ( t, Ω 0 ) and τ α ( t, Ω 0 ) are b oth zero. F ollowing this intuitiv e characterization, we assume τ α  µ α for all α ∈ { 1 , 2 , . . . , ν } and for all t ∈ (0 , T ). Let τ and µ b e the (time-dep endent) total volume and mass measures, given b y τ ( t, Ω 0 ) := ν X α =1 τ α ( t, Ω 0 ) , (3.1.1) µ ( t, Ω 0 ) := ν X α =1 µ α ( t, Ω 0 ) , (3.1.2) for all Ω 0 ∈ B (Ω) and all t ∈ (0 , T ). By the definitions of τ and µ giv en in (3.1.1) and (3.1.2) and as a consequence of Postu- late 3.1.1, we ha ve µ  τ  λ d . Note that since we also assumed τ α  µ α for eac h α , it follo ws by definition that also τ  µ holds. F or any t ∈ (0 , T ), the Radon-Nik o dym Theorem provides the existence of the unique non- negativ e densities (Radon-Nikodym deriv atives) Θ( t, · ), Θ α ( t, · ), θ α ( t, · ) ∈ L 1 λ d (Ω) (for each α ∈ { 1 , 2 , . . . , ν } ): Θ := dτ dλ d , (3.1.3) Θ α := dτ α dλ d , (3.1.4) θ α := dτ α dτ . (3.1.5) The Radon-Nik o dym deriv ative θ α arising in (3.1.5) is called the volume fr action of comp onent α at time t . As a result of (3.1.1): ν X α =1 θ α = 1 , a.e. w.r.t. τ , and for all t ∈ (0 , T ) . Mo delling Cro wd Dynamics 21 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Note also that due to Part 1 of Lemma 2.3.1, w e hav e that for all α ∈ { 1 , 2 , . . . , ν } , and for all t ∈ (0 , T ) Θ α = θ α Θ , a.e. w.r.t. λ d . Also for the mass measures unique Radon-Nik o dym deriv ativ es exist. F or all α ∈ { 1 , 2 , . . . , ν } and for all t ∈ (0 , T ), w e define ˇ ρ := dµ dτ , (3.1.6) ˇ ρ α := dµ α dτ , (3.1.7) ˆ ρ α := dµ α dτ α . (3.1.8) W e call ˇ ρ the densit y of the mixture, ˇ ρ α the p artial density of comp onent α , and ˆ ρ α the intrinsic densit y of that comp onen t. By (3.1.2) we hav e that ˇ ρ = ν X α =1 ˇ ρ α , a.e. w.r.t. τ , and for all t ∈ (0 , T ) . (3.1.9) F rom (3.1.3)-(3.1.5) and (3.1.6)-(3.1.8) a num b er of iden tities can b e derived (again via Lemma 2.3.1, P art 1). F or all t ∈ (0 , T ) we hav e ˇ ρ α = ˆ ρ α θ α , a.e. w.r.t. τ , for all α ∈ { 1 , 2 , . . . , ν } , (3.1.10) ˇ ρ α Θ = ˆ ρ α Θ α , a.e. w.r.t. λ d , for all α ∈ { 1 , 2 , . . . , ν } . (3.1.11) Remark 3.1.4. Note that the latter ˇ ρ α Θ equals the Radon-Nikodym deriv ativ e dµ α /dλ d . In practice it will b e m uc h more conv enient to work with λ d than to w ork with τ . Therefore we define ρ α := dµ α dλ d = ˇ ρ α Θ , (3.1.12) and henceforth refer to ρ α (rather than to ˇ ρ α ) as the partial densit y . F urthermore ˇ ρ Θ is the Radon-Nik o dym deriv ativ e of the total mass measure µ with resp ect to λ d . Analogously to ρ α , w e define ρ := dµ dλ d = ˇ ρ Θ , and refer to this ρ as the densit y , from no w on. No w, (3.1.9) implies ρ = ν X α =1 ρ α , a.e. w.r.t. λ d , and for all t ∈ (0 , T ) . (3.1.13) The mass c onc entr ation of constituent α at time t is defined as c α := ρ α ρ , (3.1.14) and it follo ws naturally from (3.1.13) and (3.1.14) that ν X α =1 c α = 1 , a.e. w.r.t. µ, and for all t ∈ (0 , T ) . 22 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Remark 3.1.5. The mass concen tration c α is only defined if ρ > 0. F or an y t ∈ (0 , T ), the subset Ω 0 t := { x ∈ Ω   ρ ( t, x ) = 0 } ⊂ Ω is a n ull set w.r.t. the measure µ . This can easily b e seen: µ ( t, Ω 0 t ) = Z Ω 0 t ρ ( t, x ) dλ d ( x ) = Z Ω 0 t 0 dλ d ( x ) = 0 . This implies that c α is defined almost ev erywhere w.r.t. µ . W e will see later, that defining c α actually only is useful in regions where ρ > 0 is satisfied (i.e. where mass is presen t), and that there is th us no serious problem here. Remark 3.1.6. It is ob vious that ρ α ρ = ˇ ρ α Θ ˇ ρ Θ = ˇ ρ α ˇ ρ holds µ -almost everywhere. W e thus could hav e defined c α as ˇ ρ α / ˇ ρ , and w ould hav e obtained the same prop erties almost everywhere w.r.t. µ . 3.2 Kinematics The kinematics of comp onen t α of the mixture is dictated by a motion mapping χ α , such that x = χ α ( t, X α ) , denotes the p osition at time t of an α -particle, which w as initially (i.e. at t = 0) situated in X α . By assumption χ α is smo oth, and the inv erse mapping Ξ α exists, suc h that X α = Ξ α ( t, x ) . Also b y assumption, Ξ α is smo oth. Suc h smo oth motion mapping χ α , with smo oth in verse, is called a diffe omorphism . F or eac h comp onent α its v elo cit y at p osition x ∈ Ω at time t ∈ (0 , T ) is denoted by v α ( t, x ) and can b e derived from the motion mapping χ α : v α ( t, x ) := D ( α ) x D t := ∂ ∂ t χ α ( t, X α ) . (3.2.1) Remark 3.2.1. The right-hand side in (3.2.1) is indeed a function of ( t, x ), since we ha ve to read X α as X α = Ξ α ( t, x ). The deriv ativ e ∂ /∂ t is taken how ever only with resp ect to the first v ariable of χ α . Definition 3.2.2 (Time deriv ativ es) . L et g denote some (physic al) sc alar quantity asso ciate d with c omp onent α and dep ending on sp ac e and time. We write g = ˜ g ( t, X α ) if we c onsider the quantity fr om a L angr angian p oint of view, and g = ¯ g ( t, x ) fr om a Eulerian p oint of view. The time derivative of g is now define d as D ( α ) g D t := ∂ ˜ g ( t, X α ) ∂ t = ∂ ¯ g ( t, x ) ∂ t + ∇ ¯ g ( t, x ) · v α ( t, x ) . Mo delling Cro wd Dynamics 23 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology In the ab ov e definition, note that ˜ g ( t, X α ) = ¯ g  t, χ α ( t, X α )  . No w, w e define the b aryc entric velocity (i.e. the v elo cit y of the cen tre of mass of a v olume elemen t) of the mixture b y v := 1 ρ ν X α =1 ρ α v α = ν X α =1 c α v α . (3.2.2) The barycen tric velocity is also called me an velo city , or just velo city of the mixtur e . Remark 3.2.3. A similar expression as in Definition 3.2.2 holds for time deriv ativ es of scalar functions g asso ciated to the total mixture: D g D t := ∂ ˜ g ( t, X ) ∂ t = ∂ ¯ g ( t, x ) ∂ t + ∇ ¯ g ( t, x ) · v ( t, x ) . 3.3 Balance of mass W e assume that there is no mass exchange b etw een the comp onents of the mixture. The conserv ation of mass implies that for any Ω 0 ∈ B (Ω), for all t ∈ (0 , T ) and for all α ∈ { 1 , 2 , . . . , ν } d dt µ α  t, χ α ( t, Ω 0 )  = 0 . (3.3.1) Note that χ α ( t, Ω 0 ) is the configuration at time t of all particles of the comp onent α that were initially lo cated in Ω 0 . F or µ α  t, χ α ( t, Ω 0 )  in (3.3.1) to b e well-defined, w e need that χ α ( t, Ω 0 ) is an elemen t of B (Ω) for eac h t ∈ (0 , T ) and for all Ω 0 ∈ B (Ω). Lemma 3.3.1. If for e ach t ∈ (0 , T ) , the mapping χ α ( t, · ) : Ω → Ω is a diffe omorphism, then χ α ( t, Ω 0 ) ∈ B (Ω) , for e ach t ∈ (0 , T ) and for al l Ω 0 ∈ B (Ω) . Pr o of. Given that for any t ∈ (0 , T ) fixed, χ α ( t, · ) is a diffeomorphism from Ω to Ω, we ha ve in particular that its inv erse Ξ α ( t, · ) is a contin uous mapping from Ω to Ω. By definition of con tinuous functions (cf. [46], p. 8), (Ξ α ) − 1 ( t, Ω 0 ) is an op en set, for any Ω 0 ⊂ Ω op en. The Borel σ -algebra is defined as the smallest σ -algebra containing all op en subsets of Ω. It follo ws that we ha ve that (Ξ α ) − 1 ( t, Ω 0 ) ∈ B (Ω) (i.e. it is a measurable set). By definition of measurable functions (cf. [46], p. 8), w e no w ha v e that Ξ α ( t, · ) is measurable. Theorem 1.12 (b) ([46], p. 13) provides that (Ξ α ) − 1 ( t, Ω 0 ) ∈ B (Ω) not only for all Ω 0 ⊂ Ω op en, but also for all Ω 0 ∈ B (Ω). That is, Ξ α is a Borel function (cf. [46], p. 12). Ξ α ( t, · ) is the in verse of χ α ( t, · ), so the ab ov e statement also states that for all t ∈ (0 , T ) fixed χ α ( t, Ω 0 ) ∈ B (Ω) for all Ω 0 ∈ B (Ω). W e now present a few calculations inv olving Reynolds’ transp ort theorem, that are based on [49] (pp. 18–21). W e use the notation det( ∇ χ α ) for the determinan t of the Jacobian matrix of the motion mapping. The follo wing iden tit y is deriv ed e.g. in [49] (p. 20): d dt det( ∇ χ α ) = ∇ · v α det( ∇ χ α ) . 24 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology F rom the conserv ation of mass statemen t, formulated in (3.3.1), we derive 0 = d dt µ α  t, χ α ( t, Ω 0 )  = d dt Z χ α ( t, Ω 0 ) ρ α ( t, x ) dλ d ( x ) = d dt Z Ω 0 ρ α  t, χ α ( t, X )  det( ∇ χ α )( t, X ) dλ d ( X ) = Z Ω 0 ∂ ρ α ∂ t det( ∇ χ α ) dλ d + Z Ω 0 ∇ ρ α · ∂ χ α ∂ t det( ∇ χ α ) dλ d + Z Ω 0 ρ α d dt det( ∇ χ α ) dλ d = Z Ω 0  ∂ ρ α ∂ t + ∇ ρ α · v α + ρ α ∇ · v α  det( ∇ χ α ) dλ d = Z Ω 0  ∂ ρ α ∂ t + ∇ ·  ρ α v α   det( ∇ χ α ) dλ d = Z χ α ( t, Ω 0 )  ∂ ρ α ∂ t + ∇ ·  ρ α v α   dλ d . (3.3.2) Since t and Ω 0 can b e c hosen arbitrarily , we conclude from the expression in the seven th line of (3.3.2) that for all α ∈ { 1 , 2 , . . . , ν } and for all t ∈ (0 , T )  ∂ ρ α ∂ t + ∇ ·  ρ α v α   det( ∇ χ α ) = 0 , λ d -a.e. in Ω . (3.3.3) Since the in v erse of χ α is assumed to be differen tiable, w e know that det( ∇ χ α ) 6 = 0 (cf. remark on p. 4 of [49]). Th us, w e deduce from (3.3.3) that ∂ ρ α ∂ t + ∇ ·  ρ α v α  = 0 , λ d -a.e. in Ω . (3.3.4) The w eak formulation of (3.3.4) leads to d dt Z Ω ψ α ρ α dλ d = Z Ω v α · ∇ ψ α ρ α dλ d , (3.3.5) for all α ∈ { 1 , 2 , . . . , ν } , for all t ∈ (0 , T ), and for all ψ α ∈ C 1 0 ( ¯ Ω). Remark 3.3.2. F rom (3.3.4) w e can deriv e the lo cal mass balance for the mixture as a whole. T o achiev e this, we sum (3.3.4) ov er all α and take (3.1.13) and (3.2.2) in to consideration. This yields that for all t ∈ (0 , T ) ∂ ρ ∂ t + ∇ ·  ρv  = 0 , λ d -a.e. in Ω . (3.3.6) Mo delling Cro wd Dynamics 25 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology The w eak form ulation of (3.3.6) can b e either deduced directly from (3.3.6), or derived from (3.3.5). F or the latter wa y , we take the same test function ψ α = ψ ∈ C 1 0 ( ¯ Ω) for eac h α , again sum o ver all α and use (3.1.13) and (3.2.2). This pro cedure results in d dt Z Ω ψ ρdλ d = Z Ω v · ∇ ψ ρdλ d . (3.3.7) In Remark 3.1.5 w e indicated that defining c α is actually only useful in regions where ρ > 0. This is made clear by (3.3.6) and (3.3.7). The mass concen trations c α are incorp orated in the definition of v . How ever v only app ears in com bination with ρ as the pro duct ρv . If ρ = 0 the pro duct is zero any wa y , and thus (at least physically) it do es not matter how (or whether) c α is defined. 3.4 Generalization W e w ould like to extend the ideas presented in Sections 3.1 – 3.3, such that they hold not only for those measures that are absolutely contin uous w.r.t. λ d . Indeed we hav e seen in Section 2.1 that in general, a measure might also con tain a singular part. The mixture-theoretical concepts presen ted so far do ho wev er not hold for singular measures. Let µ α ( t, Ω 0 ), lik e b efore, denote the mass of constituent α presen t in Ω 0 ∈ B (Ω) at time t ∈ (0 , T ). T o obtain a more general framew ork, we need to revok e the assumption that µ α  τ α for all α ∈ { 1 , 2 , . . . , ν } . In this setting we thus p ostulate the follo wing: P ostulate 3.4.1 (Prop erties of µ α ) . F or al l α ∈ { 1 , 2 , . . . , ν } and al l t ∈ (0 , T ) we p ostulate: 1. µ α > 0 . 2. µ α is σ -additive. F urthermore, as in (3.1.2), w e define the total mass b y µ ( t, Ω 0 ) := ν X α =1 µ α ( t, Ω 0 ) , (3.4.1) for all Ω 0 ∈ B (Ω) and all t ∈ (0 , T ). This new setting, without the absolute con tinuit y demand, implies that in most cases of Section 3.1 w e cannot apply the Radon-Nikodym Theorem anymore. Ho w ev er, note that due to (3.4.1) and Part 1 of Postulate 3.4.1 w e hav e that µ α  µ for all α ∈ { 1 , . . . , ν } . Thus for eac h µ α a unique Radon-Nik o dym deriv ativ e exists with resp ect to µ . Let us denote c α := dµ α dµ , (3.4.2) whic h is also called the mass c onc entr ation of constituent α at time t . 26 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Remark 3.4.2. If we still would ha v e b ee n in the absolute contin uous case of Section 3.1, both µ  τ and τ  µ were true. Due to P art 3 of Lemma 2.3.1, this implies that 1 / ˇ ρ = dτ /dµ , whic h is defined µ -almost everywhere. P art 1 of the same lemma then implies that c α = dµ α dµ = dµ α dτ dτ dµ = ˇ ρ α ˇ ρ . Also: ˇ ρ α ˇ ρ = ˇ ρ α Θ ˇ ρ Θ = ρ α ρ , and since all expressions ab ov e are defined µ -a.e., w e hav e that c α = ρ α ρ , µ -almost ev erywhere. Hence, in the absolutely contin uous case (3.4.2) is equiv alen t to the definition of c α giv en in Section 3.1. It follo ws from (3.4.1) that: ν X α =1 c α = 1 , a.e. w.r.t. µ, and for all t ∈ (0 , T ) . (3.4.3) Up to the definition of the barycentric velocity , in Section 3.2 we hav e not used the absolute con tinuit y of any of the mass measures. The ideas presented there are also applicable in the generalized setting. W e only hav e to redefine (in a very natural wa y) the barycentric v elo city of the mixture b y v := ν X α =1 c α v α . (3.4.4) F rom a notational point of view it is not difficult to generalize the w eak form ulation of the balance of mass concept, as given in (3.3.5). Since we kno w that ρ α = dµ α /dλ d , the w eak form can b e written as d dt Z Ω ψ α dµ α = Z Ω v α · ∇ ψ α dµ α , (3.4.5) for all α ∈ { 1 , 2 , . . . , ν } , for all t ∈ (0 , T ), and for all ψ α ∈ C 1 0 ( ¯ Ω). This ‘trick’ is merely a matter of notation, and the ab o ve is equiv alent to (3.3.5) for absolutely con tinuous mass measures. How ever, we still need to make sure that this expression also makes sense for more general mass measures. W e are running ahead by mentioning this now, but w e will see later (see e.g. Section 4.8) that we are mainly interested in measures that are either absolutely con tinuous, or discrete (i.e. sum of Dirac distributions), or p ossibly a combination of the t wo. This means that we explicitly exclude the singular contin uous part from the measure, and only show that (3.4.5) mak es sense for discrete measures. Note that the measures we do allow, are (com binations of ) exactly those measures introduced in Section 2.4. W e consider µ α to be a single Dirac measure centered at a time-dep endent p osition x ( t ) ∈ Ω Mo delling Cro wd Dynamics 27 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology for each t ∈ (0 , T ), and assume its motion to b e describ ed b y dx ( t ) /dt = v α ( t, x ( t )). W e c ho ose an arbitrary function ψ α ∈ C 1 0 ( ¯ Ω) and take the inner pro duct with this function (ev al- uated in x ( t )) on b oth sides of the equalit y . In the resulting left-hand side, w e recognize the c hain rule (recall that ψ α is once con tinuously differen tiable), so ∇ ψ α ( x ( t )) · d dt x ( t ) = d dt ψ α ( x ( t )) . No w, we ha ve obtained d dt ψ α ( x ( t )) = ∇ ψ α ( x ( t )) · v α ( t, x ( t )) . (3.4.6) W e apply the definition of the Dirac measure, and since µ α = δ x ( t ) , w e find that (3.4.6) can also b e written as d dt Z Ω ψ α dµ α = Z Ω v α · ∇ ψ α dµ α . T o extend this approac h, now let µ α b e a linear com bination of Dirac deltas centered at x i ( t ), and let eac h of thes e mo ve according to d dt x i ( t ) = v α ( t, x i ( t )) . (3.4.7) Here the index i is taken from a coun table, p ossibly infinite, index set J . If J is infinite, the co efficien ts of the Dirac deltas m ust hav e finite sum. Similar arguments as b efore yield that, for eac h i ∈ J , we hav e d dt ψ α ( x i ( t )) = ∇ ψ α ( x i ( t )) · v α ( t, x i ( t )) . If µ α = P i ∈J α i δ x i ( t ) with nonnegativ e co efficien ts such that P i ∈J α i < ∞ , then we hav e X i ∈J α i d dt ψ α ( x i ( t )) = X i ∈J α i ∇ ψ α ( x i ( t )) · v α ( t, x i ( t )) , whic h is, by the definition of the Dirac measure, again equal to (3.4.5). Note that the w eak form ulation (3.4.5) also mak es sense for linear com binations of abso- lutely contin uous and discrete measures. F or the absolutely con tin uous part w e ha ve the balance of mass-in terpretation; for the discrete part w e interpret it via (3.4.7). Remark 3.4.3. W e hav e seen in Lemma 2.1.6 that any discrete measure can b e written as a linear com bination of Dirac distributions. How ever, if (for every t ) µ α ( t, · ) is discrete, not only the centres of the Dirac deltas, but also the co efficients migh t b e time-dep endent. It will be shown later that the time-dep endent discrete measures w e work with ha ve constant co efficien ts (see Corollary 4.5.8). This means that the time-dep endence is only presen t in the lo cation of the Dirac deltas. It will also b e shown that these p ositions x i ( t ) satisfy (3.4.7); cf. Lemma 4.5.9. In this spirit, (3.4.5) mak es sense for the t yp es of measures that are relev an t for us. 28 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Remark 3.4.4. Analogously to Section 3.3, we can deduce from (3.4.5) the structure of a w eak formulation that applies to the total mass measure µ . W e take the same test function ψ α = ψ ∈ C 1 0 ( ¯ Ω) for eac h α . Each in tegral w.r.t. µ α is transformed in to an integral w.r.t. µ using (3.4.2). Consequen tly , summing o v er all α , and using (3.4.3) and (3.4.4), yields d dt Z Ω ψ dµ = Z Ω v · ∇ ψ dµ. (3.4.8) 3.5 En trop y inequalit y In this section, w e place the mixture concepts discussed in Sections 3.1 – 3.4 in to a thermo dy- namical context. Our aim is to derive an entr opy ine quality . This inequalit y is strongly related to concepts lik e the second axiom of thermo dynamics, or the Clausius-Duhem Inequalit y . F or more information on thermo dynamics and on the role of en tropy , the reader is referred to e.g. [52] or [27]. T o eac h constituen t of the mixture we assign a function η α : (0 , T ) × Ω → R , whic h is called the entr opy density (p er unit mass) of comp onen t α . The entrop y density for the total mixture is defined as η := ν X α =1 c α η α . (3.5.1) Note that this definition has the same structure as the definition of the barycentric velocity in (3.2.2). The entrop y S Ω 0 ( t ) assigned to Ω 0 ⊂ Ω at time t is defined b y integration of η with resp ect to the density ρ , that is S Ω 0 ( t ) := Z Ω 0 η ρdλ d . (3.5.2) T o eac h comp onen t we also assign a temp erature T α : (0 , T ) × Ω → R + . F ollo wing the ideas of [42] (p. 7), [20] (p. 347), and [11] (p. 28), we now p ostulate the follo wing: P ostulate 3.5.1 (Lo cal entrop y inequalit y) . L o c al ly the fol lowing ine quality holds: ρ D η D t + ∇ · j η − ν X α =1 ρ α r α T α > 0 . (3.5.3) Her e j η is the entr opy flux ve ctor, and r α is the volume supply of external he at to c onstituent α . Remark 3.5.2. In the term P ν α =1 ρ α r α /T α , which is the v olume supply of external entrop y to constituent α , w e recognize a similar structure as in (3.2.2) and (3.5.1). It can therefore b e considered as some ”barycentric quantit y”. Remark 3.5.3. Sometimes (3.5.3) is written as ∂ ∂ t ( ρη ) + ∇ · ( ρη v + j η ) − ν X α =1 ρ α r α T α > 0 , Mo delling Cro wd Dynamics 29 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology cf. e.g. [42], p. 7. Using the lo cal balance of mass (3.3.6) and Definition 3.2.2, we can indeed sho w that these tw o formulations are equiv alent: ∂ ∂ t ( ρη ) + ∇ · ( ρη v ) = ∂ ∂ t ( ρη ) + ∇ ( ρη ) · v + ρη ∇ · v = D D t ( ρη ) + ρη ∇ · v = D ρ D t η + ρ D η D t + η  ∇ · ( ρv ) − ∇ ρ · v  = ρ D η D t + η  D ρ D t − ∇ ρ · v  + η ∇ · ( ρv ) = ρ D η D t + η ∂ ρ ∂ t + η ∇ · ( ρv ) = ρ D η D t + η  ∂ ρ ∂ t + ∇ · ( ρv )  = ρ D η D t . (3.5.4) F or an arbitrarily fixed Ω 0 ⊂ Ω (such that ∂ Ω 0 is sufficiently regular) w e now calculate the time deriv ativ e (at fixed time t ∈ (0 , T )) of the total entrop y contained in Ω 0 : ∂ ∂ t Z Ω 0 η ρdλ d = Z Ω 0 ∂ ∂ t ( η ρ ) dλ d > Z Ω 0 ν X α =1 ρ α r α T α − ∇ · ( ρη v + j η ) dλ d = Z Ω 0 ν X α =1 ρ α r α T α dλ d − Z ∂ Ω 0 ( ρη v + j η ) · ndλ d − 1 , (3.5.5) where w e hav e used the lo cal entrop y inequality in the second line of (3.5.5). W e call (3.5.5) the glob al entr opy ine quality . Remark 3.5.4. It w ould hav e b een p ossible to derive the en trop y inequality for the whole mixture from the partial entrop y inequalities form ulated individually , p er constituen t; cf. e.g. [25] (pp. 475–476). Ho wev er, no w ada ys the general consensus is that this approach giv es a to o restrictive result: the motion of the mixture is ov erconstraint. F or more details on this fundamen tal issue, the reader is referred to [32] (pp. I I.12–13), and [8] (p. 865). Remark 3.5.5. In general we are also interested in mass measures that are not exclusiv ely absolutely contin uous. This naturally leads to the question whether w e can generalize the concept of entrop y and the accompanying entrop y inequalit y . How ever, at this p oint, w e should ask ourselves what the ph ysical meaning of such generalization is. More understand- ing is needed in order to judge the physical relev ance of an extension to a broader class of measures. There are also practical ob jections. F or example, caution is needed when general- izing b oundary terms. It is all but obvious in what w ay the in tegral o ver ∂ Ω 0 in (3.5.5) should b e defined for a general mass measure, that may also contain a discrete part. In Section 4.3 we introduce an explicit entrop y densit y , and deriv e the corresp onding en tropy inequalit y . Despite of the aforementioned difficulties, we sho w afterwards (see Section 4.3.3) 30 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology that structurally the same inequalit y can be obtained for discrete measures (compare the statemen t of Theorem 4.3.3 with (4.3.30)). Mo delling Cro wd Dynamics 31 Chapter 4 Application to cro wd dynamics In this section we explain and extend the measure-theoretical approach, developed b y Bene- detto Piccoli et al. (cf. e.g. [44, 45, 17]). W e in tend to fit some of their ideas to our framework presen ted in Section 3. W e consider a p opulation lo cated in a given domain Ω ⊂ R d . T o capture physically real- istic situations we take d ∈ { 1 , 2 , 3 } . Let T ∈ (0 , ∞ ) b e the fixed final time of the pro cess. W e define a measure µ : (0 , T ) × B (Ω) → R + , suc h that µ ( t, Ω 0 ) represents the mass of the part of the p opulation present in a region Ω 0 ∈ B (Ω) at time t ∈ (0 , T ). W e assume that the population consists of a fixed num b er of subp opulations (these w ere called c onstituents in Section 3), indexed α ∈ { 1 , 2 , . . . , ν } . The mass of each subpopulation present in Ω 0 ∈ B (Ω) at time t ∈ (0 , T ) is giv en by the time-dependent mass measure µ α ( t, Ω 0 ). The mass measures µ and µ α are related via µ = P ν α =1 µ α . W e explicitly assume that µ α ( t, · ) is a finite measure for all t ∈ (0 , T ) and α ∈ { 1 , 2 , . . . , ν } . As a result, µ ( t, · ) b ecomes a finite measure for all t ∈ (0 , T ). 4.1 W eak form ulation Eac h subp opulation mo ves according to its o wn motion mapping χ α , from which a v elo cit y field v α ( t, x ) follows. Note that for each α , the dep endence of v α on t indicates the functional dep endence on all time-dep endent measures µ α , as w e will see in Section 4.2. Remark 4.1.1. W e henceforth disregard the assumptions with resp ect to the motion map- pings χ α and velocity fields v α , which were done in the deriv ation of the balance of mass equations presen ted in Section 3. The result, Equation (3.4.5), is taken here as a starting p oin t for further mo delling, without considering the underlying assumptions. Recapitulating: the fact that here w e deal with ν time-dep enden t mass measures µ α , transp orted with corresp onding velocities v α , translates in to ∂ µ α ∂ t + ∇ · ( µ α v α ) = 0 , for all α ∈ { 1 , 2 , . . . , ν } . (4.1.1) Equations (4.1.1) are accompanied b y the follo wing initial conditions: µ α (0 , · ) = µ α 0 , for eac h α ∈ { 1 , 2 , . . . , ν } , (4.1.2) 32 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology for giv en µ α 0 . This partial differential equation in terms of measures is a shorthand notation for the weak form ulation presented in Sections 3.3 and 3.4. Namely , for all test functions ψ α ∈ C 1 0 ( ¯ Ω), where α ∈ { 1 , 2 , . . . , ν } and for almost every t ∈ (0 , T ) the following holds: d dt Z Ω ψ α ( x ) dµ α ( t, x ) = Z Ω v α ( t, x ) · ∇ ψ α ( x ) dµ α ( t, x ) , α ∈ { 1 , 2 , . . . , ν } . (4.1.3) Remark 4.1.2. If, for some α ∈ { 1 , 2 , . . . , ν } , the mapping t 7→ R Ω ψ α ( x ) dµ α ( t, x ) is abso- lutely con tinuous on (0 , T ), for an y c hoice of ψ α ∈ C 1 0 ( ¯ Ω), then it is differentiable at almost ev ery t ∈ (0 , T ). See e.g. Theorem 7.20 in [46], p. 148. This means that the left-hand side of (4.1.3) exists for almost ev ery t . Remark 4.1.3. If we demand that v α ∈ L 1  (0 , T ); L 1 µ α (Ω)  for all α ∈ { 1 , 2 , . . . , ν } , then the righ t-hand side of (4.1.3) is well-defined. Indeed, since ψ α ∈ C 1 0 ( ¯ Ω), w e ha ve that k∇ ψ α k L ∞ (Ω) is finite for eac h α ∈ { 1 , 2 , . . . , ν } . W e thus hav e for each α ∈ { 1 , 2 , . . . , ν } that    T Z 0 Z Ω v α ( t, x ) · ∇ ψ α ( x ) dµ α ( t, x ) dt    6 T Z 0    Z Ω v α ( t, x ) · ∇ ψ α ( x ) dµ α ( t, x )    dt 6 T Z 0 k∇ ψ α k L ∞ (Ω) Z Ω | v α ( t, x ) | dµ α ( t, x ) dt = T Z 0 k∇ ψ α k L ∞ (Ω) k v α ( t, · ) k L 1 µ α (Ω) dt = k∇ ψ α k L ∞ (Ω) T Z 0 k v α ( t, · ) k L 1 µ α (Ω) dt = k∇ ψ α k L ∞ (Ω) k v α k L 1  [0 ,T ]; L 1 µ α (Ω)  < ∞ . (4.1.4) In particular, it follows that R Ω v α ( t, x ) · ∇ ψ α ( x ) dµ α ( t, x ) is finite for almost ev ery t ∈ (0 , T ) and for all α ∈ { 1 , 2 , . . . , ν } , and thus the right-hand side of (4.1.3) is well-defined. Definition 4.1.4 (W eak solution of (4.1.1)) . The ve ctor of time-dep endent me asur es  ( µ 1 ) t > 0 , ( µ 2 ) t > 0 , . . . , ( µ ν ) t > 0  , wher e e ach µ α satisfies the initial c ondition (4.1.2), is c al le d a we ak solution of (4.1.1), if for al l α ∈ { 1 , 2 , . . . , ν } the fol lowing pr op erties hold: (i) for al l ψ α ∈ C 1 0 ( ¯ Ω) , the mappings t 7→ R Ω ψ α ( x ) dµ α ( t, x ) ar e absolutely c ontinuous, (ii) v α ∈ L 1  (0 , T ); L 1 µ α (Ω)  , (iii) (4.1.3) is satisfie d. Mo delling Cro wd Dynamics 33 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology W e deduced a w eak formulation with resp ect to the total mass measure µ in Section 3.4. W e use the shorthand notation ∂ µ ∂ t + ∇ · ( µv ) = 0 , (4.1.5) accompanied, for giv en µ 0 , b y the initial condition µ (0 , · ) = µ 0 . (4.1.6) This shorthand notation should, again, b e interpreted as (cf. (3.4.8)) d dt Z Ω ψ ( x ) dµ ( t, x ) = Z Ω v ( t, x ) · ∇ ψ ( x ) dµ ( t, x ) , (4.1.7) for all ψ ∈ C 1 0 ( ¯ Ω) and almost ev ery t ∈ (0 , T ). A weak solution to (4.1.5) will b e understo o d in this section alwa ys in the sense of Defi- nition 4.1.5: Definition 4.1.5 (W eak solution of (4.1.5)) . The time-dep endent me asur e ( µ ) t > 0 , satisfying the initial c ondition (4.1.6), is c al le d a we ak solution of (4.1.5), if the fol lowing statements hold: (i) for al l ψ ∈ C 1 0 ( ¯ Ω) , the mapping t 7→ R Ω ψ ( x ) dµ ( t, x ) is absolutely c ontinuous, (ii) v ∈ L 1  (0 , T ); L 1 µ (Ω)  , (iii) (4.1.7) is fulfil le d. The main difference b et ween our approach here and the one presented in [17], is that w e tak e the freedom to allow eac h subp opulation to ha ve its o wn velocity field v α . Note that the crucial mo delling steps take place when we decide what v α actually lo oks lik e. W e then c haracterize the motion of the corresp onding subp opulation. In particular, we define the wa y in which this motion is influenced b y the crowd surrounding it (b elonging to an y of the com- p onen ts). In [17] merely the setting of transporting the total mass measure is considered (see Defi- nition 4.1.5). This means that only the barycen tric v elo city v of the crowd seen as a whole can b e prescrib ed. How ever, we thus lose the ability of mo delling the interaction b etw een subp opulations of different types. Distinct b eha viour of subp opulations is namely driven b y underlying partial v elo cities of distinct nature. T o b e more specific, [17] considers a combination of an absolutely con tinuous measure (a ‘cloud’ of p eople, in whic h individuals are indistinguishable) and a sum of Dirac measures (p oin t masses). Only v is prescrib ed in the framew ork of that pap er, partial velocities b eing not included. This implies that on the macroscopic scale (the cro wd) the absolutely contin u- ous and discrete parts essen tially mov e according to the same velocity field. Corresp ondingly , the Dirac masses cannot ev olve indep endently from the cloud; they are in fact part of the cloud with some ”p ointer” attached to them. Note that this situation is incorp orated in our model as a sp ecial case, where w e tak e one µ α to b e a combination of Dirac measures and an absolutely contin uous part. How ever, w e feel that it is of no use drawing atten tion to these p oint masses, if they cannot b ehav e indep endently . 34 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 4.2 F urther sp ecification of the v elo cit y fields Un til no w, we ha v e not explicitly defined the velocity fields v α ( α ∈ { 1 , 2 , . . . , ν } ). In Section 4.1 w e hav e already remark ed that the crucial mo delling step tak es place at the momen t when w e decide on the explicit form of v α . By the c hoices we make here, we define the character- istics of a subp opulation. That is, we assign, so to sa y , a p ersonality to the members of that subp opulation. In the definition of v α w e can incorp orate the w ay in which an individual is influenced by the p eople around him. This individual might b e sh y , w anting to keep distance from others. The individual’s c haracter migh t also b e quite the opp osite, driving him to come close to other p eople. W e even ha ve the freedom to distinguish b etw een the wa y the individual resp onds to the presence of others, based on the subp opulation that other individual b elongs to. F or example, one subp opulation migh t consist of acquaintances (triggering attractive b ehaviour), while a second subp opulation consists of ‘enemies’ (from which the individual is rep elled). In this section, we show the wa y we wan t to mo del the v elo city fields. V ery muc h inspired by the so cial for c e mo del b y Dirk Helbing et al. [31], the v elo city of a p edestrian is mo delled as a desir e d velo city v α des p erturb ed b y a comp onent v α soc . The latter comp onent, which is called so cial velo city , is due to the presence of other individuals, b oth from the p edestrian’s own subp opulation and from the other subp opulation. This effect is mo delled by functional de- p endence of v α soc on the set of measures  µ 1 , µ 2 , . . . , µ ν  . The desired velocity is indep endent of these mass measures, and represents the velocity a p edestrian would ha v e had in absence of other p eople; this implies that v α des is indep enden t of t , provided the environmen t of an individual do es not change in time. Cf. Remark 4.2.1. The velocity v α is thus for α ∈ { 1 , 2 , . . . , ν }} defined by sup erp osing v α des and its p ertur- bation v α soc : v α ( t, x ) := v α des ( x ) + v α soc ( t, x ) , for all t ∈ (0 , T ) and x ∈ Ω . (4.2.1) The component v α soc mo dels the effect of in teractions with other p edestrians on the current v elo cit y . This is essentially a nonlo cal contribution. Since the effects (in general) differ from one subp opulation to the other, we assume that v α soc has the form: v α soc ( t, x ) := ν X β =1 Z Ω \{ x } f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dµ β ( t, y ) , for all α ∈ { 1 , 2 , . . . , ν } . (4.2.2) Remark 4.2.1. Note thus that the t -dependence of v α and v α soc is not an explicit time- dep endence. As (4.2.2) shows, the so cial velocity depends functionally on the time-dep enden t mass measures µ β . Although we ha ve not done so, we are also allow ed to c ho ose v des = v des ( t, x ). This ought to b e an explicit dep endence on t due to the interpretation giv en to v des : the v elo city a p edestrian would hav e if he was the only p erson in the ro om. Ob viously , this term should dep end only on the geometry of the domain (and not on the cro wd it contains). If v des dep ends explicitly on the time, this reflects that the domain is non-static. F or example, such situation o ccurs when certain areas are not alw ays accessible (op ening hours), or are temp orarily obstructed. F or the moment this is to o farfetched. In (4.2.2), w e hav e used the following mo delling ans¨ atze: Mo delling Cro wd Dynamics 35 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology • f α β is a function from R + to R , describing the effect of the mutual distance b etw een individuals on their in teraction. The subscript and sup erscript denote that this sp ecific function incorp orates the influence of the mem b ers of subp opulation β on subp opulation α . In principle we distinguish b etw een tw o kinds of effects: – Attr action-r epulsion : this is the interaction b etw een acquaintances (and similar kinds of interaction). In this case, f α β is a comp osition of t w o effects: at short distance individuals are repelled, since they wan t to av oid collisions and congestion, but if their mutual distance increases they are attracted to other group mates, in order not to get separated from the group. – R epulsion : i.e. ‘shy’ b ehaviour. Here, f α β con tains only the repulsive part, since the individual in subp opulation α do es not like to come close to subpopulation β . Graphical represen tations of these tw o choices of f α β are depicted in Figure 4.1. • θ α xy denotes the angle b etw een y − x and v α des ( x ): the angle under whic h x sees y if it w ere moving in the direction of v α des ( x ). • g is a function from [ − π, π ] to [0 , 1] that enco des the fact that an individual’s perception is not equal in all directions. W e c ho ose: g ( θ ) := σ + (1 − σ ) 1 + cos( θ ) 2 , for θ ∈ [ − π , π ] . (4.2.3) This definition ensures that an individual experiences the strongest influence from some- one straight ahead, since g (0) = 1 for an y σ ∈ [0 , 1]. The constant σ is a tuning param- eter, called p otential of anisotr opy . It determines how strongly a pedestrian is fo cussed on what happ ens in front of him, and how large the influence is of people at his sides or b ehind him. The effect of the choice of this parameter can b e seen in Figure 4.2. Note that cos( θ α xy ) can b e calculated in a conv enien t wa y: cos( θ α xy ) = ( y − x ) · v α des ( x ) | y − x | | v α des ( x ) | . Remark 4.2.2. Other choices for v α soc are also p ossible. Based on the current velocities we can make an individual an ticipate the distance he exp ects to b e from another p edestrian in the (near) future. Although this is probably more realistic, it increases the complexity of the mo del dramatically . If these changes are made, v α namely dep ends on the velocities v β ( β ∈ { 1 , 2 , . . . , ν } ); in particular on v α itself. The definition of v α b ecomes thus implicit, and is m uch harder to w ork with. 4.3 Deriv ation of an en tropy inequalit y In this section, w e derive an entrop y inequalit y concept in the spirit of (3.5.5) in Section 3.5. Here, we also sp ecify under which conditions this inequality holds. F or simplicity and clarity , calculations are p erformed for a crowd without distinct subpopulations first; see Section 4.3.1. A multi-component crowd is considered afterwards in Section 4.3.2. The inspiration for the one-p opulation case was provided by [16] (but we slightly deviate from their definitions). 36 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology R r R a s f H s L R r s f H s L Figure 4.1: Graphical representation of typical examples of the function f α β : attraction- repulsion (left) and repulsion only (right). Here, R r is the r adius of r epulsion : an individual is rep elled if its distance to someone else is smaller than R r , while R a is the r adius of attr action : an individual is attracted to an ‘acquaintance’ if their mutual distance is b etw een R r and R a . Note that R r migh t b e c hosen differently in either of the tw o c hoices. T ypically R r will b e larger in the repulsion case than in the attraction-repulsion case. -Π - Π 2 Π 2 Π Θ 1 2 1 g H Θ L Figure 4.2: Graphical representation of the function g . W e ha ve g (0) = 1 alwa ys, and g ( − π ) = g ( π ) = σ . F urthermore g is symmetric around θ = 0 and increasing on ( − π , 0) (thus decreasing on (0 , π )). The plot has b een made for σ = 0, 0 . 3, 0 . 5, 0 . 7, 0 . 9, 1 (b ottom to top). W e work with a mass measure that is absolutely contin uous w.r.t. the Leb esgue measure, that is: w e do ha v e a mass densit y . If one wan ts to deviate from the setting of [16], an alternativ e approach is suggested in App endix D. There, the entrop y density follo ws, if the so-called fr e e ener gy F is explicitly pro vided. The crucial decision is thus the choice of this free energy . In App endix D, this is done for the example of the ideal gas. If one would find a go o d choice of F , the same argumen ts could b e used for cro wds. 4.3.1 One p opulation W e consider the presence of a single p opulation in the domain Ω ⊂ R d . The time-dep enden t densit y is ρ : (0 , T ) × Ω → R + . W e recall that it satisfies the balance of mass equation (3.3.6): ∂ ρ ∂ t + ∇ ·  ρv  = 0 , a.e. in Ω . (4.3.1) Assumption 4.3.1. We assume that the velo city field v : (0 , T ) × Ω → R d is of the form v := ∇ V + ∇ W ? ρ, (4.3.2) Mo delling Cro wd Dynamics 37 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology wher e V : Ω → R is c al le d c onfinement p otential, and W : R d → R is c al le d inter action p otential. We also assume that W is symmetric, that is W ( ξ ) = W ( − ξ ) , for al l ξ ∈ R d . Note that in [16] a min us-sign is added to the definition in (4.3.2). The con volution ∇ W ? ρ is defined as ( ∇ W ? ρ )( t, x ) := Z Ω ∇ W ( x − y ) ρ ( t, y ) dλ d ( y ) , for all t ∈ (0 , T ) , and x ∈ Ω , (4.3.3) and th us this part is time-dep endent via ρ . F or brevit y we often write ρ ( x ) instead of ρ ( t, x ) in the sequel. The time-dep endency is to b e understo o d implicitly . Analogously , if we write ( ∇ W ? ρ )( x ), we actually mean ( ∇ W ? ρ )( t, x ). Remark 4.3.2. The comp onent ∇ V is the desir e d velo city v des of Section 4.2. Similarly , the comp onen t ∇ W ? ρ is the so cial velo city v soc . W e already prop osed a velocity field of the following form (cf. Section 4.2): v ( t, x ) := v des ( x ) + Z Ω f ( | y − x | ) g ( θ xy ) y − x | y − x | ρ ( t, y ) dλ d ( y ) . (4.3.4) W e can take Ω as our domain of integration here, instead of Ω \ { x } (cf. (4.2.2)). This is b e- cause { x } is a λ d -negligible set; the in tegral has the same v alue with or without the exclusion of this set from the domain. This holds for absolutely contin uous measures, but not for any measure in general. The v elo cit y as defined in (4.3.4) fits to the structure of Assumption 4.3.1, if: • v des can b e written as ∇ V for some p otential V . Note that in [44] a comparable situation is co vered. The p otential is found by solving the Laplace equation ∆ V = 0 in the domain Ω. F or us this would mean that v des is divergence-free. How ever, the difference with our situation is that [44] normalizes and rescales ∇ V afterwards. • we disregard here the factor g ( θ xy ), that is, we set g ≡ 1 (or σ = 1). This is necessary at this stage, because the inclusion of g mak es it impossible to find a symmetric interaction p oten tial W . 1 • there is a function F : R + → R , such that f = − F 0 . Then: f ( | y − x | ) y − x | y − x | = − f ( | x − y | ) x − y | x − y | = F 0 ( | x − y | ) x − y | x − y | = ∇ F ( | x − y | ) , (4.3.5) where the last step of the calculations is due to the c hain rule, and the fact that ∇| x − y | = x − y | x − y | . Now write W ( x − y ) := F ( | x − y | ); note that this indeed implies that W ( ξ ) = W ( − ξ ) for all ξ ∈ R d . 1 This is an inevitable, but disapp ointing choice. W e thus recommend further research in this direction. 38 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology It is, for example, p ossible to ha v e v des ≡ a ∈ R d ; that is, the desired v elo city has constant magnitude and direction. T o comply with the assumption that v des can be written as ∇ V , w e hav e to take V ( x ) = a · x . It follows from the aforementioned assumptions on g and f (and the subsequen t calcula- tions) that v soc = ∇ W ? ρ . In Figure 4.3 an impression is given of the functions F that corresp ond (via the relation f = − F 0 ) to those functions f plotted in Figure 4.3. These functions are unique up to additional constan ts. R r R a s F H s L R r s F H s L Figure 4.3: Graphical impression of the function F corresp onding, via the relation f = − F 0 , to the functions plotted in Figure 4.1: attraction-repulsion (left) and repulsion only (right). As in [16], w e define the en tropy density η as η ( t, x ) := V ( x ) + 1 2 ( W ? ρ )( t, x ) . (4.3.6) The corresp onding entrop y of the system in Ω is then, according to (3.5.2), giv en b y: S ( t ) = Z Ω η ( t, x ) ρ ( t, x ) dλ d ( x ) . (4.3.7) The cen tral question here is: Do es the choice (4.3.6) – (4.3.7) satisfy the Clausius-Duhem Inequalit y? W e answer this question in Theorem 4.3.3. Theorem 4.3.3 (En tropy inequalit y) . Assume that v satisfies Assumption 4.3.1, that the entr opy S is given by (4.3.6) – (4.3.7), and also that the system is isolate d. 2 Then the fol lowing ine quality holds: dS dt > 0 . Pr o of. W e inv estigate directly the time-deriv ative of the entrop y S , i.e. we hav e: dS dt = Z Ω V ( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) + 1 2 Z Ω ∂ ∂ t ( W ? ρ )( x ) ρ ( x ) dλ d ( x ) + 1 2 Z Ω ( W ? ρ )( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) = Z Ω V ( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) + 1 2 Z Ω ( W ? ∂ ρ ∂ t )( x ) ρ ( x ) dλ d ( x ) + 1 2 Z Ω ( W ? ρ )( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) . (4.3.8) 2 The statement that the system is isolated means that there is no flux through the b oundary of Ω, i.e. ρv · n = 0 at ∂ Ω. Mo delling Cro wd Dynamics 39 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Note that Z Ω ( W ? ∂ ρ ∂ t )( x ) ρ ( x ) dλ d ( x ) = Z Ω Z Ω W ( x − y ) ∂ ρ ∂ t ( y ) dλ d ( y ) ρ ( x ) dλ d ( x ) = Z Ω Z Ω W ( x − y ) ∂ ρ ∂ t ( y ) ρ ( x ) dλ d ( y ) dλ d ( x ) = Z Ω Z Ω W ( y − x ) ∂ ρ ∂ t ( x ) ρ ( y ) dλ d ( x ) dλ d ( y ) = Z Ω Z Ω W ( x − y ) ρ ( y ) dλ d ( y ) ∂ ρ ∂ t ( x ) dλ d ( x ) = Z Ω ( W ? ρ )( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) . (4.3.9) W e ha ve replaced x by y , and vic e versa , to obtain the third equality (this is solely a matter of notation). T o obtain the fourth one, we used that for all x, y ∈ Ω, W ( y − x ) = W ( x − y ) and in terchanged the order of in tegration. F rom (4.3.9), w e conclude that (4.3.8) can b e written as dS dt = Z Ω V ( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) + Z Ω ( W ? ρ )( x ) ∂ ρ ∂ t ( x ) dλ d ( x ) = Z Ω  V ( x ) + ( W ? ρ )( x )  ∂ ρ ∂ t ( x ) dλ d ( x ) . (4.3.10) Substitution of the balance of mass (4.3.1) in (4.3.10) yields dS dt = − Z Ω  V ( x ) + ( W ? ρ )( x )  ∇ ·  ρ ( x ) v ( x )  dλ d ( x ) = − Z ∂ Ω  V ( x ) + ( W ? ρ )( x )  ρ ( x ) v ( x ) · ndλ d − 1 ( x ) + Z Ω ∇  V ( x ) + ( W ? ρ )( x )  · ρ ( x ) v ( x ) dλ d ( x ) . (4.3.11) By the h yp othesis that ρv · n = 0 at the boundary ∂ Ω, the b oundary term in (4.3.11) v anishes. 40 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology W e pro ceed as follows: dS dt = Z Ω ∇  V ( x ) + ( W ? ρ )( x )  · ρ ( x ) v ( x ) dλ d ( x ) = Z Ω  ∇ V ( x ) + ∇ ( W ? ρ )( x )  · ρ ( x ) v ( x ) dλ d ( x ) = Z Ω  ∇ V ( x ) + ( ∇ W ? ρ )( x )  · ρ ( x ) v ( x ) dλ d ( x ) = Z Ω v ( x ) · ρ ( x ) v ( x ) dλ d ( x ) = Z Ω | v ( x ) | 2 ρ ( x ) dλ d ( x ) > 0 . (4.3.12) Note that the relation ∇ ( W ? ρ )( x ) = ∇ x Z Ω W ( x − y ) ρ ( y ) dλ d ( y ) = Z Ω ∇ x W ( x − y ) ρ ( y ) dλ d ( y ) = Z Ω ( ∇ W )( x − y ) ρ ( y ) dλ d ( y ) =( ∇ W ? ρ )( x ) , w as used in the third step. In (4.3.12), w e recognize the en tropy inequality w e were lo oking for: dS dt > 0 . (4.3.13) Remark 4.3.4. In Theorem 4.3.3 and in its pro of we ha ve implicitly assumed a sufficient amoun t of regularit y of the b oundary ∂ Ω. This is also imp ortant for the steps we are ab out to tak e. Let us comment on the situation in which ρv · n = 0 do es not necessarily hold at ∂ Ω. This means that w e allow mass (and consequently also entrop y) to escap e from or enter the domain of our fo cus. Instead of (4.3.12), this would yield dS dt > − Z ∂ Ω  V ( x ) + ( W ? ρ )( x )  ρ ( x ) v ( x ) · ndλ d − 1 ( x ) = − Z ∂ Ω η ( x ) ρ ( x ) v ( x ) · ndλ d − 1 ( x ) − Z ∂ Ω 1 2 ( W ? ρ )( x ) ρ ( x ) v ( x ) · ndλ d − 1 ( x ) . (4.3.14) Mo delling Cro wd Dynamics 41 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology After defining the entr opy flux j η as j η := 1 2 ( W ? ρ ) ρv , w e are thus in the setting of (3.5.5) if there is no external v olume supply of heat: dS dt > − Z ∂ Ω  η ρv + j η  · ndλ d − 1 . 4.3.2 Multi-comp onen t crowd In this section, we extend the results of Section 4.3.1 to the situation in which the crowd consists of a giv en n umber of subp opulations. Let the crowd in the domain Ω ⊂ R d consist of ν subp opulations, indexed b y α ∈ { 1 , 2 , . . . , ν } . The time-dep endent density of comp onent α is denoted by ρ α : (0 , T ) × Ω → R + . Cf. the definition (3.1.12) of ρ α as a Radon-Nik o dym deriv ativ e in Section 3.1. Assumption 4.3.5. 1. The velo city field of c omp onent α , denote d by v α : (0 , T ) × Ω → R d is assume d to b e of the form v α := ∇ V α + ν X β =1 ∇ W α β ? ρ β , for al l α ∈ { 1 , 2 , . . . , ν } , (4.3.15) wher e V α : Ω → R is c al le d the c onfinement p otential of c omp onent α , and W α β : R d → R is c al le d the inter action p otential of c omp onent β (affe cting α ). 2. F or e ach α, β ∈ { 1 , 2 , . . . , ν } we assume that W α β ( ξ ) = W α β ( − ξ ) holds for al l ξ ∈ R d . 3. We assume symmetric inter actions, that is W α β ≡ W β α , for al l α, β ∈ { 1 , 2 , . . . , ν } . (4.3.16) Remark 4.3.6. P art 3 of Assumption 4.3.5 implies for example that w e do not allo w a ”predator-prey relation” b etw een tw o subp opulations. In suc h relation a predator should b e attracted to the prey-p opulation, but a prey should b e rep elled from the predators. These in teractions are asymmetric . The follo wing balance of mass equation is satisfied for eac h α ∈ { 1 , 2 , . . . , ν } : ∂ ρ α ∂ t + ∇ ·  ρ α v α  = 0 , a.e. in Ω . (4.3.17) W e recall from Sections 3.1 – 3.2 the definitions of the total densit y ρ and barycen tric v elo city v : ρ := ν X α =1 ρ α , v := ν X α =1 ρ α ρ v α . 42 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology On the macroscopic scale, the follo wing balance of mass equation is satisfied: ∂ ρ ∂ t + ∇ ·  ρv  = 0 , a.e. in Ω . (4.3.18) F or all t ∈ (0 , T ) and x ∈ Ω we define the partial en tropy density of comp onen t α as η α ( t, x ) := V α ( x ) + 1 2 ν X β =1 ( W α β ? ρ β )( t, x ) . (4.3.19) The en tropy densit y of the whole crowd is defined as η := ν X α =1 ρ α ρ η α . Consequen tly , the entrop y of the system at time t is given by S ( t ) = Z Ω η ( t, x ) ρ ( t, x ) dλ d ( x ) = ν X α =1 Z Ω  V α ( x ) + 1 2 ν X β =1 ( W α β ? ρ β )( t, x )  ρ α ( t, x ) dλ d ( x ) . (4.3.20) In the spirit of Theorem 4.3.3 w e can form ulate the follo wing theorem: Theorem 4.3.7 (En tropy inequalit y) . Assume that for e ach α we have that v α satisfies Assumption 4.3.5. Mor e over, assume that the entr opy is given by (4.3.20) and that the system is isolate d. Then the fol lowing ine quality holds: dS dt > 0 . Pr o of. The approac h here is very muc h in the spirit of the pro of of Theorem 4.3.3. W e thus consider the time-deriv ativ e of S : dS dt = ν X α =1 Z Ω V α ( x ) ∂ ρ α ∂ t ( x ) dλ d ( x ) + 1 2 ν X α =1 Z Ω ν X β =1 ∂ ∂ t ( W α β ? ρ β )( x ) ρ α ( x ) dλ d ( x ) + 1 2 ν X α =1 Z Ω ν X β =1 ( W α β ? ρ β )( x ) ∂ ρ α ∂ t ( x ) dλ d ( x ) = ν X α =1 Z Ω V α ( x ) ∂ ρ α ∂ t ( x ) dλ d ( x ) + 1 2 ν X α =1 ν X β =1 Z Ω ( W α β ? ∂ ρ β ∂ t )( x ) ρ α ( x ) dλ d ( x ) + 1 2 ν X α =1 ν X β =1 Z Ω ( W α β ? ρ β )( x ) ∂ ρ α ∂ t ( x ) dλ d ( x ) (4.3.21) Mo delling Cro wd Dynamics 43 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology F ollo wing the idea of (4.3.9), we deriv e ν X α =1 ν X β =1 Z Ω ( W α β ? ∂ ρ β ∂ t )( x ) ρ α ( x ) dλ d ( x ) = ν X α =1 ν X β =1 Z Ω Z Ω W α β ( x − y ) ∂ ρ β ∂ t ( y ) dλ d ( y ) ρ α ( x ) dλ d ( x ) = ν X α =1 ν X β =1 Z Ω Z Ω W α β ( x − y ) ∂ ρ β ∂ t ( y ) ρ α ( x ) dλ d ( y ) dλ d ( x ) = ν X β =1 ν X α =1 Z Ω Z Ω W β α ( y − x ) ∂ ρ α ∂ t ( x ) ρ β ( y ) dλ d ( x ) dλ d ( y ) = ν X α =1 ν X β =1 Z Ω Z Ω W β α ( x − y ) ρ β ( y ) dλ d ( y ) ∂ ρ α ∂ t ( x ) dλ d ( x ) = ν X α =1 ν X β =1 Z Ω ( W β α ? ρ β )( x ) ∂ ρ α ∂ t ( x ) dλ d ( x ) . (4.3.22) W e ha v e replaced x by y , α b y β , and vic e versa , to obtain the third equality (this is again solely a matter of notation). T o obtain the fourth one, we used the fact that W β α ( y − x ) = W β α ( x − y ) for all α, β ∈ { 1 , 2 , . . . } . Moreov er, we interc hanged the order of summation and in tegration. W e no w combine (4.3.21) and (4.3.22), and conclude that dS dt = ν X α =1 Z Ω h V α ( x ) + ν X β =1  1 2 ( W α β + W β α ) ? ρ β  ( x ) i ∂ ρ α ∂ t ( x ) dλ d ( x ) . (4.3.23) W e substitute (4.3.17), the balance of mass p er comp onent, in (4.3.23), by which we obtain dS dt = − ν X α =1 Z Ω h V α ( x ) + ν X β =1  1 2 ( W α β + W β α ) ? ρ β  ( x ) i ∇ ·  ρ α ( x ) v α ( x )  dλ d ( x ) = − ν X α =1 Z ∂ Ω h V α ( x ) + ν X β =1  1 2 ( W α β + W β α ) ? ρ β  ( x ) i ρ α ( x ) v α ( x ) · ndλ d − 1 ( x ) + ν X α =1 Z Ω ∇ h V α ( x ) + ν X β =1  1 2 ( W α β + W β α ) ? ρ β  ( x ) i · ρ α ( x ) v α ( x ) dλ d ( x ) . (4.3.24) Under the assumption that there is no flux of mass through the b oundary of Ω for an y of the constituen ts α , w e hav e ρ α v α · n = 0 at ∂ Ω for all α ∈ { 1 , 2 , . . . , ν } . This means that the b oundary term in (4.3.24) v anishes. T aking also Part 3 of Assumption 4.3.5 (i.e. symmetric 44 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology in teractions) into consideration, (4.3.24) reads dS dt = ν X α =1 Z Ω ∇ h V α ( x ) + ν X β =1 ( W α β ? ρ β )( x ) i · ρ α ( x ) v α ( x ) dλ d ( x ) = ν X α =1 Z Ω h ∇ V α ( x ) + ν X β =1 ( ∇ W α β ? ρ β )( x ) i · ρ α ( x ) v α ( x ) dλ d ( x ) = ν X α =1 Z Ω v α ( x ) · ρ α ( x ) v α ( x ) dλ d ( x ) = ν X α =1 Z Ω | v α ( x ) | 2 ρ α ( x ) dλ d ( x ) > 0 . (4.3.25) By (4.3.25) w e thus ha ve the following entrop y inequality: dS dt > 0 . (4.3.26) If w e allow mass (and consequen tly also entrop y) to escap e or enter through the boundary of Ω, w e should c onsider the b oundary term in (4.3.24). Let us define the entrop y flux j η as j η := 1 2 ν X α =1 ν X β =1  W β α ? ρ β  ρ α v α , Still assuming symmetric in teractions, and using (4.3.19), (4.3.24) w ould now read dS dt > − ν X α =1 Z ∂ Ω h V α ( x ) + ν X β =1  1 2 ( W α β + W β α ) ? ρ β  ( x ) i ρ α ( x ) v α ( x ) · ndλ d − 1 ( x ) = − Z ∂ Ω ν X α =1 η α ( x ) ρ α ( x ) v α ( x ) · ndλ d − 1 ( x ) − Z ∂ Ω j η ( x ) · ndλ d − 1 ( x ) . (4.3.27) Remark 4.3.8. Note that this do es not bring us to the setting of (3.5.5) if there is no external v olume supply of heat. This is b ecause, in general ν X α =1 η α ρ α v α 6 = η ρv . Compare this to Remark 3.5.4. Using P ν α =1 η α ρ α v α in the en tropy inequality fits the approach of [25]. This is how ever rejected by [8, 32], who go for using η ρv , as it is mentioned in (3.5.5). Remark 4.3.9. The symmetry of the in teractions, as imposed b y P art 3 of Assumption 4.3.5, is crucial in the pro of of Theorem 4.3.7. How ev er, it is a quite restrictive assumption, that disallo ws man y in teresting settings and thus deserves further researc h. At a later stage we hop e to include a drift in the in teractions, which is such that we can still form ulate an en tropy inequalit y . Mo delling Cro wd Dynamics 45 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 4.3.3 Generalization to discrete measures In this section, we make feasible that it is p ossible to deriv e (at least from a mathematical p oin t of view) an analogon for discrete measures of the entrop y inequalities mentioned in Sections 4.3.1 and 4.3.2. W e already announced that w e w ould do so in Remark 3.5.5. W e consider a single p opulation with corresp onding discrete mass measure of the form µ := X i ∈J δ x i ( t ) , where J ⊂ N is some index set. The evolution in time of the cen tres x i is gov erned b y the v elo cit y field v via d dt x i ( t ) = v ( t, x i ( t )) . This v elo cit y field is assumed to b e of the form (cf. (4.3.2)) v := ∇ V + ∇ W ? µ, (4.3.28) The con volution ∇ W ? µ is a generalized form of (4.3.3), defined as ( ∇ W ? µ )( t, x ) := Z Ω \{ x } ∇ W ( x − y ) dµ ( t, y ) , for all t ∈ (0 , T ) , and x ∈ Ω . (4.3.29) The p oint x itself has b een excluded from the domain of in tegration to av oid in teraction of a p oin t mass/p edestrian with itself. W e assume again that W is symmetric: W ( ξ ) = W ( − ξ ) , for all ξ ∈ R d . In the spirit of (4.3.6) w e define the en tropy densit y η as η ( t, x ) := V ( x ) + 1 2 ( W ? µ )( t, x ) . The corresp onding entrop y of the system in Ω is S ( t ) = Z Ω η ( t, x ) dµ ( t, x ) . W e explicitly restrict ourselves to the situation that all point masses remain in Ω. 3 This restriction implies that an integral with resp ect to the measure µ can b e represented b y a sum, in whic h all ce n tres x i con tribute. W e hav e that S ( t ) = X i ∈J n V ( x i ) + 1 2 X j ∈J x j 6 = x i W ( x i − x j ) 1 x j ∈ Ω o 1 x i ∈ Ω , where all p ositions x i are time-dep endent. The indicator function 1 x i ∈ Ω is 1 if x i ∈ Ω is true, and 0 otherwise. 3 Allo wing point masses to lea ve (or enter) the domain, means that a b oundary measure, or a trace of the mass measure on the b oundary , needs to b e defined prop erly . It is all but trivial to do so for general measures. 46 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology The requirement that all centres x i remain in Ω, makes that 1 x i ∈ Ω nev er b ecomes 0. F or all i ∈ J thus 1 x i ∈ Ω ≡ 1 holds, for all t ∈ [0 , T ]. As a result, w e lose time-dep endence in these indicator functions, and w e can just write S ( t ) = X i ∈J n V ( x i ) + 1 2 X j ∈J x j 6 = x i W ( x i − x j ) o . Remark 4.3.10. T o derive an entrop y inequality we examine the time deriv ative of S . Con- sider the ‘forbidden’ situation that x i ( t ) leav es (or enters) the domain, say at time t = t ∗ . Then 1 x i ( t ) ∈ Ω is discon tinuous in t = t ∗ . W e disallo w this situation, b ecause the time deriv a- tiv e of S do es not exist in suc h p oin t (not ev en in a w eak sense). W e tak e the time deriv ative of S ( t ): dS dt = X i ∈J n ∇ V ( x i ) · dx i dt + 1 2 X j ∈J x j 6 = x i ∇ x i W ( x i − x j ) · dx i dt + 1 2 X j ∈J x j 6 = x i ∇ x j W ( x i − x j ) · dx j dt o = X i ∈J n ∇ V ( x i ) · dx i dt + 1 2 X j ∈J x j 6 = x i ∇ W ( x i − x j ) · dx i dt − 1 2 X j ∈J x j 6 = x i ∇ W ( x i − x j ) · dx j dt o = X i ∈J ∇ V ( x i ) · v ( t, x i ) + 1 2 X i,j ∈J x j 6 = x i ∇ W ( x i − x j ) · v ( t, x i ) − 1 2 X i,j ∈J x j 6 = x i ∇ W ( x i − x j ) · v ( t, x j ) = X i ∈J ∇ V ( x i ) · v ( t, x i ) + 1 2 X i,j ∈J x j 6 = x i ∇ W ( x i − x j ) · v ( t, x i ) + 1 2 X i,j ∈J x j 6 = x i ∇ W ( x j − x i ) · v ( t, x j ) = X i ∈J ∇ V ( x i ) · v ( t, x i ) + X i,j ∈J x j 6 = x i ∇ W ( x i − x j ) · v ( t, x i ) = X i ∈J v ( t, x i ) · n ∇ V ( x i ) + X j ∈J x j 6 = x i ∇ W ( x i − x j ) o = X i ∈J v ( t, x i ) · n ∇ V ( x i ) + ( ∇ W ? µ )( x i ) o = Z Ω | v ( t, x ) | 2 dµ ( t, x ) > 0 . (4.3.30) The seven th equalit y is due to (4.3.29). By (4.3.30) we ha ve derived an equiv alent statemen t as in Theorem 4.3.3. F ollo wing similar lines of argumen t, an entrop y inequality as in Theorem 4.3.7 can b e deriv ed also if w e allo w subp opulations with b oth discrete and absolutely con tinuous mass measures. W e will, ho wev er, not go into further details in this direction. Mo delling Cro wd Dynamics 47 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 4.4 Deriv ation of the time-discrete mo del Inspired by [17, 45], we derive in this section a discrete-in-time counterpart of the w eak for- m ulation presented in (4.1.3) and Definition 4.1.4. F or this aim, we in tro duce a strictly increasing sequence of discrete p oints in time { t n } n ∈N ⊂ [0 , T ]. Here, N := { 0 , 1 , . . . , N T } is the index set, with N T ∈ N . The set of p oin ts in time is c hosen such that t 0 = 0 and t N T = T . W e define ∆ t n := t n +1 − t n . Since { t n } n ∈N is a strictly increasing sequence, ∆ t n > 0 holds for all n . T o emphasize that w e are working in a time-discrete setting, we write µ α n ( · ) as the time-discrete equiv alen t of µ α ( t n , · ) : B (Ω) → R + for all n ∈ N and all α ∈ { 1 , 2 , . . . , ν } . Analogously , we write v α n ( · ) for v α ( t n , · ) : Ω → R d . F or arbitrary n ∈ N , integration in time of (4.1.3) o v er the interv al ( t n , t n +1 ) yields for all α ∈ { 1 , 2 , . . . , ν } and all ψ α ∈ C 1 0 ( ¯ Ω): Z Ω ψ α ( x ) dµ α ( t n +1 , x ) − Z Ω ψ α ( x ) dµ α ( t n , x ) = t n +1 Z t n Z Ω v α ( t, x ) · ∇ ψ α ( x ) dµ α ( t, x ) dt. (4.4.1) Assuming all necessary regularity , we can expand the righ t-hand term in a T a ylor series around t n . F or the sak e of brevity we define A ( t ) := R t t n R Ω v α ( ˜ t, x ) · ∇ ψ α ( x ) dµ α ( ˜ t, x ) d ˜ t . Note that A ( t n +1 ) is the righ t-hand side of (4.4.1). No w A ( t n +1 ) = A ( t n ) + ∆ t n dA dt    t = t n + O  ∆ t 2 n  . Note that A ( t n ) = 0 and d dt A ( t ) = R Ω v α ( t, x ) · ∇ ψ α ( x ) dµ α ( t, x ). By writing µ α n ( · ) instead of µ α ( t n , · ), and v α n ( · ) instead of v α ( t n , · ) (ab ov e w e already announced to do so), (4.4.1) transforms in to Z Ω ψ α ( x ) dµ α n +1 ( x ) − Z Ω ψ α ( x ) dµ α n ( x ) = ∆ t n Z Ω v α n ( x ) · ∇ ψ α ( x ) dµ α n ( x ) + O  ∆ t 2 n  . This can also b e written as Z Ω ψ α ( x ) dµ α n +1 ( x ) = Z Ω  ψ α ( x ) + ∆ t n v α n ( x ) · ∇ ψ α ( x )  dµ α n ( x ) + O  ∆ t 2 n  . (4.4.2) If v α n is ‘w ell-b eha ved’ w e expand ψ α  x + ∆ t n v α n ( x )  = ψ α ( x ) + ∆ t n v α n ( x ) · ∇ ψ α ( x ) + O  ∆ t 2 n  , (4.4.3) or ψ α ( x ) + ∆ t n v α n ( x ) · ∇ ψ α ( x ) = ψ α  x + ∆ t n v α n ( x )  + O  ∆ t 2 n  . (4.4.4) By ‘well-behav ed’ we mean that v α n is (at least) µ α n -uniformly b ounded. That is, for fixed n ∈ N there exists a non-negative constant M n suc h that | v α n ( x ) | < M n for µ α n -almost every 48 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology x , b y which µ α n -almost ev erywhere: ∆ t n v α n ( x ) = O (∆ t n ). W e substitute this in (4.4.2) to obtain Z Ω ψ α ( x ) dµ α n +1 ( x ) = Z Ω  ψ α  x + ∆ t n v α n ( x )  + O  ∆ t 2 n   dµ α n ( x ) + O  ∆ t 2 n  = Z Ω ψ α  x + ∆ t n v α n ( x )  dµ α n ( x ) + O  ∆ t 2 n  , (4.4.5) where w e hav e tak en the O  ∆ t 2 n  -terms outside the integral, and used that µ α n (Ω) is finite, since µ α ( t n , · ) is a finite measure for all c hoices of n ∈ N . W e neglect the O  ∆ t 2 n  -part and obtain Z Ω ψ α ( x ) dµ α n +1 ( x ) ≈ Z Ω ψ α  χ α n ( x )  dµ α n ( x ) . (4.4.6) Although (4.4.6) is an appro ximation, w e treat it as an equality from now on. Whenever w e refer to (4.4.6), we thus mean the equality rather than the approximation. In (4.4.6) we hav e moreo ver used the definition: Definition 4.4.1 (One-step motion mapping) . The one-step motion mapping χ α n is define d by χ α n ( x ) := x + ∆ t n v α n ( x ) , (4.4.7) for al l n ∈ { 0 , 1 , . . . , N T − 1 } . F or simplicity, it wil l henc eforth just b e c al le d motion mapping. It pr ovides the p osition at time step n + 1 of the p oint lo c ate d in x at time step n . Assumption 4.4.2 (Prop erties of the motion mappings) . F or al l n ∈ { 0 , 1 , . . . , N T − 1 } and e ach α ∈ { 1 , 2 , . . . , ν } we assume that χ α n : Ω → Ω is a home omorphism. This me ans that: (i) χ α n is invertible, (ii) χ α n is c ontinuous, (iii) ( χ α n ) − 1 is c ontinuous. Remark 4.4.3. F rom the pro of of Lemma 3.3.1 w e extract the statemen t that a contin uous mapping is Borel. Regarding Assumption 4.4.2, this implies that χ α n and ( χ α n ) − 1 are Borel. Resp ectiv ely , that is: (i) for all Ω 0 ∈ B (Ω) w e hav e that ( χ α n ) − 1 (Ω 0 ) ∈ B (Ω); (ii) for all Ω 0 ∈ B (Ω) w e hav e that χ α n (Ω 0 ) ∈ B (Ω). Note that Assumption 4.4.2 is the somewhat weak er counterpart of the assumptions on the motion mappings in Section 3.2. Remark 4.4.4. The expression in (4.4.6) makes sense even if we do not restrict ourselv es to taking only ψ α ∈ C 1 0 ( ¯ Ω). In the sequel w e allo w ψ α to b e any function that is in tegrable on Ω with resp ect to the measure µ α n +1 , that is: ψ α ∈ L 1 µ α n +1 (Ω). Mo delling Cro wd Dynamics 49 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology F ollo wing Remark 4.4.4, we can tak e ψ α = 1 Ω 0 the c haracteristic function for an y Ω 0 ∈ B (Ω). As a result, (4.4.6) reduces to µ α n +1 (Ω 0 ) = µ α n  ( χ α n ) − 1 (Ω 0 )  . (4.4.8) Definition 4.4.5 (Push forward) . The me asur e η n +1 is c al le d the push forwar d of the me asur e η n via the motion mapping χ α n , notation: η n +1 = χ α n # η n , (4.4.9) if η n +1 (Ω 0 ) = η n  ( χ α n ) − 1 (Ω 0 )  , is satisfie d for al l Ω 0 ∈ B (Ω) . W e ha ve no w derived the time-discrete version of the problem formulated in Section 4.1: Definition 4.4.6 (Time-discrete solution) . The ve ctor of time-discr ete me asur es:  ( µ 1 n ) n ∈N , ( µ 2 n ) n ∈N , . . . , ( µ ν n ) n ∈N  , is c al le d time-discr ete solution, if: (i) µ α n =0 = µ α 0 is satisfie d for e ach α ∈ { 1 , 2 , . . . , ν } , for some given set of initial me asur es µ α 0 that ar e p ositive and finite, (ii) the evolution of µ α n is for e ach α determine d by the push forwar d µ α n +1 = χ α n # µ α n , (iii) for e ach α , µ α n is p ositive and finite for al l n . W e refer to finding a time-discrete solution as Problem ( P ). 4.5 Solv abilit y of Problem ( P ) and prop erties of the solution In this section, w e prov e that there exists a unique time-discrete solution in the sense of Definition 4.4.6. 4.5.1 Solv ability Theorem 4.5.1 (Global existence of time-discrete solutions) . Supp ose that for e ach n ∈ N ther e exist c onstants c n > 0 and C n > 0 , such that for e ach α ∈ { 1 , 2 , . . . , ν } c n λ d (Ω 0 ) 6 λ d  ( χ α n ) − 1 (Ω 0 )  6 C n λ d (Ω 0 ) , for e ach Ω 0 ∈ B (Ω) . (4.5.1) Supp ose furthermor e that for e ach α the initial me asur e µ α 0 is given in its r efine d L eb esgue de c omp osition (cf. Cor ol lary 2.1.10) µ α 0 = µ α ac , 0 + µ α d , 0 + µ α sc , 0 , wher e µ α ac , 0  λ d , µ α d , 0 is discr ete w.r.t. λ d and µ α sc , 0 is singular c ontinuous w.r.t. λ d . Assume that µ α ac , 0 , µ α d , 0 and µ α sc , 0 ar e p ositive and finite me asur es. 50 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Then a time-discr ete solution as define d in Definition 4.4.6 exists and it is of the form µ α n = µ α ac ,n + µ α d ,n + µ α sc ,n , (4.5.2) wher e µ α ac ,n  λ d , µ α d ,n is discr ete w.r.t. λ d , µ α sc ,n is singular c ontinuous w.r.t. λ d , for al l n ∈ N , and e ach c omp onent is p ositive and finite. Pr o of. The pro of of Theorem 4.5.1 partly follo ws the lines of arguments of [17]. Let α ∈ { 1 , 2 , . . . , ν } b e fixed but arbitrary . The pro of go es b y induction. The statement in (4.5.2) is true for n = 0 due to the given initial condition. Moreov er, each of the comp onents µ α ac , 0 , µ α d , 0 and µ α sc , 0 is p ositiv e and finite. The induction h yp othesis is that for some (arbitrary , fixed) n ∈ { 0 , 1 , . . . , N T − 1 } the time- discrete solution exists, that it is of the form (4.5.2), and that each of the three measures µ α ac ,n , µ α d ,n and µ α sc ,n in the corresp onding refined Leb esgue decomp osition is p ositive and finite. W e no w prov e that if the induction hypothesis holds for this n , then it also holds for n + 1. 1. By the definition of the push forw ard op erator formulated in (4.4.8), for an y Ω 0 ∈ B (Ω) the follo wing holds: µ α n +1 (Ω 0 ) = µ α n  ( χ α n ) − 1 (Ω 0 )  = µ α ac ,n  ( χ α n ) − 1 (Ω 0 )  + µ α d ,n  ( χ α n ) − 1 (Ω 0 )  + µ α sc ,n  ( χ α n ) − 1 (Ω 0 )  . W e can th us write µ α n +1 = χ α n # µ α ac ,n + χ α n # µ α d ,n + χ α n # µ α sc ,n . 2. W e now define µ α ac ,n +1 := χ α n # µ α ac ,n . W e wish to show that µ α ac ,n +1 is absolutely contin uous w.r.t. λ d . Let Ω 0 ∈ B (Ω) b e such that λ d (Ω 0 ) = 0. By (4.5.1): λ d  ( χ α n ) − 1 (Ω 0 )  6 C n λ d (Ω 0 ) , and th us λ d (Ω 0 ) = 0 implies λ d  ( χ α n ) − 1 (Ω 0 )  = 0. It is part of the induction h yp othesis that µ α ac ,n  λ d , and thus it follows from λ d  ( χ α n ) − 1 (Ω 0 )  = 0 that µ α ac ,n  ( χ α n ) − 1 (Ω 0 )  = 0. Th us µ α ac ,n +1 (Ω 0 ) = µ α ac ,n  ( χ α n ) − 1 (Ω 0 )  = 0 , b y which w e hav e prov en that µ α ac ,n +1  λ d . Since µ α ac ,n is p ositiv e b y the induction h yp othesis, w e also ha v e that µ α ac ,n +1 (Ω 0 ) = µ α ac ,n  ( χ α n ) − 1 (Ω 0 )  > 0 , for all Ω 0 ∈ B (Ω) , and th us µ α ac ,n +1 is p ositiv e. Similarly , finiteness of µ α ac ,n +1 follo ws from finiteness of µ α ac ,n : µ α ac ,n +1 (Ω) = µ α ac ,n  ( χ α n ) − 1 (Ω)  = µ α ac ,n (Ω) < ∞ , where ( χ α n ) − 1 (Ω) = Ω due to the assumed in vertibilit y of the motion mapping. Mo delling Cro wd Dynamics 51 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 3. Similarly , define µ α d ,n +1 := χ α n # µ α d ,n . By the induction h yp othesis µ α d ,n is a positive, finite and discrete measure. Lemma 2.1.6 pro vides that we can th us write µ α d ,n = P i ∈J a i δ x i , for some coun table index set J ⊂ N , a set { x i } i ∈J ⊂ Ω and a set of corresponding nonnegative co efficients { a i } i ∈J ⊂ R , suc h that P i ∈J a i < ∞ . By definition of µ α d ,n +1 , w e hav e that for any Ω 0 ∈ B (Ω) µ α d ,n +1 (Ω 0 ) = µ α d ,n  ( χ α n ) − 1 (Ω 0 )  = X i ∈J a i δ x i  ( χ α n ) − 1 (Ω 0 )  = X i ∈J a i 1 x i ∈ (( χ α n ) − 1 (Ω 0 )) = X i ∈J a i 1 χ α n ( x i ) ∈ Ω 0 = X i ∈J a i δ χ α n ( x i ) (Ω 0 ) . This pro ves that µ α d ,n +1 is a discrete measure with resp ect to λ d . P ositivity of µ α d ,n +1 follo ws from p ositivity of the Dirac measure and the fact that a i > 0 for all i ∈ J . Since χ α n maps homeomorphically from Ω to Ω, obviously { x i } i ∈J ⊂ Ω implies { χ α n ( x i ) } i ∈J ⊂ Ω. As a result µ α d ,n +1 (Ω) = X i ∈J a i δ χ α n ( x i ) (Ω) = X i ∈J a i < ∞ , and th us µ α d ,n +1 is also finite. 4. Finally , we define µ α sc ,n +1 := χ α n # µ α sc ,n . W e w ant to pro ve that this measure is singular contin uous w.r.t. λ d . F or any x ∈ Ω the definition of the push forward implies that µ α sc ,n +1 ( x ) = µ α sc ,n  ( χ α n ) − 1 ( x )  . As µ α sc ,n is b y the induction hypothesis singular con tinuous, µ α sc ,n  ( χ α n ) − 1 ( x )  = 0 , for an y ( χ α n ) − 1 ( x ) ∈ Ω . Th us µ α sc ,n +1 ( x ) = 0 for all x ∈ Ω. Let the set B n ∈ B (Ω) b e such that µ α sc ,n (Ω \ B n ) = λ d ( B n ) = 0, which exists b y definition of singular con tinuous measures (see Definition 2.1.7). Because Ω \ B n =  χ α n  − 1  χ α n  Ω \ B n   the follo wing identit y is true: µ α sc ,n  Ω \ B n  = µ α sc ,n   χ α n  − 1  χ α n  Ω \ B n   = µ α sc ,n +1  χ α n  Ω \ B n   = µ α sc ,n +1  Ω \ χ α n  B n   . 52 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology In the last step w e used that χ α n (Ω) = Ω (due to inv ertibility of χ α n ). The b ottom line is that µ α sc ,n +1  Ω \ χ α n  B n   = µ α sc ,n  Ω \ B n  = 0 . The last equalit y follows from the w ay we hav e chosen B n . By h yp othesis of the theorem, see (4.5.1), w e ha ve that λ d  χ α n  B n   6 1 c n λ d   χ α n  − 1  χ α n  B n   = 1 c n λ d ( B n ) = 0 , (4.5.3) where it is imp ortant that c n > 0, and λ d ( B n ) = 0 b y definition of B n . Consequently , (4.5.3) pro ves that λ d  χ α n  B n   = 0. Define B n +1 := χ α n  B n  . F rom the fact that µ α sc ,n +1 ( x ) = 0 for all x ∈ Ω, and µ α sc ,n +1  Ω \ B n +1  = λ d  B n +1  = 0 , it follo ws that µ α sc ,n +1 is a singular con tinuous measure w.r.t. λ d . Since µ α sc ,n is p ositiv e b y the induction h yp othesis, w e obtain µ α sc ,n +1 (Ω 0 ) = µ α sc ,n  ( χ α n ) − 1 (Ω 0 )  > 0 , for all Ω 0 ∈ B (Ω). Thus, µ α sc ,n +1 is p ositive. Similarly , µ α sc ,n +1 is finite b ecause µ α sc ,n is finite: µ α sc ,n +1 (Ω) = µ α sc ,n  ( χ α n ) − 1 (Ω)  = µ α sc ,n (Ω) < ∞ . 5. W e ha v e no w pro ven that if a time-discrete solution of the form (4.5.2) exists for n , it also exists for n + 1. P ositivity and finiteness of µ α n +1 follo w from p ositivity and finiteness of its three comp onents, whic h we ha ve pro v en ab ov e. The induction argument guaran tees that (4.5.2) holds for all n ∈ N . Theorem 4.5.2 (Uniqueness of global time-discrete solutions) . Assume the hyp otheses of The or em 4.5.1. Then the glob al time-discr ete solution: µ α n = µ α ac ,n + µ α d ,n + µ α sc ,n , for al l n ∈ N is unique. Pr o of. Uniqueness of the time-discrete solution follows from the fact that µ α n +1 = χ α n # µ α n defines the push forward unambiguously . T o see this, assume that a unique µ α n exists, and there are tw o measures µ α 1 ,n +1 and µ α 2 ,n +1 suc h that µ α i,n +1 = χ α n # µ α n for eac h i ∈ { 1 , 2 } . Then w e hav e that for any Ω 0 ∈ B (Ω) the follo wing holds: µ α 1 ,n +1 (Ω 0 ) = µ α n  ( χ α n ) − 1 (Ω 0 )  = µ α 2 ,n +1 (Ω 0 ) , th us µ α 1 ,n +1 ≡ µ α 2 ,n +1 . It no w follows by induction that if the initial measure µ α 0 is unique, consequen tly µ α n is defined uniquely for all n ∈ N . Note that the decomp osition of µ α n in its three parts is unique by Corollary 2.1.10. Mo delling Cro wd Dynamics 53 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Remark 4.5.3 (Connections to Reference [17]) . The results of [17] are a sp ecial case of our setting; w e already said so in our commen ts at the end of Section 4.1. Let m 0 b e a discrete measure: m 0 := P N j =1 δ P j , and let M 0 b e an absolutely con tinuous measure (with density ρ ( t, x )). There are tw o wa ys to recov er their situation. One wa y is to set ν = 1 and consider µ 1 0 = θ m 0 + (1 − θ ) M 0 , where θ ∈ [0 , 1] is a tuning parameter. The second w ay to reco ver the results of [17] is by setting ν = 2, and then by taking µ 1 0 = θ m 0 , µ 2 0 = (1 − θ ) M 0 , henceforth only considering the total mass measure and the barycen tric v elo cit y . 4.5.2 Basic prop erties of the time-discrete solution In this section we will form ulate and pro ve a n um b er of properties of the time-discrete solution pro vided b y Theorems 4.5.1 and 4.5.2. These prop erties follo w mainly from the subsequen t steps done in the pro ofs of those theorems. Corollary 4.5.4 (Conserv ation of mass) . Assume the hyp otheses of The or em 4.5.1. F or e ach n ∈ N the initial mass is c onserve d by e ach of the thr e e c omp onents of the time-discr ete solu- tion pr ovide d by The or ems 4.5.1 and 4.5.2. That is, for al l n ∈ N and e ach α ∈ { 1 , 2 , . . . , ν } : (i) µ α ac ,n (Ω) = µ α ac , 0 (Ω) , (ii) µ α d ,n (Ω) = µ α d , 0 (Ω) , (iii) µ α sc ,n (Ω) = µ α sc , 0 (Ω) . As a r esult, also µ α n (Ω) = µ α 0 (Ω) . Pr o of. F or eac h n ∈ { 0 , 1 , . . . , N T − 1 } and α ∈ { 1 , 2 , . . . , ν } , consider the measure µ α ω ,n , where ω ∈ { ac, d, sc } . By definition of µ α ω ,n +1 (see the constructive pro of of Theorem 4.5.1, P arts 2, 3 and 4): µ α ω ,n +1 := χ α n # µ α ω ,n . F or eac h n we th us hav e µ α ω ,n +1 (Ω) = µ α ω ,n  ( χ α n ) − 1 (Ω)  , and ( χ α n ) − 1 (Ω) = Ω due to the inv ertibilit y of χ α n . This implies that µ α ω ,n +1 (Ω) = µ α ω ,n (Ω) for all n ∈ { 0 , 1 , . . . , N T } , and b y an inductive argumen t: µ α ω ,n (Ω) = µ α ω , 0 (Ω) for eac h n . Since µ α n = µ α ac ,n + µ α d ,n + µ α sc ,n for all n ∈ N , the abov e implies trivially that µ α n (Ω) = µ α 0 (Ω) , for all n ∈ N . Corollary 4.5.5. Assume the hyp otheses of The or em 4.5.1. If µ α ac , 0 ≡ 0 , µ α d , 0 ≡ 0 , or µ α sc , 0 ≡ 0 r esp e ctively, then for e ach n the c orr esp onding c omp onent of µ α n vanishes. That is, for e ach α , the fol lowing statements ar e true: (i) µ α ac , 0 ≡ 0 implies µ α ac ,n ≡ 0 for al l n ∈ N , 54 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology (ii) µ α d , 0 ≡ 0 implies µ α d ,n ≡ 0 for al l n ∈ N , (iii) µ α sc , 0 ≡ 0 implies µ α sc ,n ≡ 0 for al l n ∈ N . Pr o of. F or any α ∈ { 1 , 2 , . . . , ν } , let ω ∈ { ac, d, sc } b e arbitrary . Then the corresp onding part µ α ω ,n +1 in the refined Leb esgue decomp osition of µ α n +1 follo ws uniquely from µ α ω ,n b y the push forw ard µ α ω ,n +1 = χ α n # µ α ω ,n . See P arts 2, 3 and 4 of the pro of of Theorem 4.5.1. Assume that µ α ω ,n ≡ 0. Then for each Ω 0 ∈ B (Ω) µ α ω ,n +1 (Ω 0 ) = µ α ω ,n  ( χ α n ) − 1 (Ω 0 )  = 0 . W e th us conclude that µ α ω ,n +1 ≡ 0. It follo ws by an inductiv e argument that µ α ω ,n ≡ 0 for all n ∈ N , if µ α ω , 0 ≡ 0 is given. Remark 4.5.6. If µ α ac , 0 ≡ 0, then the assumption that there is a C n > 0 such that λ d  ( χ α n ) − 1 (Ω 0 )  6 C n λ d (Ω 0 ) can b e remov ed from the theore m. This is b ecause, even without this assumption, the push forw ard of µ α ac ,n ≡ 0 is an absolutely contin uous measure (of course, more sp ecifically µ α ac ,n +1 ≡ 0). Due to similar arguments, µ α sc , 0 ≡ 0 allo ws us to remo ve the assumption that there exists a c n > 0 for which c n λ d (Ω 0 ) 6 λ d  ( χ α n ) − 1 (Ω 0 )  . Remark 4.5.7. If w e s et µ α ac , 0 ≡ 0 and µ α sc , 0 ≡ 0, then Theorem 4.5.1 provides that µ α n is discrete w.r.t. λ d for all n ∈ N . W e ha ve shown in Lemma 2.1.6 that any discrete measure w.r.t. λ d can be written as a linear com bination of Dirac measures. Note ho wev er, that in µ α n b oth the centres of the Dirac masses and the co efficients might in principle dep end on n . (Since the index set J is necessarily c ountable , is suffices to allow only the p ositions and co efficien ts to b e n -dep endent.) Ho wev er, due to the specific structure of the push forward, w e hav e the following corollary of Theorem 4.5.1: Corollary 4.5.8. If the initial me asur e is a line ar c ombination of Dir ac me asur es: µ α 0 = P i ∈J a i δ x i, 0 for some c ountable index set J , then for e ach n the me asur e is of the form µ α n = P i ∈J a i δ x i,n , with the same (c onstant) c o efficients a i . Pr o of. W e extract from Part 3 of the pro of of Theorem 4.5.1 that for any n ∈ { 0 , 1 , . . . , N T − 1 } : µ α n = P i ∈J a i δ x i implies µ α n +1 (Ω 0 ) = P i ∈J a i δ χ α n ( x i ) (Ω 0 ). Assume that µ α 0 = P i ∈J a i δ x i, 0 has b een giv en. By inductive reasoning we can now deduce that µ α n = P i ∈J a i δ x i,n for all n ∈ N , where x i,n :=  χ α n − 1 ◦ χ α n − 2 ◦ . . . ◦ χ α 0  ( x i, 0 ) . Lemma 4.5.9. If µ α n = P i ∈J a i δ x i,n for e ach n ∈ N then, for any i ∈ J , the p osition x i,n satisfies a discr etize d version of dx i ( t ) /dt = v α ( t, x i ( t )) , evaluate d at t = t n for al l n ∈ { 0 , 1 , . . . , N T − 1 } . Mo delling Cro wd Dynamics 55 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Pr o of. F or any i ∈ J , a T aylor series expansion of x i around t = t n yields (ev aluated in t = t n +1 ): x i ( t n +1 ) = x i ( t n ) + ∆ t n d dt x i ( t n ) + O (∆ t 2 n ) , b y which w e find the expression d dt x i ( t n ) = x i ( t n +1 ) − x i ( t n ) ∆ t n + O (∆ t 2 n ) . (4.5.4) The discretized version of dx i ( t ) /dt = v α ( t, x i ( t )) is now obtained by substitution of (4.5.4), and after neglecting the O (∆ t 2 n )-part. W rite x i,n as the discrete-in-time equiv alen t of x i ( t n ), and furthermore use v α n ( x i,n ) = v α ( t n , x i,n ) lik e b efore. The discretized v ersion b ecomes x i,n +1 − x i,n ∆ t n = v α n ( x i,n ) , or x i,n +1 = x i,n + ∆ t n v α n ( x i,n ) . (4.5.5) W e hav e seen in the pro ofs of Theorem 4.5.1 and Corollary 4.5.8 that x i,n +1 and x i,n are related via x i,n +1 = χ α n ( x i,n ) , and b y definition of the motion mapping (see Definition 4.4.1) this is exactly (4.5.5). Remark 4.5.10. In Remark 3.4.3 we hav e already an ticipated the statements of Corollary 4.5.8 and Lemma 4.5.9. Lemma 4.5.11. Assume that, for e ach n ∈ N and α ∈ { 1 , 2 , . . . , ν } , the velo city field v α n is Lipschitz c ontinuous, with Lipschitz c onstant strictly smal ler than 1 / ∆ t n . That is, ther e is a c onstant 0 6 K n < 1 / ∆ t n , such that | v α n ( x ) − v α n ( y ) | 6 K n | x − y | , for al l x, y ∈ Ω . F or e ach n ∈ N , the me asur e µ α d ,n is discr ete, and thus it is the line ar c ombination of Dir ac masses (L emma 2.1.6). L et { x i,n } i ∈J b e the c orr esp onding set of c entr es (at time-slic e n ). Now, if { x i, 0 } i ∈J c onsists of distinct elements, then also { x i,n } i ∈J c onsists of distinct ele- ments, for e ach n ∈ N . Pr o of. The pro of go es b y con tradiction. Assume that n ∈ { 1 , 2 , . . . , N T } is suc h that not all elements of { x i,n } i ∈J are distinct, and, more sp ecifically , let j, k ∈ J satisfy j 6 = k and x j,n = x k,n . W e kno w that x i,n =  χ α n − 1 ◦ χ α n − 2 ◦ . . . ◦ χ α 0  ( x i, 0 ) for all i ∈ J , as this was sho wn in the pro of of Corollary 4.5.8. By assumption of the lemma x j, 0 6 = x k, 0 . F or x j,n = x k,n to hold, there must therefore b e an ˜ n ∈ { 0 , 1 , . . . , n − 1 } such that x j, ˜ n 6 = x k, ˜ n , but χ α ˜ n ( x j, ˜ n ) = χ α ˜ n ( x k, ˜ n ) . By definition of the push forw ard, the last equiv alence can be written as x j, ˜ n + ∆ t ˜ n v α ˜ n ( x j, ˜ n ) = x k, ˜ n + ∆ t ˜ n v α ˜ n ( x k, ˜ n ) , 56 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology or ∆ t ˜ n  v α ˜ n ( x j, ˜ n ) − v α ˜ n ( x k, ˜ n )  = x k, ˜ n − x j, ˜ n . (4.5.6) By assumption of the lemma, v α ˜ n is Lipschitz con tinuous, and there is a K ˜ n < 1 / ∆ t ˜ n suc h that | v α ˜ n ( x ) − v α ˜ n ( y ) | 6 K ˜ n | x − y | . In particular, this implies that | v α ˜ n ( x j, ˜ n ) − v α ˜ n ( x k, ˜ n ) | 6 K ˜ n | x j, ˜ n − x k, ˜ n | < 1 ∆ t ˜ n | x j, ˜ n − x k, ˜ n | . Ho wev er, as a result of (4.5.6) we already hav e that | v α ˜ n ( x j, ˜ n ) − v α ˜ n ( x k, ˜ n ) | = 1 ∆ t ˜ n | x j, ˜ n − x k, ˜ n | , and th us we ha ve a contradiction. This finishes the pro of. 4.5.3 Relaxing the conditions on the motion mapping and velocity fields Up to no w w e ha ve made a num b er of assumptions with resp ect to the one-step motion map- ping χ α n , and the (discretized) velocity field v α n . W e first give a short recapitulation of these assumptions here. Afterw ards we indicate which of these assumptions can b e relaxed, and in what w ay this can b e done. The results of this section ha ve a preliminary c haracter: w e exp ect that more is to b e disco v ered as so on as more analysis effort is inv ested in this direction. The follo wing restrictions on the motion mapping and v elo cit y field were imposed so far: 1. In Section 4.4, we assumed v α n to b e µ α n -uniformly b ounded. W e needed this to b e able to write the T aylor expansion in (4.4.3). This assumption means that, for eac h n ∈ N and α ∈ { 1 , 2 , . . . , ν } , there exists a constant M n for whic h | v α n ( x ) | < M n , for µ α n -almost ev ery x ∈ Ω . 2. By Assumption 4.4.2, we p ostulated that χ α n : Ω → Ω is a homeomorphism, for all n ∈ { 0 , 1 , . . . , N T − 1 } and α ∈ { 1 , 2 , . . . , ν } . 3. In case µ α ac , 0 6≡ 0 and µ α sc , 0 6≡ 0, w e need for all n ∈ { 0 , 1 , . . . , N T − 1 } strictly p ositive constan ts c n and C n , suc h that c n λ d (Ω 0 ) 6 λ d  ( χ α n ) − 1 (Ω 0 )  6 C n λ d (Ω 0 ) , for eac h Ω 0 ∈ B (Ω) , to prov e existence and uniqueness of the time-discrete solution (see Theorems 4.5.1 and 4.5.2). 4. The Lipsc hitz con tin uity of the v elo city field v α n is demanded in the h yp othesis of Lemma 4.5.11. More sp ecifically , w e assume that for each n ∈ { 0 , 1 , . . . , N T − 1 } a constan t K n < 1 / ∆ t n exists, suc h that | v α n ( x ) − v α n ( y ) | 6 K n | x − y | , for all x, y ∈ Ω . This restriction is only needed if µ α d , 0 6≡ 0, to make sure that no t w o centres of Dirac masses can ‘merge’. Mo delling Cro wd Dynamics 57 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Before w e can relax these conditions, w e introduce the following concept: Definition 4.5.12 (Supp ort of a measure) . The supp ort of a (p ositive, finite) me asur e µ on B (Ω) is define d as the set supp µ := { x ∈ Ω | µ (Ω 0 ) > 0 for al l Ω 0 ∈ B (Ω) op en such that x ∈ Ω 0 } . One can sho w (see [43], pp. 27–28) that the supp ort is a closed set of full measure, that is µ (supp µ ) = µ (Ω) . As a result µ (Ω \ supp µ ) = µ (Ω) − µ (Ω ∩ supp µ ) = µ (Ω) − µ (supp µ ) = 0 . (4.5.7) F or an y measurable set Ω 0 ⊂ Ω \ supp µ it follows that µ (Ω 0 ) 6 µ (Ω \ supp µ ) = 0 . (4.5.8) W e ha ve introduced the concept of a supp ort, b ecause it is in fact only imp ortant to consider the prop erties of the motion mapping χ α n on supp µ α n . As (4.5.8) sho ws, there is no mass con- tained in any region outside the supp ort. F or our purp oses, it is irrelev an t whether the image of such Ω 0 ⊂ Ω \ supp µ α n is again a subset of Ω or not. In order to ha ve mass conserv ation within Ω the only thing that matters is whether χ α n (Ω 0 ) ⊂ Ω for all Ω 0 ⊂ supp µ α n . W e do not c hange the restriction in Part 1 of the list ab ov e. Ho wev er, we remark that if w e demand that | v α n ( x ) | < M n holds for all x ∈ supp µ α n , then it also holds for µ α n -a.e. x . Indeed, if | v α n ( x ) | < M n for all x ∈ supp µ α n , then Ω M :=  x ∈ Ω   M n 6 | v α n ( x ) |  ⊂ Ω \ supp µ α n , th us, cf. (4.5.7), µ α n (Ω M ) 6 µ α n (Ω \ supp µ α n ) = 0 . In P art 2 of the list of assumptions, we pay sp ecial atten tion to the restriction of χ α n to supp µ α n , that is, ¯ χ α n := χ α n   supp µ α n . Moreo ver, we replace the assumption that the motion mapping is a homeomorphism b y: Assumption 4.5.13. F or al l n ∈ { 0 , 1 , . . . , N T − 1 } and e ach α ∈ { 1 , 2 , . . . , ν } we assume that the motion mapping χ α n : Ω → R d is such that the fol lowing pr op erties ar e satisfie d: (i) the r ange of ¯ χ α n := χ α n   supp µ α n is c ontaine d in Ω . That is: ¯ χ α n : supp µ α n → Ω ; (ii) for al l Ω 0 ∈ B (Ω) we have that ( χ α n ) − 1 (Ω 0 ) is me asur able; (iii) for al l Ω 0 ∈ B (Ω) we have that χ α n (Ω 0 ) is me asur able. Her e, ( χ α n ) − 1 should not b e understo o d as the inverse mapping. The set ( χ α n ) − 1 (Ω 0 ) is the pr e-image of Ω 0 . We do not assume invertibility of the motion mapping. 58 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Note that throughout this thesis, an ywhere we used the inv erse mapping ( χ α n ) − 1 , w e should no w read it in the pre-image sense. This is e.g. also the case in the definition of the push forw ard, Definition 4.4.5. Under the new set of restrictions on the motion mapping, we still wan t the Theorems and Lemmas of Section 4.5 to hold. W e wan t this esp ecially for Theorems 4.5.1 and 4.5.2 (global existence and uniqueness of the time-discrete solution). T o achiev e this, Part 3 of the list of assumptions needs reconsideration. W e mak e it sligh tly stronger: Assumption 4.5.14. Supp ose that for e ach n ∈ { 0 , 1 , . . . , N T − 1 } ther e exist c onstants c n > 0 and C n > 0 , such that for e ach α ∈ { 1 , 2 , . . . , ν } c n λ d  χ α n (Ω 0 )  6 λ d (Ω 0 ) , (4.5.9) λ d  ( χ α n ) − 1 (Ω 0 )  6 C n λ d (Ω 0 ) , (4.5.10) hold for al l Ω 0 ∈ B (Ω) . A gain, ( χ α n ) − 1 (Ω 0 ) is the pr e-image of Ω 0 . Note that (4.5.9) implies the left inequalit y of c n λ d (Ω 0 ) 6 λ d  ( χ α n ) − 1 (Ω 0 )  6 C n λ d (Ω 0 ) , b y substituting ( χ α n ) − 1 (Ω 0 ) for Ω 0 , and b y using χ α n  ( χ α n ) − 1 (Ω 0 )  = Ω 0 . W e still demand that the velocity field is a Lipschitz contin uous function if µ α d , 0 6≡ 0 (Re- striction 4). W e relax this restriction in this sense, that it is no longer required for all x, y ∈ Ω. It suffices to demand this prop ert y only for all x, y ∈ supp µ α d ,n . Note that supp µ α d ,n consists exactly of those p oints in whic h the Dirac measures of µ α d ,n are centered. In the pro of of Lemma 4.5.11 v α n is only ev aluated in these p oin ts. The pro of is therefore still v alid under the prop osed weak er assumption. The results of Section 4.5 are still v alid under the new assumptions prop osed in this Section. W e only need to mo dify some minor details in the pro ofs of Theorem 4.5.1 and Corollary 4.5.4. These mo difications are addressed in App endix E. Remark 4.5.15. In Lemma 4.5.11 we hav e shown that the centres of the Dirac measures cannot merge if the v elo cit y field is a Lipsc hitz con tin uous function. W e ac knowledge that it is equally imp ortant that t w o Dirac masses that b elong to distinct subp opulations cannot merge. Moreo ver, the in tegrand in the so cial v elo cit y terms, see (4.2.2), typically has a singularity in 0. This means that at timeslice n , the velocity field v α n ( x ) is unbounded if x is arbitrarily close to a Dirac mass. W e thus prop ose as an extra demand that for all α ∈ { 1 , 2 , . . . , ν } and for eac h n ∈ N supp µ α n ∩ supp µ β d ,n = ∅ , for all β ∈ { 1 , 2 , . . . , ν } such that β 6 = α, and supp µ α d ,n ∩ (supp µ α ac ,n ∩ supp µ α sc ,n ) = ∅ . If this condition is ob eyed, no Dirac measures from distinct p opulations can b e centred in the same p oint. Since the supp ort of a measure is a closed set, we furthermore preven t v α n ( x ) from b eing unbounded for all x ∈ supp µ α n . What happ ens outside supp µ α n is unimp ortan t. W e come bac k to this issue in Section 5. Mo delling Cro wd Dynamics 59 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 4.6 Reform ulation of the prop osed v elo cit y fields in the time- discrete setting In Sections 4.4 and 4.5 we hav e used the velocity fields v α n without sp ecifying their actual form. In this section w e wan t to do so. It is quite self-evident that w e just take the prop osed form of Section 4.2 and determine its time-discrete coun terpart. F or α ∈ { 1 , 2 , . . . , ν } , w e thus take v α n as the sup erp osition of the desired and the so cial v elo cit y like in (4.2.1): v α n ( x ) := v α des ( x ) + v α soc ,n ( x ) , for all x ∈ Ω . F or v α soc ,n w e hav e the following expression: v α soc ,n ( x ) := ν X β =1 Z Ω \{ x } f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dµ β n ( y ) , for all α ∈ { 1 , 2 , . . . , ν } . 4.7 En trop y inequalit y for the time-discrete problem In Section 4.4 we ha ve derived a time-discrete version of our problem. W e deriv ed an entrop y inequalit y for the con tinuous-in-time problem in Section 4.3. Its discrete-in-time counterpart is presen ted in this section. W e again work with absolutely contin uous mass measures only , and treat (for clarit y) the single-p opulation case first. 4.7.1 One p opulation In Section 4.4 we derived, omitting O (∆ t 2 n )-terms, (4.4.6): an equation relating the situa- tion at time t n to the one at time t n +1 . If the time-discrete mass measures are absolutely con tinuous, (4.4.6) reads Z Ω ψ ( x ) ρ n +1 ( x ) dλ d ( x ) = Z Ω ψ  χ n ( x )  ρ n ( x ) dλ d ( x ) , (4.7.1) where we can take an y ψ ∈ L 1 µ n +1 (Ω) (cf. Remark 4.4.4). W e deduce our discrete-in-time en tropy inequalit y here, taking (4.7.1) as a starting p oint. The subscript n in the sequel denotes that w e consider the time-discrete version of a function at time t n . Recall that χ n denotes the one-step push forw ard (see Definition 4.4.1), defined b y χ n ( x ) := x + ∆ t n v n ( x ) , Let S n b e the time-discrete equiv alen t of the entrop y S ( t n ). The follo wing theorem holds: Theorem 4.7.1 (Discrete-in-time entrop y inequalit y) . Assume that the time-discr ete velo city v n is of the form v n := ∇ V + ∇ W ? ρ n , (4.7.2) 60 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology wher e W is such that W ( ξ ) = W ( − ξ ) for al l ξ ∈ R d . Assume mor e over that the evolution of the system is governe d by (4.7.1), and that the entr opy is define d by S n := Z Ω  V ( x ) + 1 2 ( W ? ρ n )( x )  ρ n ( x ) dλ d ( x ) , for al l n ∈ N . (4.7.3) Then for e ach n ∈ { 0 , 1 , . . . , N T − 1 } the fol lowing ine quality holds, up to O (∆ t 2 n ) -terms: S n +1 > S n . (4.7.4) Pr o of. Fix n ∈ { 0 , 1 , . . . , N T − 1 } and take ψ := V + 1 2 ( W ? ρ n +1 ) . By (4.7.1), w e derive S n +1 = Z Ω  V ( x ) + 1 2 ( W ? ρ n +1 )( x )  ρ n +1 ( x ) dλ d ( x ) = Z Ω  V ( χ n ( x )) + 1 2 ( W ? ρ n +1 )( χ n ( x ))  ρ n ( x ) dλ d ( x ) . (4.7.5) Note that, cf. (4.4.3) – (4.4.4): V ( χ n ( x )) = V ( x ) + ∆ t n v n ( x ) · ∇ V ( x ) + O (∆ t 2 n ) , (4.7.6) ( W ? ρ n +1 )( χ n ( x )) =( W ? ρ n +1 )( x ) + ∆ t n v n ( x ) · ∇ ( W ? ρ n +1 )( x ) + O (∆ t 2 n ) =( W ? ρ n +1 )( x ) + ∆ t n v n ( x ) · ( ∇ W ? ρ n +1 )( x ) + O (∆ t 2 n ) , (4.7.7) ( W ? ρ n +1 )( x ) = Z Ω W ( x − y ) ρ n +1 ( y ) dλ d ( y ) = Z Ω W ( x − χ n ( y )) ρ n ( y ) dλ d ( y ) = Z Ω W ( x − y − ∆ t n v n ( y )) ρ n ( y ) dλ d ( y ) = Z Ω W ( x − y ) ρ n ( y ) dλ d ( y ) − ∆ t n Z Ω v n ( y ) · ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) + O (∆ t 2 n ) =( W ? ρ n )( x ) − ∆ t n Z Ω v n ( y ) · ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) + O (∆ t 2 n ) . (4.7.8) In the second equality of (4.7.8) w e hav e used (4.7.1) with ψ ( y ) = W ( x − y ) (in this scope, x is regarded as a parameter). Note that (4.7.8) can also b e written as ( W ? ρ n +1 )( x ) = ( W ? ρ n )( x ) + O (∆ t n ) , (4.7.9) Mo delling Cro wd Dynamics 61 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology whic h we use no w. Combination of (4.7.7) – (4.7.9) yields ( W ? ρ n +1 )( χ n ( x )) =( W ? ρ n +1 )( x ) + ∆ t n v n ( x ) · ∇ ( W ? ρ n +1 )( x ) + O (∆ t 2 n ) =( W ? ρ n )( x ) − ∆ t n Z Ω v n ( y ) · ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) + ∆ t n v n ( x ) · ∇  ( W ? ρ n )( x ) + O (∆ t n )  + O (∆ t 2 n ) =( W ? ρ n )( x ) − ∆ t n Z Ω v n ( y ) · ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) + ∆ t n v n ( x ) · Z Ω ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) + O (∆ t 2 n ) . (4.7.10) Substituting (4.7.6) and (4.7.10) in to (4.7.5), w e find S n +1 = Z Ω  V ( χ n ( x )) + 1 2 ( W ? ρ n +1 )( χ n ( x ))  ρ n ( x ) dλ d ( x ) = Z Ω  V ( x ) + ∆ t n v n ( x ) · ∇ V ( x )  ρ n ( x ) dλ d ( x ) + 1 2 Z Ω ( W ? ρ n )( x ) ρ n ( x ) dλ d ( x ) − 1 2 ∆ t n Z Ω Z Ω v n ( y ) · ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) ρ n ( x ) dλ d ( x ) + 1 2 ∆ t n Z Ω v n ( x ) · Z Ω ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) ρ n ( x ) dλ d ( x ) + O (∆ t 2 n ) . (4.7.11) Since W ( ξ ) = W ( − ξ ) for all ξ ∈ R d , we ha ve that ∇ W ( ξ ) = −∇ W ( − ξ ). Using some elemen tary calculus, we deriv e − 1 2 ∆ t n Z Ω Z Ω v n ( y ) · ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) ρ n ( x ) dλ d ( x ) = − 1 2 ∆ t n Z Ω v n ( y ) · Z Ω ∇ W ( x − y ) ρ n ( x ) dλ d ( x ) ρ n ( y ) dλ d ( y ) = 1 2 ∆ t n Z Ω v n ( y ) · Z Ω ∇ W ( y − x ) ρ n ( x ) dλ d ( x ) ρ n ( y ) dλ d ( y ) = 1 2 ∆ t n Z Ω v n ( x ) · Z Ω ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) ρ n ( x ) dλ d ( x ) . (4.7.12) In the last step we changed notation, replacing x b y y and vic e versa . In this final expression w e recognize the last term of (4.7.11), whic h can in short b e written as 1 2 ∆ t n Z Ω v n ( x ) · ( ∇ W ? ρ n )( x ) ρ n ( x ) dλ d ( x ) . 62 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology The expression for S n +1 in (4.7.11) no w transforms into S n +1 = Z Ω  V ( x ) + ∆ t n v n ( x ) · ∇ V ( x )  ρ n ( x ) dλ d ( x ) + 1 2 Z Ω ( W ? ρ n )( x ) ρ n ( x ) dλ d ( x ) + ∆ t n Z Ω v n ( x ) · Z Ω ∇ W ( x − y ) ρ n ( y ) dλ d ( y ) ρ n ( x ) dλ d ( x ) + O (∆ t 2 n ) = Z Ω  V ( x ) + ∆ t n v n ( x ) · ∇ V ( x )  ρ n ( x ) dλ d ( x ) + 1 2 Z Ω ( W ? ρ n )( x ) ρ n ( x ) dλ d ( x ) + ∆ t n Z Ω v n ( x ) · ( ∇ W ? ρ n )( x ) ρ n ( x ) dλ d ( x ) + O (∆ t 2 n ) = Z Ω  V ( x ) + 1 2 ( W ? ρ n )( x )  ρ n ( x ) dλ d ( x ) + ∆ t n Z Ω v n ( x ) ·  ∇ V ( x ) + ( ∇ W ? ρ n )( x )  ρ n ( x ) dλ d ( x ) + O (∆ t 2 n ) . (4.7.13) No w we recognize the definitions of the en tropy S n , (4.7.3), and the v elo city v n , (4.7.2), in the last equalit y of (4.7.13). W e thus pro ceed: S n +1 = S n + ∆ t n Z Ω | v n ( x ) | 2 ρ n ( x ) dλ d ( x ) + O (∆ t 2 n ) > S n + O (∆ t 2 n ) . (4.7.14) Neglecting O (∆ t 2 n )-terms, w e thus ha ve that S n +1 > S n , for all n ∈ { 0 , 1 , . . . , N T − 1 } . Remark 4.7.2. Note that (4.7.4) is the discrete-in-time coun terpart of (4.3.13), up to O (∆ t 2 n )-appro ximation. Remark 4.7.3. Unlike the contin uous-in-time case, no b oundary terms app ear in this time- discrete con text. In Section 4.3.1, indeed these terms did app ear, and w e either assumed them to v anish (isolated system), or we incorp orated them in the en tropy inequality . How ever, in the pro of of Theorem 4.7.1 no b oundary terms are encoun tered. This is because, originally , the test functions ψ were taken such that they v anish on ∂ Ω. This implicitly assumes that the time-discrete mo del is more suitable for describing an isolated system. Mo delling Cro wd Dynamics 63 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 4.7.2 Multi-comp onen t crowd F or a crowd that consists of multiple subp opulations, for eac h constituent of the mixture a go verning equation lik e (4.7.1) holds: Z Ω ψ α ( x ) ρ α n +1 ( x ) dλ d ( x ) = Z Ω ψ α  χ α n ( x )  ρ α n ( x ) dλ d ( x ) , (4.7.15) for an y ψ α ∈ L 1 µ α n +1 (Ω). The one-step push forw ard χ α n , is defined b y χ α n ( x ) := x + ∆ t n v α n ( x ) , where w e take the time-discrete v elo cities v α n of the form v α n := ∇ V α + ν X β =1 ∇ W α β ? ρ β n , for all α ∈ { 1 , 2 , . . . , ν } . (4.7.16) W e assume that for all α, β ∈ { 1 , 2 , . . . , ν } : W α β ( ξ ) = W α β ( − ξ ) holds for all ξ ∈ R d , and W α β ≡ W β α . V ery m uch as in Section 4.7.1 w e define the time-discrete en trop y for the m ulti-comp onen t cro wd to b e S n := ν X α =1 Z Ω h V α ( x ) + 1 2 ν X β =1 ( W α β ? ρ β n )( x ) i ρ α n ( x ) dλ d ( x ) , for all n ∈ N . (4.7.17) Theorem 4.7.4 (Discrete-in-time entrop y inequality) . Assume that v α n satisfies (4.7.16) and the ac c omp anying c onditions on W α β . Assume furthermor e that the evolution of the system is governe d by (4.7.15), and that the entr opy is define d by (4.7.17). Then for e ach n ∈ { 0 , 1 , . . . , N T − 1 } the fol lowing ine quality holds, up to O (∆ t 2 n ) -terms: S n +1 > S n . (4.7.18) Pr o of. Fix n ∈ { 0 , 1 , . . . , N T − 1 } and take ψ α := V + 1 2 P ν β =1 ( W α β ? ρ β n ) for eac h α ∈ { 1 , 2 , . . . , ν } . Using (4.7.15), we derive that S n +1 = ν X α =1 Z Ω h V α ( x ) + 1 2 ν X β =1 ( W α β ? ρ β n +1 )( x ) i ρ α n +1 ( x ) dλ d ( x ) = ν X α =1 Z Ω h V α ( χ α n ( x )) + 1 2 ν X β =1 ( W α β ? ρ β n +1 )( χ α n ( x )) i ρ α n ( x ) dλ d ( x ) . (4.7.19) F or fixed α, β ∈ { 1 , 2 , . . . , ν } similar expressions as in (4.7.6), (4.7.7), (4.7.8) hold. They can 64 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology b e deriv ed in an iden tical manner. Therefore, we list here only the results: V α ( χ α n ( x )) = V α ( x ) + ∆ t n v α n ( x ) · ∇ V α ( x ) + O (∆ t 2 n ) , (4.7.20) ( W α β ? ρ β n +1 )( χ α n ( x )) =( W α β ? ρ β n +1 )( x ) + ∆ t n v α n ( x ) · ∇ ( W α β ? ρ β n +1 )( x ) + O (∆ t 2 n ) =( W α β ? ρ β n +1 )( x ) + ∆ t n v α n ( x ) · ( ∇ W α β ? ρ β n +1 )( x ) + O (∆ t 2 n ) , (4.7.21) ( W α β ? ρ β n +1 )( x ) =( W α β ? ρ β n )( x ) − ∆ t n Z Ω v β n ( y ) · ∇ W α β ( x − y ) ρ β n ( y ) dλ d ( y ) + O (∆ t 2 n ) . (4.7.22) W e combine (4.7.21) and (4.7.22), and (omitting the details) obtain the analogon of (4.7.10): ( W α β ? ρ β n +1 )( χ α n ( x )) =( W α β ? ρ β n )( x ) − ∆ t n Z Ω v β n ( y ) · ∇ W α β ( x − y ) ρ β n ( y ) dλ d ( y ) + ∆ t n v α n ( x ) · Z Ω ∇ W α β ( x − y ) ρ β n ( y ) dλ d ( y ) + O (∆ t 2 n ) . (4.7.23) By substitution of (4.7.20) and (4.7.23) in to (4.7.19) w e get S n +1 = ν X α =1 Z Ω h V α ( χ α n ( x )) + 1 2 ν X β =1 ( W α β ? ρ β n +1 )( χ α n ( x )) i ρ α n ( x ) dλ d ( x ) = ν X α =1 Z Ω  V α ( x ) + ∆ t n v α n ( x ) · ∇ V α ( x )  ρ α n ( x ) dλ d ( x ) + 1 2 ν X α =1 Z Ω ν X β =1 ( W α β ? ρ β n )( x ) ρ α n ( x ) dλ d ( x ) − 1 2 ∆ t n ν X α =1 Z Ω ν X β =1 Z Ω v β n ( y ) · ∇ W α β ( x − y ) ρ β n ( y ) dλ d ( y ) ρ α n ( x ) dλ d ( x ) + 1 2 ∆ t n ν X α =1 Z Ω ν X β =1 v α n ( x ) · Z Ω ∇ W α β ( x − y ) ρ β n ( y ) dλ d ( y ) ρ α n ( x ) dλ d ( x ) + O (∆ t 2 n ) . (4.7.24) W e now use that W α β ( ξ ) = W α β ( − ξ ) for all ξ ∈ R d . This implies that ∇ W α β ( ξ ) = −∇ W α β ( − ξ ). Mo delling Cro wd Dynamics 65 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology W e obtain − 1 2 ∆ t n ν X α =1 Z Ω ν X β =1 Z Ω v β n ( y ) · ∇ W α β ( x − y ) ρ β n ( y ) dλ d ( y ) ρ α n ( x ) dλ d ( x ) = − 1 2 ∆ t n ν X α =1 ν X β =1 Z Ω v β n ( y ) · Z Ω ∇ W α β ( x − y ) ρ α n ( x ) dλ d ( x ) ρ β n ( y ) dλ d ( y ) = 1 2 ∆ t n ν X α =1 ν X β =1 Z Ω v β n ( y ) · Z Ω ∇ W α β ( y − x ) ρ α n ( x ) dλ d ( x ) ρ β n ( y ) dλ d ( y ) = 1 2 ∆ t n ν X β =1 ν X α =1 Z Ω v α n ( x ) · Z Ω ∇ W β α ( x − y ) ρ β n ( y ) dλ d ( y ) ρ α n ( x ) dλ d ( x ) = 1 2 ∆ t n ν X α =1 Z Ω ν X β =1 v α n ( x ) · ( ∇ W β α ? ρ β n )( x ) ρ α n ( x ) dλ d ( x ) . (4.7.25) In the third step w e c hanged notation: w e replaced x by y , α b y β , and vic e versa . The expression for S n +1 in (4.7.24) transforms in to S n +1 = ν X α =1 Z Ω  V α ( x ) + ∆ t n v α n ( x ) · ∇ V α ( x )  ρ α n ( x ) dλ d ( x ) + 1 2 ν X α =1 Z Ω ν X β =1 ( W α β ? ρ β n )( x ) ρ α n ( x ) dλ d ( x ) + ∆ t n ν X α =1 Z Ω ν X β =1 v α n ( x ) ·  1 2 ( ∇ W α β + ∇ W β α ) ? ρ β n  ( x ) ρ α n ( x ) dλ d ( x ) + O (∆ t 2 n ) = ν X α =1 Z Ω h V α ( x ) + 1 2 ν X β =1 ( W α β ? ρ β n )( x ) i ρ α n ( x ) dλ d ( x ) + ∆ t n ν X α =1 Z Ω v α n ( x ) · h ∇ V α ( x ) + 1 2 ν X β =1  ( ∇ W α β + ∇ W β α ) ? ρ β n  ( x ) i ρ α n ( x ) dλ d ( x ) + O (∆ t 2 n ) = S n + ∆ t n ν X α =1 Z Ω | v α n ( x ) | 2 ρ α n ( x ) dλ d ( x ) + O (∆ t 2 n ) , (4.7.26) where the last step is only v alid under the assumption that the in teractions W α β are symmetric (this is a h yp othesis of the theorem). If w e omit the O (∆ t 2 n )-terms, w e deduce from (4.7.26) that S n +1 > S n , for all n ∈ { 0 , 1 , . . . , N T − 1 } , whic h finishes the pro of. Remark 4.7.5. Note that (4.7.18) is the discrete-in-time counterpart of (4.3.26), again up to O (∆ t 2 n )-appro ximation. 66 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 4.7.3 Generalization to discrete measures F ollo wing the lines of Section 4.3.3, we generalize the entrop y inequalit y for the time-discrete mo del (cf. Theorem 4.7.1) to the situation in which the discrete-in-time mass measure is of a discrete t yp e. W e consider a single p opulation with discrete-in-time mass measure µ n := X i ∈J δ x i,n . The push forw ard of the cen tres x i,n is giv en by (4.4.7) suc h that x i,n +1 := χ n ( x i,n ) = x i,n + ∆ t n v n ( x i,n ) , for all i ∈ J . (4.7.27) Here, the v elo cit y field is the time-discrete v ersion of (4.3.28): v n := ∇ V + ∇ W ? µ n , where the con volution is giv en by ( ∇ W ? µ n )( x ) := Z Ω \{ x } ∇ W ( x − y ) dµ n ( y ) , for all x ∈ Ω . W e assume again that W is symmetric: W ( ξ ) = W ( − ξ ) , for all ξ ∈ R d . The corresp onding entrop y of the system in Ω is S n = Z Ω n V ( x ) + 1 2 ( W ? µ n )( x ) o dµ n ( x ) , for all n ∈ N . W e again explicitly restrict ourselves to the situation that x i,n ∈ Ω for all i ∈ J and all n ∈ N ; cf. Section 4.3.3. Our aim is to deriv e the entrop y inequality of Section 4.7.1 for the proposed microscopic measure. That is, up to O (∆ t 2 n )-terms, w e hav e that S n +1 > S n , for all n ∈ { 0 , 1 , . . . , N T − 1 } . Using the definition of µ n (in terms of the sum of Dirac measures), the relation b etw een x i,n +1 and x i,n in (4.7.27), w e derive that S n +1 = X i ∈J n V ( x i,n +1 ) + 1 2 X j ∈J x j,n +1 6 = x i,n +1 W ( x i,n +1 − x j,n +1 ) o = X i ∈J n V  x i,n + ∆ t n v n ( x i,n )  + 1 2 X j ∈J x j,n +1 6 = x i,n +1 W  x i,n + ∆ t n v n ( x i,n ) − x j,n − ∆ t n v n ( x j,n )  o . (4.7.28) Mo delling Cro wd Dynamics 67 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Remark 4.7.6. Note that the statement x j,n +1 6 = x i,n +1 is equiv alen t to x j,n 6 = x i,n , if v n is a Lipschitz contin uous function with the Lipsc hitz constan t strictly smaller than 1 / ∆ t n . The implication  x j,n +1 6 = x i,n +1  = ⇒  x j,n 6 = x i,n  follo ws trivially from the unambiguit y of the push forward, whereas the implication  x j,n 6 = x i,n  = ⇒  x j,n +1 6 = x i,n +1  is a consequence of similar argumen ts as in the pro of of Lemma 4.5.11. W e can th us pro ceed on (4.7.28): S n +1 = X i ∈J n V  x i,n + ∆ t n v n ( x i,n )  + 1 2 X j ∈J x j,n 6 = x i,n W  x i,n + ∆ t n v n ( x i,n ) − x j,n − ∆ t n v n ( x j,n )  o = X i ∈J n V ( x i,n ) + ∆ t n v n ( x i,n ) · ∇ V ( x i,n ) + O (∆ t 2 n ) o + 1 2 X i,j ∈J x j,n 6 = x i,n n W ( x i,n − x j,n ) + ∆ t n  v n ( x i,n ) − v n ( x j,n )  · ∇ W ( x i,n − x j,n ) + O (∆ t 2 n ) o . (4.7.29) Here w e hav e used T aylor series expansions around x i,n (in V ) and around x i,n − x j,n (in W ), resp ectiv ely . Note also that ∇ W ( x i,n − x j,n ) = −∇ W ( x j,n − x i,n ). W e can th us write: S n +1 = X i ∈J n V ( x i,n ) + 1 2 X j ∈J x j,n 6 = x i,n W ( x i,n − x j,n ) o +∆ t n X i ∈J v n ( x i,n ) · ∇ V ( x i,n ) + 1 2 ∆ t n X i,j ∈J x j,n 6 = x i,n v n ( x i,n ) · ∇ W ( x i,n − x j,n ) + 1 2 ∆ t n X i,j ∈J x j,n 6 = x i,n v n ( x j,n ) · ∇ W ( x j,n − x i,n ) + O (∆ t 2 n ) = S n + ∆ t n X i ∈J v n ( x i,n ) · ∇ V ( x i,n ) +∆ t n X i,j ∈J x j,n 6 = x i,n v n ( x i,n ) · ∇ W ( x i,n − x j,n ) + O (∆ t 2 n ) = S n + ∆ t n X i ∈J v n ( x i,n ) · n ∇ V ( x i,n ) + X j ∈J x j,n 6 = x i,n ∇ W ( x i,n − x j,n ) o + O (∆ t 2 n ) (4.7.30) 68 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Omitting O (∆ t 2 n )-terms, (4.7.30) w e arrive at S n +1 = S n + ∆ t n X i ∈J v n ( x i,n ) · n ∇ V ( x i,n ) + ( ∇ W ? µ n )( x i,n ) o = S n + ∆ t n Z Ω | v n ( x ) | 2 dµ n ( x ) > S n . (4.7.31) The inequality in (4.7.31) is an analogon of the statement of Theorem 4.7.1 for a discrete mass measure. W e do not giv e details on the m ulti-comp onent case. 4.8 Tw o-scale phenomena A t this stage, we wan t to p oint out which sp ecific types of measures we hav e in mind to use in our mo del. Throughout the previous sections w e ha ve already giv en clues ab out the c hoices we intend to make, for example in Sections 1.3, 2.4, 3.4 and 4.1. As anticipated, we are mainly in terested in tw o sorts of measures. The first option is a discrete measure (or: microscopic mass measure), as introduced already in Section 2.4.1. In general, a discrete measure may b e the (weigh ted) sum of a countable yet infinite num b er of Dirac measures (cf. Lemma 2.1.6). W e restrict ourselves here to a finite - but arbitrary - num b er of p oint masses (read: p eople). This approac h enables us to trace the exact motion of a particular individual. Secondly , if we work with an absolutely contin uous mass measure (w.r.t. the Leb esgue mea- sure), then we are pro vided with a density by the Radon-Nikodym Theorem. This option (a macroscopic mass measure) has already b een introduced in Section 2.4.2. In fact, this p ersp ective corresp onds to considering the cro wd as a fluid (or a large ‘cloud’). W e decide to do so if the n umber of p eople is large, and if we, moreo ver, are not in terested in what happ ens exactly at the individual’s lev el. Suc h lo cal fluctuations hav e b een av eraged out. W e only allow these t wo options, which means that from no w on w e explicitly exclude singular con tinuous measures from our area of in terest. This is b ecause it is not self-eviden t what in- terpretation we should give to suc h measure. This decision has b een an ticipated in Section 3.4. In Section 1, w e hav e already indicated that w e are esp ecially interested in the in terplay b et w een microscopic and macroscopic mass measures. Piccoli et al. [17] do so by including a discrete and an absolutely con tin uous part in one measure. F urthermore a tuning parame- ter (taking v alues from 0 to 1) is used to enable a transition from fully microscopic to fully macroscopic. Sp eaking in terms of mixture theory (cf. Section 3), this means that the dis- crete and absolutely contin uous part together describ e a s ingle constituent. They can not act indep enden tly . W e already remarked in Section 4.1 that we consider this to b e not very useful. In the numerical scheme that is describ ed in Section 5, we therefore do not include this sup erp osition of a micro and a macro part in one measure. If desired, it can how ever b e incorp orated without to o muc h difficult y . Mo delling Cro wd Dynamics 69 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Our inten tion is a bit differen t: W e namely wan t to distinguish b etw een subp opulations and assign to them a measure that corresp onds to their ‘c haracter’. More sp ecifically , w e w ant to use a discrete measure only if this subp opulation consists of a limited num b er of individuals whic h are of s p ecial in terest. This is wh y we only consider discrete measures that are consti- tuted of a finite n umber of Dirac measures. Large crowds are in our framework automatically represen ted b y an absolutely con tinuous mass measure. Suc h subpopulations consist of p eople that are not sp ecial (otherwise they should hav e b een included in an indep endent discrete measure), and therefore it suffices to b e able to observ e only global b ehaviour. It is of no use to ‘do effort’ to capture information from the microscopic lev el in a discrete measure, if we are not interested in that information an y wa y . The use of b oth microscopic and macroscopic mass measures in one unified framew ork, makes that w e w ork in a two-sc ale setting. Our p ersp ective is tw ofold: on one hand w e observ e from a macroscopic p oin t of view, while on the other hand we can detect b ehaviour at an individual’s lev el if we hav e identified this individual as ‘sp ecial’. Remark 4.8.1. In the future, w e p ossibly w ant to come bac k to mo delling the cloud as a particle system, consisting of a large n umber of individuals. W e then aim at comparing the particle system’s results to the results of this thesis. Note that these tw o approaches are incorp orated in the measure-theoretical framework we are dealing with. By introducing separate mass measures, we allow the subp opulations to evolv e differ- en tly . F or example, w e assign to each subp opulation its o wn desired v elo cit y field. That is, for eac h subp opulation there is a separate goal that it w ants to ac hiev e. Moreo v er, it is p ossible to ha v e asymmetric in teractions. W e hav e already shortly referred to what we called ”predator-prey relations” in Remark 4.3.6. In suc h asymmetric situation preys are only re- p elled from predators, but predators are attracted to preys. Nature turns out to pro vide v ery clear illustrations of systems that are of a t wo-scale c haracter and at the same time display asymmetric interactions of the ”predator-prey” type. W e p oin t out t wo examples. F ascinating interaction tak es place b etw een flo cks of small birds (starlings, sa y) and one or more larger predator birds (e.g. ha wks), as is illustrated by Figure 4.4. The h uge num b er of starlings makes it nearly imp ossible to distinguish b etw een individuals in the flo c k. One clearly p erceiv es a contin uum-like cloud of birds; mo delling this cloud b y means of an absolutely con tinuous measure thus seems natural. As the hawk attacks the starlings, the flo ck reacts to the approaching enem y as if there were some macroscopic co ordination. Increase and decrease of the density in the flo ck can b e observed. It is striking to see that the group do es not fall apart. 4 Similar effects can b e seen in shoals of fish b eing attack ed. The in teraction b et w een the predator fish and the shoal triggers complex structures to app ear and enables sudden transitions b etw een macroscopic patterns. The phenomena we refer to are describ ed e.g. in [36, 51]. Tw o-scale effects also o ccur in human crowds, when there is sp ecial interaction b etw een a sp ecific individual (or a limited n umber of them) and the rest of the crowd. These phenom- ena might b e harder to visualize (b ecause the individual and the group members hav e the same size), but this do es not mean that they are not there. Phenomena of the ”predator- prey” t yp e o ccur, for example when a cro wd is attack ed by some criminal or terrorist. More 4 Mo vies of suc h phenomena are a v ailable on the internet, see e.g. www.youtube.com/watch?v=b8eZJnbDHIg&feature=related . 70 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology ‘friendly’ interpla y is present when the sp ecial individuals hav e the role of tourist guides, leaders, firemen, safety guards et cetera. Our aim in Section 5 and further is to capture these t wo-scale phenomena with our mo del. Figure 4.4: Illustration of t wo-scale, asymmetric in teraction b etw een a single predator bird and a flo ck of smaller individuals. Mo delling Cro wd Dynamics 71 Chapter 5 Numerical sc heme In this section, w e propose a n umerical sc heme for finding solutions of the time-discrete model of Section 4.4. See Definition 4.4.6. The scheme was originally developed in [17]. W e consider the set Ω ⊂ R 2 suc h that Ω := [0 , L ] × [0 , W ], where L , W ∈ (0 , ∞ ). W e restrict ourselv es to the situation in whic h eac h comp onen t of the cro wd is either discrete (concen- trated in a finite num b er of p oints) or absolutely contin uous (w.r.t. λ d ), as was motiv ated in Section 4.8. Note that we hav e seen in the pro of of Theorem 4.5.1 that the push forward of a discrete measure is again discrete (and similar for the push forward of an absolutely con tin u- ous measure). The mass measure of eac h constituent is th us of the same type throughout time. 5.1 T yp es of measures 5.1.1 Discrete measure Supp ose that we hav e a discrete mass measure for constituent α , consisting of N α distinct Dirac distributions. W e c ho ose the particular form µ α n := M α N α X i =1 δ x α i,n , for the mass measure at time slice n . Here, M α ∈ R + is a prop ortionality constant that mak es it p ossible to compare a sum of Dirac measures (they measure the numb er of p e ople ) to an absolutely contin uous measure (that measures kilo gr ams of p e ople ). By { x α i,n } N α i =1 ⊂ Ω w e denote the set of (time-dep enden t) cen tres at time t = t n . 5.1.2 Absolutely contin uous measure: spatial approximation of densit y If the constituen t α has a corresponding density ρ α n , then we need to approximate ρ α n in space. Therefore, w e fix N L , N W ∈ N and we sub divide Ω into N L · N W cells. Define h L := L / N L and h W := W / N W , the horizon tal and v ertical grid size, respectively . The size of a cell is th us h L h W . Eac h of the cells E j,k is given an index ( j, k ) ∈ K :=  1 , 2 , . . . , N L  ×  1 , 2 , . . . , N W  , suc h that E j,k :=  ( j − 1) h L , j h L  ×  ( k − 1) h W , k h W  . 72 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Note that the b oundaries of these cells o v erlap each other. This is, how ev er, not a serious problem, since the b oundaries are n ull sets w.r.t. λ d . Up to a null set the cells are mutually disjoin t. W e sk etch suc h grid in Figure 5.1. h L h W L W E 1 , 1 E j,k E N L , 1 E 1 ,N W E N L ,N W Figure 5.1: The spatial discretization of the domain by sub division into N L · N W cells. W e approximate the density ρ α n b y a piecewise constant function ˜ ρ α n . This means that for eac h ( j , k ) ∈ K there is a ρ α ( j,k ) ,n ∈ R + suc h that ˜ ρ α n ( x ) := ρ α ( j,k ) ,n , for all x in the interior of E j,k . Since the b oundaries of the cells form a null set, it is not imp ortan t ho w w e define ˜ ρ α n there. 5.2 Calculating v elo cities In Section 4.6 w e ha ve prop osed time-discrete v elo cit y fields v α n defined b y v α n ( x ) := v α des ( x ) + ν X β =1 Z Ω \{ x } f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dµ β n ( y ) , for all x ∈ Ω . (5.2.1) W e appro ximate v α n b y ˜ v α n . Where and ho w an appro ximation is needed, dep ends on the types of measures asso ciated to α and β . W e discuss this asp ect in more detail in Sections 5.2.1 and 5.2.2. 5.2.1 Discrete µ α n If µ α n is a discrete measure, then the ev aluation of v α n is required in eac h of the points x α i,n ∈ Ω. The desired v elo cit y v α des can b e ev aluated in these p oin ts without an y difficult y . Ho wev er, Mo delling Cro wd Dynamics 73 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology atten tion has to b e paid regarding the in tegral terms with resp ect to µ β n that arise in the so cial v elo city part of (5.2.1). W e distinguish b etw een the cases: (i) µ β n is discrete; (ii) µ β n is absolutely con tinuous. Firstly , let µ β n for eac h n b e giv en as µ β n := M β N β X i =1 δ x β i,n , for some M β ∈ R + , N β ∈ N and a set of cen tres { x β i,n } N β i =1 ⊂ Ω. The corresponding in teraction term can b e calculated exactly as follows: Z Ω \{ x } f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dµ β n ( y ) = M β N β X i =1 x β i,n 6 = x f α β ( | x β i,n − x | ) g ( θ α xx β i,n ) x β i,n − x | x β i,n − x | . (5.2.2) An y of the p oin ts x α i,n can b e taken as a choice of x . Remark 5.2. 1. Note that the exclusion of { x } from the domain of integration is intended to av oid interaction of a p oin t mass with itself for the case α = β . How ever, in the right- hand side of (5.2.2) the contribution of interactions are also excluded if the p ositions of tw o distinct point masses coincide. That is, if w e ha ve x α i,n = x β j,n for α = β but i 6 = j (t wo distinct p oin ts from the same subp opulation), or x α i,n = x β j,n where α 6 = β (tw o coinciding p oints from differen t subp opulations). W e require our v elo cit y field and initial conditions to b e such that neither of these t wo situations o ccur; see also Remark 4.5.15. Secondly , we consider the in teraction integral for the case that µ β n is absolutely con tinuous w.r.t. λ d . The appro ximation ˜ ρ β n of the densit y is use d to obtain Z Ω \{ x } f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dµ β n ( y ) = Z Ω f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | ρ β n ( y ) dλ d ( y ) ≈ Z Ω f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | ˜ ρ β n ( y ) dλ d ( y ) = X ( j,k ) ∈K ρ β ( j,k ) ,n Z E j,k f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dλ d ( y ) . (5.2.3) The integrals o ver E j,k are approximated b y a tw o-dimensional form of the trap ezoid rule, using the four v ertices of the rectangle (Newton-Cotes). Let these vertices b e called y i j,k , for i ∈ { 1 , 2 , 3 , 4 } . F or example, if y 4 j,k is the top righ t vertex, then y 4 j,k :=  j h L , k h W  . W e obtain Z E j,k f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dλ d ( y ) ≈ h L h W 4 4 X i =1 f α β ( | y i j,k − x | ) g ( θ α xy i j,k ) y i j,k − x | y i j,k − x | . (5.2.4) 74 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Ho wev er, if x = y i j,k for some i , we need to adapt this appro ximation, since typically f α β has a singularit y at 0. W e then choose to use the in teraction with the midp oin t of the cell instead. Let y (m) j,k :=  ( j − 1 2 ) h L , ( k − 1 2 ) h W  denote the midp oint of E j,k . Consequently , we use the appro ximation Z E j,k f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dλ d ( y ) ≈ h L h W f α β ( | y (m) j,k − x | ) g ( θ α xy (m) j,k ) y (m) j,k − x | y (m) j,k − x | , (5.2.5) for an y x coinciding with a v ertex of the cell E j,k . Remark 5.2.2. If the position x coincides with a v ertex of a cell, this causes a problem solely from the numerical point of view. The general sc heme (trap ezoid rule using the four vertices) is in that case no longer applicable for appro ximating the v alue of the integral ov er that cell. The concerning singularity do es not cause a problem from the p ersp ectiv e of mathematical analysis. Namely , if w e demand f α β ( s ) ∼ 1 /s , and assume that the density is uniformly b ounded, then    Z Ω \{ x } f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | dµ β n ( y )    6 Z Ω    f α β ( | y − x | ) g ( θ α xy ) y − x | y − x | ρ β n ( y )    dλ d ( y ) = Z Ω    f α β ( | y − x | )       g ( θ α xy )    | {z } 6 1    y − x | y − x |    | {z } =1    ρ β n ( y )    | {z } 6 k ρ β n k ∞ dλ d ( y ) 6 k ρ β n k ∞ Z Ω    f α β ( | y − x | )    dλ d ( y ) 6 k ρ β n k ∞ Z B ( x,R )    f α β ( | y − x | )    dλ d ( y ) . (5.2.6) Here, B ( x, R ) is the unit ball in R 2 around x with radius R := √ L 2 + W 2 . Due to the c hoice of this sp ecific radius it is guaranteed that Ω ⊂ B ( x, R ) for eac h x ∈ Ω. Let us no w only tak e the repulsive part around x in to consideration. The contribution of the attraction-part (if presen t, cf. Section 4.2 and Figure 4.1) is finite any w ay . Let R r denote the radius of repulsion. W e ha v e k ρ β n k ∞ Z B ( x,R r )    f α β ( | y − x | )    dλ d ( y ) = 2 π k ρ β n k ∞ R r Z 0    f α β ( r )    r dr = 2 π k ρ β n k ∞ R r Z 0  R r r − 1  r dr = π R 2 r k ρ β n k ∞ < ∞ . (5.2.7) W e hav e taken the repulsive part of f α β in its most simple form, but y et satisfying f α β ( s ) ∼ 1 /s . 1 A m ultiplicative constan t can of course b e added without lo osing finiteness of the in tegral. 1 In fact, it suffices to imp ose the weak er demand that f α β ( s ) ∼ s − γ for any γ < 2 Mo delling Cro wd Dynamics 75 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Remark 5.2.3. Contrary to what is stated in Remark 5.2.1, here we do not wish to forbid the situation that a p oint mass coincides with the v ertex of a cell. If tw o distinct point masses are lo cated in the same p osition (the case considered in Remark 5.2.1), this corresp onds to a ph ysically impossible situation. Although probably only rarely a p oin t mass will b e lo cated exactly on a cell’s vertex, this is not ph ysically undesirable, and we do w ant to allow for it. F or those situations, we ha ve therefore in tro duced an alternativ e approximation (5.2.5) to circum ven t the o ccurring problems in the numerical sc heme. 5.2.2 Absolutely contin uous µ α n If µ α n is absolutely contin uous with resp ect to λ d , then v α n is approximated by a piecewise constan t function. Let ˜ v α n ( y (m) j,k ) b e (an appro ximation of ) the v elo cit y of subp opulation α in the midp oin t of cell E j,k . Then for eac h x in the interior of E j,k , the appro ximated v elo cit y field ˜ v α n ( x ) is giv en by ˜ v α n ( x ) ≡ ˜ v α n ( y (m) j,k ) . In each midp oin t y (m) j,k ev aluating the desired velocity v α des do es not cause any problem. In order to deal with the integral terms in the so cial velocity , w e again take into consideration whic h type of measure µ β n is. Firstly , w e treat the case when µ β n is a discrete measure. W e sp ecifically supp ose that µ β n has the form µ β n := M β N β X i =1 δ x β i,n . If, for a given cell E j,k , the midp oint y (m) j,k do es not coincide with any of the centre points x β i,n , then Z Ω \  y (m) j,k  f α β ( | y − y (m) j,k | ) g ( θ α y (m) j,k y ) y − y (m) j,k | y − y (m) j,k | dµ β n ( y ) = M β N β X i =1 f α β ( | x β i,n − y (m) j,k | ) g ( θ α y (m) j,k x β i,n ) x β i,n − y (m) j,k | x β i,n − y (m) j,k | . (5.2.8) If by chance one of the cen tres, sa y x β m,n , of a Dirac mass coincides with the midp oint y (m) j,k , then we hav e to adapt our scheme. W e replace the direct in teraction betw een these tw o p oin ts, b y an av erage o v er the interaction b etw een x β m,n and each of the four vertices y i j,k of the cell. That is, w e use 1 4 4 X i =1 f α β ( | x β m,n − y i j,k | ) g ( θ α y i j,k x β m,n ) x β m,n − y i j,k | x β m,n − y i j,k | . 76 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Using this expression, w e obtain the follo wing approximation: Z Ω \  y (m) j,k  f α β ( | y − y (m) j,k | ) g ( θ α y (m) j,k y ) y − y (m) j,k | y − y (m) j,k | dµ β n ( y ) ≈ M β N β X i =1 i 6 = m f α β ( | x β i,n − y (m) j,k | ) g ( θ α y (m) j,k x β i,n ) x β i,n − y (m) j,k | x β i,n − y (m) j,k | + M β 4 4 X i =1 f α β ( | x β m,n − y i j,k | ) g ( θ α y i j,k x β m,n ) x β m,n − y i j,k | x β m,n − y i j,k | . (5.2.9) W e hav e merely ov ercome problems that migh t o ccur in particular if the spatial grid is to o coarse. F or a sufficien tly fine grid, the condition in Remark 4.5.15 makes sure that there is a neigh b ourho o d of zero density around the p osition of eac h Dirac mass. It is not imp ortan t whether the v elo cit y is prop erly defined in regions of zero densit y , or not. W e lo ok now to the case when µ β n is an absolutely contin uous measure. W e follow (5.2.3) and (5.2.4), and deriv e Z Ω \  y (m) j,k  f α β ( | y − y (m) j,k | ) g ( θ α y (m) j,k y ) y − y (m) j,k | y − y (m) j,k | dµ β n ( y ) ≈ h L h W 4 X ( j,k ) ∈K ρ β ( j,k ) ,n 4 X i =1 f α β ( | y i j,k − y (m) j,k | ) g ( θ α y (m) j,k y i j,k ) y i j,k − y (m) j,k | y i j,k − y (m) j,k | . (5.2.10) Keep in mind that the midp oin t y (m) j,k can never coincide with any of the vertices of a cell in Ω. 5.2.3 Summary: The appro ximation ˜ v α n W e no w shortly summarize the w ay we ha ve defined ˜ v α n in Sections 5.2.1 and 5.2.2. • If µ α n is discrete, w e need the v alue of ˜ v α n in the corresp onding centres x α i,n of the Dirac masses. Calculation of the v alue of v α des is straigh tforward. Those terms in the so cial velocity for which the measure µ β n is discrete are calculated according to (5.2.2). If, on the other hand, the measure µ β n is absolutely contin uous, then the appro ximation of (5.2.3) is used. Regarding the approximation of the integrals o ver all cells E j,k , we distinguish b et w een t wo cases: – x α i,n do es not coincide with one of the four v ertices of cell E j,k . Use the t w o- dimensional trap ezoid rule as given by (5.2.4). – x α i,n coincides with one of the four vertices of cell E j,k . Use the midp oint of the cell to appro ximate the integral; see (5.2.5). Mo delling Cro wd Dynamics 77 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology • If µ α n is absolutely contin uous, the velocity in eac h interior point of a cell E j,k is approx- imated by the velocity in the midp oint of that cell. Ev aluation of v α des in the midp oin t y (m) j,k is again straightforw ard. If the measure µ β n in the so cial velocity is discrete, then w e distinguish b et ween t wo cases: – None of the Dirac-centres x β i,n coincides with the midp oint y (m) j,k . Then use (5.2.8) to ev aluate the in tegral. – One of the cen tres x β i,n is lo cated exactly in the midp oint y (m) j,k . Use the approxi- mation giv en in (5.2.9). If µ β n is absolutely con tinuous, then the appro ximation as given by (5.2.10). 5.3 Push forw ard of the mass measures The only thing that is still to b e included in our numerical sc heme, is the push forward of the measure µ α n to obtain µ α n +1 . W e use ˜ v α n (see Section 5.2, in particular Section 5.2.3) to obtain a mo dified version of the motion mapping (cf. Definition 4.4.1): ˜ χ α n ( x ) := x + ∆ t n ˜ v α n ( x ) . If µ α n is a discrete measure, the push forward can b e found in a natural wa y , as was already suggested in Corollary 4.5.8. If µ α n is giv en by µ α n := M α N α X i =1 δ x α i,n , then the n umerical approximation of the push forw ard is µ α n := M α N α X i =1 δ ˜ χ α n ( x α i,n ) . Determining the push forw ard of this discrete mass measure boils down to updating the cen tres of the Dirac masses in the follo wing wa y: x α i,n +1 := ˜ χ α n ( x α i,n ) = x α i,n + ∆ t n ˜ v α n ( x α i,n ) . If µ α n is absolutely contin uous, then a little more effort is required. Since ˜ v α n is defined such that it is constant within a cell, the push forw ard of a cell E j,k is a translation by the vector ∆ t n ˜ v α n ( y (m) j,k ). Indeed w e ha v e ˜ χ α n ( E j,k ) =  ˜ χ α n ( x )   x ∈ E j,k  =  x + ∆ t n ˜ v α n ( x )   x ∈ E j,k  =  x + ∆ t n ˜ v α n ( y (m) j,k )   x ∈ E j,k  = E j,k + ∆ t n ˜ v α n ( y (m) j,k ) . The push forw ard of E j,k is indicated in Figure 5.2. W e wan t to k eep the spatial grid fixed in time. Also, we require our appro ximated densit y ˜ ρ α n to b e constant within a cell, for ev ery 78 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology ∆ t n ˜ v α n ( y (m) j,k ) E j,k ˜ χ α n ( E j,k ) Figure 5.2: The push forward of a cell E j,k go verned b y the approximated v elo cit y field. The ‘image’ ˜ χ α n ( E j,k ) clearly lies within four cells of the spatial domain. n . In general, the push forward of E j,k will not exactly coincide with a cell of our grid: cf. Figure 5.2, where ˜ χ α n ( E j,k ) lies in four cells. W e th us prop ose the following up date of the densit y: ˜ ρ α n +1 ( x ) ≡ ρ α ( j,k ) ,n +1 := 1 λ d  E j,k  X ( p,q ) ∈K ρ α ( p,q ) ,n λ d  E j,k ∩ ˜ χ α n ( E p,q )  , (5.3.1) for all x in the in terior of E j,k . A p ositive con tribution is given by eac h cell E p,q that is (partially) mapp ed into E j,k , since only then E j,k ∩ ˜ χ α n ( E p,q ) is non-empt y , and thus has p ositive Leb esgue measure. Note that ρ α ( p,q ) ,n λ d  E j,k ∩ ˜ χ α n ( E p,q )  is exactly the mass that is transferred from E p,q in to E j,k . The densit y of E j,k then follows from adding the mass contributions of all cells E p,q , and dividing b y the area of E j,k . This mak es (5.3.1) a natural w a y of up dating densities. The numerical approximation of the push forward of µ α n is fully determined by the itera- tiv e scheme for its densit y ˜ ρ α n as giv en by (5.3.1). Remark 5.3.1. Using the up date of the densit y in (5.3.1), ˜ ρ α n +1 is b y definition non-negativ e, as it is a sum of non-negative terms. F urthermore, the total mass con tained in Ω is conserved Mo delling Cro wd Dynamics 79 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology in time. W e namely ha v e that Z Ω ˜ ρ α n +1 ( x ) dλ d ( x ) = X ( j,k ) ∈K ρ α ( j,k ) ,n +1 λ d  E j,k  = X ( j,k ) ∈K λ d  E j,k  1 λ d  E j,k  X ( p,q ) ∈K ρ α ( p,q ) ,n λ d  E j,k ∩ ˜ χ α n ( E p,q )  = X ( p,q ) ∈K ρ α ( p,q ) ,n X ( j,k ) ∈K λ d  E j,k ∩ ˜ χ α n ( E p,q )  = X ( p,q ) ∈K ρ α ( p,q ) ,n λ d  ˜ χ α n ( E p,q )  = X ( p,q ) ∈K ρ α ( p,q ) ,n λ d  E p,q  = Z Ω ˜ ρ α n ( x ) dλ d ( x ) . W e ha ve used in the fifth step that ˜ χ α n is a translation, due to which λ d  ˜ χ α n ( E p,q )  = λ d  E p,q  . Conserv ation of total mass in Ω is only guaranteed under the assumption that no cell (of non- zero densit y) is mapp ed outside Ω; cf. Par t (i) of Assumption 4.5.13. 80 Mo delling Cro wd Dynamics Chapter 6 Numerical illustration: sim ulation results In this section we present the results of our simulation experiments. Except for some simple test cases, we only consider tw o-scale situations. That is, in eac h exp eriment there are tw o subp opulations, one of which is discrete and the other is absolutely con tinuous. The comp o- nen t indexed α = 1 is a collection of discrete individuals. In our sim ulations, comp onent 1 consists mostly of one individual only , and exceptionally of tw o. The comp onen t with index α = 2 is a macroscopic crowd. T o av oid effects at the boundaries (whic h w e ha ve not sp ecified so far), the domain Ω is ‘sufficien tly large’. That is, the size of Ω is such that no mass reac hes the b oundary . No problems o ccur as long as the cells of the spatial grid on the p eriphery of Ω hav e zero density , and the p ositions of the individuals remain in Ω. 1 In the following sections, the results of the simulation are presen ted b y giving a graphical represen tation of the cro wd in the domain. The p ositions of individuals are marked by red bullets, whereas the density of the macroscopic crowd is indicated by a grey shading. The dark er the colour, the higher the density; a colour bar at the righ t-hand side of the graph sho ws what shading corresp onds to a certain densit y . Note that we essentially use the scheme of [17, 45]. The conv ergence and stabilit y of the n umerical solution is prov en in these pap ers. W e exp ect similar prop erties to hold for our setting. W e, ho wev er, omit to give further details in this direction and focus directly on sim ulation results. The results we presen t will b e interpreted only on a qualitative basis. At a later stage we will try to make some of these results quantitativ e by reco vering exp erimental data by S. Ho ogendo orn, W. Daamen and M. Campanella (Delft Universit y of T echnology , see Section 1.4 of the In tro duction) regarding exp eriments in a corridor. 1 W e admit that this is a rather pragmatic solution, but for the moment it is the best w e can do. 81 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 6.1 Reference setting F or attraction-repulsion interactions, we use the following expression in the interaction inte- gral: f AR ( s ) :=                    F AR  1 − R AR r s  , if 0 < s 6 R AR r ; − F AR R AR r ( R AR a − R AR r )  s − R AR r  s − R AR a  , if R AR r < s 6 R AR a ; 0 , if s > R AR a . (6.1.1) The radii R AR r and R AR a are suc h that 0 < R AR r < R AR a . The factor F AR ∈ [0 , ∞ ) is a p ositive scaling constan t. Note that this is exactly the function plotted in Figure 4.1 (left). Note that f AR is differen tiable in s = R AR r . Similarly , w e define for purely repulsive in teraction f R ( s ) :=    F R  1 − R R r s  , if 0 < s 6 R R r ; 0 , if s > R R r . (6.1.2) Again, we tak e R R r > 0 and the scaling constant F R ∈ [0 , ∞ ). This function is plotted in Figure 4.1 (righ t). Unless indicated otherwise, w e use a set of ‘standard’ parameters in our simulations. As dimensions of the domain Ω w e tak e L = W = 50 . The prop ortionalit y constan t M α is only presen t for α = 1 and is assigned the v alue M α = 60 , for α = 1 . The desired velocity is indep endent of the space v ariable, and only the direction is different for the t wo subpopulations. W e take v 1 des ( x ) ≡ − 1 . 34 e 1 , for all x ∈ Ω , v 2 des ( x ) ≡ 1 . 34 e 1 , for all x ∈ Ω . Here e 1 is the unit vector in the direction that corresponds to the horizon tal axis in our graphical represen tation. Regarding the in teraction functions, we c ho ose the following reference parameters: F AR = 0 . 03 , F R = 0 . 03 , R AR r = 1 . 5 , R AR a = 3 , R R r = 4 , σ = 0 . 5 . 82 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology A ttraction-repulsion in teractions tak e place within one subp opulation, whereas repulsion-only tak es place in the in teractions b et ween distinct subp opulations. In the sequel w e indicate clearly where w e deviate from these standard parameters. 6.2 Tw o basic critical examples: one micro, one macro W e inv estigate here whether the simulation results are mathematically and physically accept- able. Consider tw o individuals b oth ha ving desired v elo city in the direction of e 1 . The magni- tude of the desired velocities is equal and constant, but they are directed opp ositely . The initial p ositions of the t w o individuals differ only in the e 1 direction, and they are lo cated suc h that they initially w ant to mo ve to wards eac h other due to their desired velocities. Their in teraction is purely of repulsive nature, and has the same parameter v alues for an y of the t wo. W e exp ect these individuals to approac h eac h other, up to a certain distance. At this distance the desired velocity in one direction is in balance with the (opp ositely directed) re- pulsiv e effect in the so cial v elo city . The total velocity is thus zero for b oth individuals. Let x i and x j denote the p ositions of the tw o p edestrians. F or simplicity we tak e σ = 1 and M i = M j = 1. The v elo city of p edestrian i is given by v ( x i ) = v i des + f R ( | x j − x i | ) x j − x i | x j − x i | , (6.2.1) with f R as in (6.1.2). The v elo city is zero if − f R ( | x j − x i | ) x j − x i | x j − x i | = v i des . A necessary condition is    f R ( | x j − x i | ) x j − x i | x j − x i |    = | v i des | . (6.2.2) Note that ( x j − x i ) / | x j − x i | is a unit v ector. Considering (6.1.2), condition (6.2.2) reads F R  R R r | x j − x i | − 1  = | v i des | , if | x j − x i | 6 R R r . This yields that | x j − x i | = F R | v i des | + F R R R r , (6.2.3) at the p oint where the velocity is zero. In Figure 6.1 we show the outcome of this simulation. The individual placed initially on the left has desired velocity v des ≡ 1 . 34 e 1 , while the one on the righ t has desired velocity v des ≡ − 1 . 34 e 1 . F urthermore we tak e F R = 1. In Figure 6.2, the m utual distance is plotted against the time. Indeed the equilibrium distance is the one predicted by (6.2.3). Mo delling Cro wd Dynamics 83 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 20 22 24 26 28 30 24 25 26 20 22 24 26 28 30 24 25 26 Figure 6.1: Two individuals approac hing each other until a certain minimal distance is reac hed. The images are taken at t = 0 (left), t = 15 (right). 0 5 10 15 1 2 3 4 5 6 7 8 9 10 Time Mutual distance Figure 6.2: The m utual distance b etw een tw o approac hing individuals (blue). The predicted minimal distance is indicated in red. Note that the setting of Figure 6.1 is unrealistic and thus undesirable. In ev eryda y life these tw o p edestrians w ould namely b oth mov e a bit aside and then walk straight ahead without any constraints. In Section 1.3 of the In tro duction, we ha v e indicated already that a small amoun t of random noise can be used to a v oid these deadlocks. W e exp ect that the equilibrium configuration is instable, and that deadlocks do neither o ccur if w e p erturb the initial data (in the direction p erp endicular to the desired velocities). In the exp eriment presented in Figure 6.3, w e observ e that a macroscopic cro wd tends to form a circular configuration. The crowd has no desired v elo cit y: v des ≡ 0, and therefore σ = 1 is tak en. This typical b ehaviour is well-kno wn from molecular dynamics. The mutual in teractions fa v our the minimization of the ratio circumference to area. A mathematical proof is given in [24], for the same macroscopic (con tin uous-in-time) equation of mass conserv ation. The interaction p oten tial in [24] is not the same as the one used here, but we exp ect that similar results can b e derived along comparable lines of argument. 6.3 Tw o-scale in teractions of repulsiv e nature In Figure 6.4, w e show the interaction b etw een a macroscopic cro wd and an individual that wishes to approach it, and even tually forces itself a wa y through. This is a situation in which w e try to mimic the t wo-scale (‘predator-prey’) b eha viour describ ed in Section 4.8. A circular area of (nearly) zero density is formed around the individual. The size of this 84 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 15 20 25 30 35 15 20 25 30 35 2 4 6 8 10 12 14 15 20 25 30 35 15 20 25 30 35 2 4 6 8 10 12 14 15 20 25 30 35 15 20 25 30 35 2 4 6 8 10 12 14 Figure 6.3: The simulation of a macroscopic crowd’s motion. Initially , the crowd forms a square of uniform density ρ ≡ 2. Its configuration evolv es into a circular shap e. The images are tak en at t = 0 (top left), t = 30 (top right), t = 60 (bottom left). region typically dep ends on R R r . At a distance shorter than R R r mass is driven aw a y from the individual. If we decrease the radius R R r , also the ‘empty zone’ around the individual decreases; see Figure 6.5 for this effect. W e estimate the distance in front of the individual that is empt y , and use the ideas of Section 6.2 to do so. Let us consider the p oin t x in the cro wd that is right in front of the individual. Disregarding the effect of the rest of the macroscopic cro wd, the velocity of x is similar to the one giv en in (6.3.1): v ( x ) = v 2 des + f R ( | z − x | ) z − x | z − x | . Here, the v ariable z is used to denote the p osition of the individual. The desired velocity of the individual v 1 des is coupled to v 2 des via the relation: v 1 des = − v 2 des . Assume that the individual mov es at this sp eed 2 and that x remains righ t in fron t of z . Then the mass lo cated 2 This is generally not the case. Mo delling Cro wd Dynamics 85 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 Figure 6.4: The interaction b et w een a discrete individual and a macroscopic cro wd. The images are tak en at t = 0 (top left), t = 5 (top right), t = 10 (bottom left). in x is forced to mo ve also with v elo city v 1 des . This leads to the follo wing equation: v ( x ) = v 2 des + f R ( | z − x | ) z − x | z − x | = v 1 des = − v 2 des . (6.3.1) F ollo wing the lines of Section 6.2, w e derive that a nece ssary condition is that | z − x | = M F R 2 | v 2 des | + M F R R R r . (6.3.2) According to this estimate the size of the empty area around the individual scales linearly with R R r (note that | z − x | 6 R R r ). In Figure 6.5 the configuration is giv en for t wo distinct v alues of R R r . Indeed the size of the lo w density zone decreases as R R r decreases. The distance estimate in (6.3.2) is indicated b y a circle in blue. Right in front of the individual there is a sp ot of high density . This is the sp ot x w e were considering. W e observ e that x is lo cated just outside the blue circle, by which the estimate turns out to b e quite go o d. W e also observ e that the estimate loses its v alue more to the sides of the individual. After the individual passes by , w e observe in Figure 6.4 that the crowd splits in t wo and that the tw o halves do not come together again. This effect dep ends on the choice of parameters. By adjusting the parameters we can achiev e the crowd to ‘env elop e’ around the individual. T o this end, we decrease the radius of repulsion of the in teraction b etw een the individual and the crowd. Also w e increase the radius of attraction of the internal interactions in the cro wd. T o do so, we tak e R R r = 2, R AR a = 5. This mo dification makes that the macroscopic crowd do es enclose the individual. The result is sho wn in Figure 6.6. W e deduce from Figure 6.5 that the width of the empt y zone is of the order of 2 R R r (probably a little bit narro wer, as also (6.3.2) suggests). Let us define a critical radius R cr suc h that the width of the empt y zone is 2 R cr . W e exp ect the cro wd to come together again, if R AR a > 2 R cr ; 86 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 22 24 26 28 30 32 34 36 38 40 42 20 22 24 26 28 30 0 1 2 3 4 5 6 7 8 22 24 26 28 30 32 34 36 38 40 42 20 22 24 26 28 30 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 6.5: The size of the empty zone around the individual dep ends on R R r . On the left R R r = 4 (this is the top right situation in Figure 6.4), on the right R R r = 2. In blue a circle is dra wn with radius equal to the distance predicted in (6.3.2). The images are b oth tak en at t = 5 starting from identical initial configurations. 15 20 25 30 35 40 45 50 15 20 25 30 35 0 2 4 6 8 10 12 14 16 18 15 20 25 30 35 40 45 50 15 20 25 30 35 0 2 4 6 8 10 12 14 16 18 15 20 25 30 35 40 45 50 15 20 25 30 35 0 2 4 6 8 10 12 14 16 18 Figure 6.6: The interaction b etw een a discrete individual and a macroscopic crowd. Mo di- fication of the radii of in teraction mak es that the cro wd comes together after the individual has passed by . The images are taken at t = 0 (top left), t = 4 (top righ t), t = 5 (b ottom left). this is the situation in which the crowd in one half can also ‘feel’ the part on the other side of the empt y zone. Remark 6.3.1. W e hav e tried to achiev e the effect of Figure 6.6 also by adding macroscopic mass ab ov e and b elow the initial square. That is, initially the crowd now has a rectangular shap e twice as wide (in vertical direction in the graph). W e hop ed that the repulsive effects within the crowd w ould force the tw o halv es to mov e tow ards each other again. How ever, this effect w as not obtained. Remark 6.3.2. The desired v elo cit y of the individual do es not need to b e necessarily in the direction of e 1 . If the desired v elo cities of the individual and the macroscopic cro wd are at Mo delling Cro wd Dynamics 87 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology an angle (not equal to π ), the result do es ho wev er not fundamentally differ from Figure 6.4. The individual only lea ves b ehind a diagonal trace in the cro wd, instead of a horizontal one. 15 20 25 30 35 40 45 50 15 20 25 30 35 1 2 3 4 5 6 7 15 20 25 30 35 40 45 50 15 20 25 30 35 1 2 3 4 5 6 7 15 20 25 30 35 40 45 50 15 20 25 30 35 1 2 3 4 5 6 7 Figure 6.7: The interaction b et ween tw o discrete individuals and a macroscopic cro wd. The individuals are initially a distance of R AR r apart. The images are taken at t = 0 (top left), t = 5 (top right), t = 10 (b ottom left). 6.4 In teraction with t w o individuals Tw o-scale simulation experiments can inv olv e tw o individuals instead of one. The outcome of suc h a simulation can b e seen in Figure 6.7. The tw o individuals initially are at a relatively short distance from one another. Their mutual distance remains small and macroscopic mass is not able to separate the t wo. The macroscopic crowd only mov es around them. In fact, the impact of the t w o individuals is not fundamen tally different from the impact a single individual of double mass w ould hav e. If w e increase the initial distance b et ween the tw o individuals, then the macroscopic mass do es find a wa y to fill the space b etw een them. It even forces the individuals more apart. This can be seen in Figure 6.8. Note that it still requires some ‘effort’ to separate the t wo individuals. In the top right image in Figure 6.8 a region of high density is presen t at the left-hand side of the op ening betw een the individuals. W e exp ect that it dep ends on the initial distance b etw een the t wo individuals compared to R R r , whether macroscopic mass can pass through the op ening b etw een the t w o individuals or not. 88 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 15 20 25 30 35 40 45 50 15 20 25 30 35 5 10 15 20 25 15 20 25 30 35 40 45 50 15 20 25 30 35 5 10 15 20 25 15 20 25 30 35 40 45 50 15 20 25 30 35 5 10 15 20 25 Figure 6.8: The in terac tion b etw een tw o discrete individuals and a macroscopic crowd. Ini- tially the distance b etw een the individuals is slightly more than R AR a , and thus they do not ‘feel’ each other. The images are taken at t = 0 (top left), t = 5 (top righ t), t = 10 (b ottom left). 6.5 Mo delling leadership Can we capture ‘leadership’ effects via our tw o-scale mo del? More precisely , can an individ- ual b e enabled to drag macroscopic mass along? W e interpret suc h b eha viour as a tendency to follo w the individual, even though the macroscopic crowd in itself has no sp ecial desire to go in a certain direction. The individual then acts as a leader (or guide) for the group. T o achiev e this effect, in this section, we choose attraction-repulsion as the influence of the individual on the macroscopic cro wd, instead of solely repulsiv e interactions. In Figure 6.9, the individual is only driven b y its desired velocity , which is directed to the left. The crowd do es not affect its motion. Y et, the dynamics of the macroscopic cro wd is influenced b y the ‘target particle’ due to attraction-repulsion. The cro wd has no desired v elo cit y , th us σ = 1 is taken. All other conditions are as in the reference setting. Figure 6.9 shows that the individual is able to create a short slipstream of macroscopic mass, when mo ving through the crowd, but w e can not convincingly call it a leader. The striking feature app earing in Figure 6.9 are the vertical lines, alternatingly of high and low density . These app ear in the macroscopic crowd. Why do es this happen? As Figure 6.10 shows, these oscillations are apparently only a result of the numerical appro ximation. The densit y is smo othened if w e decrease the grid size. W e wish to enhance the effect of leadership (compared to Figure 6.9) by adjusting parameters. Mo delling Cro wd Dynamics 89 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 Figure 6.9: The interaction b et w een a discrete individual and a macroscopic cro wd. The attraction-repulsion influence of the individual on the macroscopic cro wd makes that the cro wd is slightly dragged along. A small amount of mass ev ades on the left-hand side of the cro wd. The images are taken at t = 0 (top left), t = 7 (top right), t = 14 (b ottom left). Firstly , w e increase the magnitude of the attraction-repulsion influence of the individual on the crowd. More sp ecifically , w e take F AR = 0 . 3 instead of F AR = 0 . 03; for interactions within the cro wd F AR = 0 . 03 is main tained. All other conditions are as in Figure 6.9. In Figure 6.11 w e see the effect of the p erformed parameter mo dification. Note that within the environmen t of the individual, the density attains v ery high v alues, whic h is physically not acceptable. Let us concentrate on qualitative b ehaviour only . The individual successfully attracts a large part of the macroscopic cro wd, at least initially . As time elapses, it loses control ov er the crowd. Maybe, this fact is due to his to o high v elo cit y . A t first, the mass in its direct en vironment is distributed more or less uniformly o ver a circle around it; this mass shifts more and more to its rear end. Also, we observ e a trace of mass left b ehind by the individual. Once this mass is at a distance greater than R AR a from the individual, it remains near the place where it loses con tact. 3 Effects of a similar kind can b e obtained not only by mo difying the interaction strength, but also by increasing the radius of attraction. In the attraction-repulsion influence of the individual on the cro wd we restore the v alue F AR = 0 . 03, and tak e R AR a = 6. Indeed this enables the individual to act as a leader, cf. Figure 6.12. The macroscopic cro wd is initially attracted to the individual, which is comparable to Figure 6.11. This situation differs from Figure 6.11 in the sense that the macroscopic crowd is m uc h less compressed. A p ossible explanation is tw ofold. The area of attraction around the leader is m uch larger, and moreo ver, the attraction tow ards the centre of this region is less strong. 3 The scaling of the grey shading has b een altered in order to b e able to distinguish low er densities from zero density . V ery high densities o ccurring in small regions dominate the scaling. The aforementioned trace of mass is therefore hardly visible, in spite of the fact the densit y is around ρ ≈ 2. 90 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 15 20 25 30 35 40 45 50 15 20 25 30 35 0 1 2 3 4 5 6 7 8 Figure 6.10: Decreasing mesh size for the setting of Figure 6.9 at time t = 14. T op left: the original grid size of Figure 6.9. T op right: the num b er of cells has b een increased b y a factor 1.5 in b oth horizontal and v ertical direction. Bottom left: increase of the n umber of cells by a factor 2 compared to the top left situation. Oscillations in the density are diminished by decreasing grid size. Note that due to the weak er attraction-repulsion (compared to the previous case F AR = 0 . 3), the leader is less successful in k eeping his follo w ers with him. Finally , the macroscopic mass has mov ed on av erage a smaller distance to the left. Ho w ev er, a p ositiv e asp ect of this setting is that the maximal densit y is nearly a factor 4 smaller than in Figure 6.11. Mo delling Cro wd Dynamics 91 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 15 20 25 30 35 40 45 50 15 20 25 30 35 0 10 20 30 40 50 60 70 80 15 20 25 30 35 40 45 50 15 20 25 30 35 0 10 20 30 40 50 60 70 80 15 20 25 30 35 40 45 50 15 20 25 30 35 0 10 20 30 40 50 60 70 80 Figure 6.11: The interaction b et w een a discrete individual and a macroscopic crowd. The influence of the individual on the crowd has b een increased, and as a result the individual is able to drag the cro wd along. The images are taken at t = 0 (top left), t = 7 (top right), t = 14 (b ottom left). 15 20 25 30 35 40 45 50 15 20 25 30 35 0 5 10 15 20 15 20 25 30 35 40 45 50 15 20 25 30 35 0 5 10 15 20 15 20 25 30 35 40 45 50 15 20 25 30 35 0 5 10 15 20 Figure 6.12: The interaction b et w een a discrete individual and a macroscopic crowd. The radius of attraction has b een increased, which enables the individual (temp orarily) to take macroscopic crowd along. The images are tak en at t = 0 (top left), t = 7 (top right), t = 14 (b ottom left). 92 Mo delling Cro wd Dynamics Chapter 7 Discussion A t this p oint, w e wish to lo ok back and list a few op en issues with resp ect to mo delling, analysis and sim ulation. W e will inv estigate some of them in the near future. • Via (mainly formal) calculations we arrived at a fully con tinuous-in-time mo del. Sub- sequen tly , w e deduced a time-discrete version of this mo del, which we analyzed math- ematically . W e prov ed global existence and uniqueness of a time-discrete solution (in Theorems 4.5.1 – 4.5.2), and show ed some prop erties of this solution (e.g. p ositivit y and conserv ation of mass). A n umber of conditions on the (one-step) motion mappings and v elo cit y fields w ere inevitably needed. Afterwards, w e prop osed a sp ecific form of the v elo cit y field, in which a dep endence on the instan taneous configuration of the mass app ears. Up to now, w e ha ve not pro ven that the chosen velocity field actually satisfies the imp osed conditions. This is still to b e done. W e exp ect this step to b e difficult, mainly due to the functional dep endence of the v elo city on the mass measures, and due to the fact that the dynamics of all subp opulations are coupled via their velocity fields. • F ormulating and proving results like existence and uniqueness of solutions to the contin uous- in-time problem, is our next goal. Ho w ev er, this will b e a muc h more difficult task than treating the discrete-in-time mo del. F or instance, the one-step motion mapping of the mass measures is linear in v α n for the time-discrete mo del, cf. Definition 4.4.1. This is b ecause we linearized by using T a ylor series approximations in order to obtain the time-discrete mo del. The motion mapping will in general no longer b e linear in the v elo cit y if we consider the con tin uous case. Moreo ver, it is all but clear whether we can even sp eak so easily ab out a one-step motion mapping and the accompanying push forw ard of the mass measure. These concepts are likely to translate into contin uous-in- time counterparts, relating the configurations at time instances that are not an a priori fixed time step apart. • W e would like to enlarge the class of in teraction p otentials for whic h an entrop y in- equalit y can b e derived. F or the moment the p otential W α β is only admissible if W α β ( ξ ) = W α β ( − ξ ) for all ξ ∈ R d , and in teractions are symmetric. The latter con- dition means that W α β ≡ W β α . See Sections 4.3 and 4.7, and esp ecially Assumption 4.3.5. These restrictions on W α β are simply to strong for many in teresting settings. An idea at least to circum v en t the condition W α β ( ξ ) = W α β ( − ξ ) is inspired b y [10] (esp ecially 93 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Section 3.1 therein). W e write W α β as the sup erp osition of a symmetric part and a drift part: W α β = W α β , symm + W α β , drift , where W α β , symm ( ξ ) = W α β , symm ( − ξ ) is satisfied for all ξ ∈ R d . The second term W α β , drift con tains the deviations from symmetry . W e are interested to see which restrictions on W α β , drift are needed in order still to b e able to derive an entrop y inequalit y . Ho wev er, some problems arise in the interpretation of such interactions. Consider the v ector ξ in its represen tation in p olar coordinates. Without the condition that W α β ( ξ ) = W α β ( − ξ ), the interaction p otential W α β certainly dep ends not only on the length of ξ but also on its angle (the azimuth). Since the velocity field in v olves ∇ W α β , it inevitably has a non-zero azimuthal comp onent. W e should ask ourselv es the question whether this is physically acceptable: are the in teractions allow ed to influence the velocity in a direction that is not parallel to the connection v ector b etw een t wo p oin ts? If we do not wan t to restrict ourselves to a gradien t (in this case ∇ W α β ), w e could replace it b y a v ector field of the form ∇ W + ∇ × ω (Helmholtz decomp osition). Under which conditions on the v ector field ω can w e still derive an en tropy inequality? • A wider range of domains can b e considered, if we kno w ho w to incorp orate w alls (outer b oundaries) and internal obstacles in our domain. W e hav e not treated this asp ect so far, but it surely is v ery imp ortant to do so. In [44], Piccoli prop oses to pro ject the obtained velocity on the space of admissible velocities. That is, those velocities that satisfy v · n > 0 at the b oundaries. In practice, this means that out ward p ointing normal comp onents of the v elo city are to b e set zero. Perhaps, w e would lik e to hav e an alternativ e wa y , which fits b etter to our measure-theoretical framew ork. Probably one needs to define b oundary measures and Cauch y fluxes, while also describing their throughput. • More fundamen tal mathematical c hallenges are related to discr ete-to-c ontinuum limits . W e would like to in v estigate in what sense w e can approximate the absolutely con tinu- ous part of our mass measures b y particle systems (discrete mass measures in the case of cro wd dynamics). Esp ecially , the limiting pro cess N → ∞ (where N is the num b er of Dirac measures in the particle system) is relev ant. Such appro ximation, in combina- tion with the W asserstein metric, might turn out to b e useful for proving existence and uniqueness prop erties for the contin uous-in-time mo del, and also to derive alternative n umerical schemes for computing the densit y based on particle simulations. The question is which scaling one needs to obtain our macroscopic p ersp ective in the limit. Moreov er, we are i n terested in whether other t yp es of scaling w ould lead to differ- en t limits, e.g. a mesosc opic Boltzmann-t yp e description (‘intermediate lev el’ b etw een micro and macro, see e.g. [15]) or the tw ofold micro-macro setting of [17]. Perhaps there ev en exists a w ay to exploit the fact that w e ha ve multiple subp opulations. That is, we might apply different scalings to distinct subp opulations, such that e.g. for one subp opulation the limit is of Boltzmann-t yp e, and for another the limit is a tw o-scale micro-macro description. What is the actual interpretation of such distinction b et ween the t wo subpopulations? • Regarding our simulation results sho wn in Section 6: W e observe that we hav e success- fully circumv en ted problems that might o ccur at the b oundary , b y considering simula- tion runs as long as all mass is con tained in the in terior of Ω. Suc h ad ho c solution can 94 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology b e av oided in the future if w e find appropriate to ols for including b oundary effects (see ab o v e). When outputting the configuration, our simulation to ol is equipp ed to chec k whether the total mass is actually conserved. This turns out to b e indeed the case for ev ery instance of our simulation. • When examining the results of our simulation, several questions arise. All of them are the sub ject for further research: – Figure 6.3 indicates that the equilibrium configuration for attraction-repulsion in- teractions is a ball. If instead, w e test the same situation for purely attractive in teractions, do es the cro wd shrink to a p oint? Is there a mathematical wa y to sho w that the limit is a Dirac mass? Note that in this case the right-hand inequality in (4.5.1) can no longer b e satisfied. – In Section 6.3, w e estimate the radius of the (nearly) empty area in front of the individual. It migh t b e p ossible to characterize the shap e and dimensions of the rest of the empt y zone induced b y the individual. In the same section, we sho w that a mo dification of parameters mak es the cro wd clog together again after the individual has gone past. What are the typical length and time scales at which this reunion tak es place? – The in teraction betw een t wo individuals and a macroscopic cro wd w as sim ulated in Section 6.4. If w e zo om out, under what conditions can w e deduce from the macro- cro wd’s b eha viour that more than one individual was present (cf. Figure 6.8)? When do w e obtain merely the same results as in the presence of one individual of double mass (cf. Figure 6.7), and how do es this effect dep end on the initial mutual distance b et w een the t wo individuals? – In Section 6.5 we tried to create leaders. W e observed (esp ecially in Figures 6.11 and 6.12) that the individual at first is able to take a part of the macroscopic cro wd along, but that he loses control as time go es b y . Regarding the long-time b eha viour of the crowd, will ev entually all mass b e lost by the leader? When do leadership effects end? In Figure 6.11, b ottom left, the ‘tail’ of mass b ehind the individual consists of t wo regions. Closest to the individual the densit y is relativ ely high, whereas after quite sudden transition the density is lo wer. What are the characteristics of b oth regions (shap e, dimensions), and why is there an apparen t strict separation b et ween them? – In all of the cases ab ov e: How exactly do the observed phenomena dep end on the parameters? What can w e say ab out the (in)stability of equilibrium configurations? Can we tune parameters in such a wa y that physically unacceptable densities do not occur? Can w e identify a set of dimensionless quan tities, that c haracterizes the dynamics in the model? Also the role of the initial conditions needs to b e tak en in to consideration. Mo delling Cro wd Dynamics 95 Ac kno wledgemen ts I wish to express m y appreciation to a n um b er of p eople. First of all, to m y sup ervisor Adrian Mun tean for his guidance and supp ort. He has shap ed me as a mathematician, a scien tist and in some sense also as a human b eing. ”How is life?” - ”Go o d and busy” has b ecome a sort of mantra. Mark P eletier was and is a source of inspiration for me. Hearing him sp eak, con vinced me again and again that there is muc h more to b e explored ab out crowd dynamics. I wou ld like to thank the Assessmen t Committee for my master’s thesis; its members are: Adrian Mun tean, Prof.dr.ir. Harald v an Brummelen (Dept. of Mathematics and Computer Science and Dept. of Mechanical Engineering), Dr.ir. Huub ten Eik elder (Dept. of Biomed- ical Engineering), Dr. Georg Prokert (Dept. of Mathematics and Computer Science) and Dr.ir. F ons v an de V en (Dept. of Mathematics and Computer Science). F ons v an de V en also deserv es a p ersonal w ord of praise for sharing his exp erience and par- ticipating v ery activ ely in the pro cess leading to this thesis. Also, I am indebted to Mic hiel Renger MSc for n umerous discussions, often on the interpretation of certain concepts. I ap ol- ogize for k eeping you from y our own work, Michiel! The P article Systems seminar at the Institute for Complex Molecular Systems (ICMS) has giv en me a b etter understanding of topics closely related to my problem. I ackno wledge the con tributions of all participants to the seminar. The help of Dr.ir. Bart Markvoort (Dept. of Biomedical Engineering) to my sim ulation co de was of vital imp ortance. Without his program (that serv ed as a basis for m y tw o-scale exp erimen ts) I still w ould ha v e b een trying. Thank y ou! My uncle Jo ep Julic her once again rendered a service to one of his relatives with his photo camera. I am most grateful for his contribu tion to m y thesis. Thanks to all m y fellow studen ts, but particularly to Patric k v an Meurs and Jero en Bogers. I will miss the ‘Three Musk eteers’ of HG 8.64, and the Cup-a-Soup ritual at 12:30. F urthermore, thanks are due to m y friends (esp ecially the ones studying h umanities), for their nev erending ridiculization of mathematics, mathematicians and science in general. Y our at- titude has truly pro vided me with the b est motiv ation to b e p ersistent. My gratitude (of course) go es to my parents, for their supp ort that has alwa ys gone without sa ying, and to Janneke and F rank, Willem and Anneke for taking care of their little brother. I am extremely happy that the ICMS giv es me the opp ortunit y to con tinue my research for another four y ears. 96 App endix A Pro of of Theorem 2.1.4 This pro of follo ws the lines indicated b y [22], p. 42. Pr o of. 1. Define the set E := { Ω 0 ∈ B (Ω)   λ (Ω \ Ω 0 ) = 0 } . Note that E is non-empty since at least Ω ∈ E . Cho ose a sequence { B k } k ∈ N + ⊂ E suc h that µ ( B k ) 6 inf Ω 0 ∈E µ (Ω 0 ) + 1 k , for all k ∈ N + . Suc h sequence exists by definition of the infim um. Define B := T ∞ k =1 B k ; B is an elemen t of B (Ω), since any σ -algebra is closed w.r.t. coun table in tersections. By one of De Morgan’s laws we hav e that Ω \  T ∞ k =1 B k  = S ∞ k =1  Ω \ B k  , for which w e should note that this is not a disjoint union. Therefore we ha ve 0 6 λ (Ω \ B ) 6 ∞ X k =1 λ (Ω \ B k ) = 0 , since λ (Ω \ B k ) = 0 for all k ∈ N + . W e ha v e th us prov en that B ∈ E . In fact w e even ha ve that µ ( B ) = inf Ω 0 ∈E µ (Ω 0 ). T o see this, note that µ ( B ) = µ  ∞ \ k =1 B k  6 µ ( B j ) 6 inf Ω 0 ∈E µ (Ω 0 ) + 1 j , for all j ∈ N + . Since this statemen t is true for an y j ∈ N + , it follo ws that µ ( B ) 6 inf Ω 0 ∈E µ (Ω 0 ). Due to the fact that B ∈ E , it is clear that also µ ( B ) > inf Ω 0 ∈E µ (Ω 0 ), thus µ ( B ) = inf Ω 0 ∈E µ (Ω 0 ). 2. Let µ ac : B (Ω) → R + b e defined by µ ac (Ω 0 ) = µ (Ω 0 ∩ B ), and similarly µ s : B (Ω) → R + b y µ s (Ω 0 ) = µ  Ω 0 ∩ (Ω \ B )  . Note that it follows from this definition that µ = µ ac + µ s . Let A ∈ B (Ω), b e suc h that A ⊂ B and λ ( A ) = 0. Assume that µ ( A ) > 0. Note that B \ A ∈ B (Ω), and λ  Ω \ ( B \ A )  = λ  (Ω \ B ) ∪ A  = λ (Ω \ B ) + λ ( A ) = 0 . Here w e use that Ω \ B and A are disjoint (since A ⊂ B ) and that λ (Ω \ B ) = 0 (since B ∈ E ). W e conclude that B \ A ∈ E . As a result of the fact that A ⊂ B , it holds that µ ( B ) = µ ( B \ A ) + µ ( B ∩ A ) = 97 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology µ ( B \ A ) + µ ( A ), and because w e assumed that µ ( A ) > 0 it follows that µ ( B \ A ) < µ ( B ). As B \ A ∈ E , this contradicts µ ( B ) = inf Ω 0 ∈E µ (Ω 0 ). Th us µ ( A ) = 0 m ust hold. Let Ω 0 ∈ B (Ω) now b e such that λ (Ω 0 ) = 0. Then µ ac (Ω 0 ) = µ (Ω 0 ∩ B ) = 0 follo ws if we set A := Ω 0 ∩ B ⊂ B in the lines of arguments ab ov e. Hence, µ ac  λ . F urthermore µ s ( B ) = µ  B ∩ (Ω \ B )  = µ ( ∅ ) = 0, and, due to the fact that B ∈ E , λ (Ω \ B ) = 0. This implies µ s ⊥ λ . 3. T o pro ve uniqueness of this decomp osition, assume that w e hav e µ = µ 1 ac + µ 1 s and µ = µ 2 ac + µ 2 s , where µ i ac  λ and µ i s ⊥ λ for i ∈ { 1 , 2 } . F or all Ω 0 ∈ B (Ω) w e thus ha ve ( µ 1 ac − µ 2 ac )(Ω 0 ) = ( µ 2 s − µ 1 s )(Ω 0 ) =: ˜ µ (Ω 0 ) . Neither in the left-hand side, nor in the right-hand side we necessarily hav e a p ositive measure. Ho wev er it can quite easily b e seen that ( µ 1 ac − µ 2 ac )  λ , b ecause λ (Ω 0 ) = 0 implies µ i ac (Ω 0 ) = 0 for i = 1 , 2 (as µ i ac  λ ), and thus also ( µ 1 ac − µ 2 ac )(Ω 0 ) = 0. F urthermore there exist B 1 , B 2 ∈ B (Ω), such that µ i s (Ω \ B i ) = λ ( B i ) = 0 for all i ∈ { 1 , 2 } and for all Ω 0 ∈ B (Ω). Define ˜ B := B 1 ∪ B 2 , and note that Ω \ ˜ B ⊂ Ω \ B i for each i ∈ { 1 , 2 } . It follo ws that 0 6 µ i s (Ω \ ˜ B ) 6 µ i s (Ω \ B i ) = 0, for i ∈ { 1 , 2 } , thus µ i s (Ω \ ˜ B ) = 0 and th us ( µ 2 s − µ 1 s )(Ω \ ˜ B ) = 0 . Also, 0 6 λ ( ˜ B ) = λ ( B 1 ) + λ ( B 2 ) = 0, and th us λ ( ˜ B ) = 0 , from whic h we conclude that ( µ 2 s − µ 1 s ) ⊥ λ . W e now hav e ˜ µ  λ and ˜ µ ⊥ λ , and will show that this implies ˜ µ ≡ 0. Use statement (ii) from Lemma 2.1.3 to c haracterize singular measures, b y whic h we hav e disjoin t A 1 , A 2 ∈ B (Ω) such that ˜ µ (Ω 0 ) = ˜ µ (Ω 0 ∩ A 1 ) and λ (Ω 0 ) = λ (Ω 0 ∩ A 2 ) for all Ω 0 ∈ B (Ω). F or an y Ω 0 ∈ B (Ω), we ha v e λ (Ω 0 ) = λ (Ω 0 \ A 2 ) + λ (Ω 0 ∩ A 2 ) = λ (Ω 0 \ A 2 ) + λ (Ω 0 ), whic h implies λ (Ω 0 \ A 2 ) = 0. Because ˜ µ is absolutely contin uous w.r.t. λ , we also hav e ˜ µ (Ω 0 \ A 2 ) = 0. Th us: 0 = ˜ µ (Ω 0 \ A 2 ) = ˜ µ  (Ω 0 \ A 2 ) ∩ A 1  = ˜ µ (Ω 0 ∩ A 1 ) = ˜ µ (Ω 0 ). Since ˜ µ (Ω 0 ) = 0 for all Ω 0 ∈ B (Ω), we ha ve that ˜ µ ≡ 0 and hence µ 1 ac ≡ µ 2 ac and µ 1 s ≡ µ 2 s . 98 Mo delling Cro wd Dynamics App endix B Pro of of Theorem 2.1.9 The inspiration for this pro of comes from [37], pp. 45–46, although the theorem presented here is more general than the one stated in [37]. Pr o of. 1. Due to the fact that µ s is singular w.r.t. λ , we are provided t wo disjoin t sets A 1 , A 2 ∈ B (Ω) such that µ s (Ω 0 ) = µ s (Ω 0 ∩ A 1 ) and λ (Ω 0 ) = λ (Ω 0 ∩ A 2 ) for all Ω 0 ∈ B (Ω). No w define the sequence of sets { B n } n ∈ N + giv en by B 1 := { x ∈ A 1   µ s ( x ) > 1 } , B n := { x ∈ A 1   1 n 6 µ s ( x ) < 1 n − 1 } , for n ∈ { 2 , 3 , . . . } . Eac h B n con tains only finitely many elements, b ecause µ s is a finite measure. It follows that B := S ∞ n =1 B n is a countable set. Note that B = { x ∈ A 1   µ s ( x ) > 0 } . Also note that x / ∈ B implies µ s ( x ) = 0. Indeed, if x ∈ A 1 \ B this is trivial. F urthermore, if x / ∈ A 1 then µ s ( x ) = µ s ( { x } ∩ A 1 ) = µ s ( ∅ ) = 0. 2. Let µ d : B (Ω) → R + b e defined by µ d (Ω 0 ) = µ s (Ω 0 ∩ B ), and similarly µ sc : B (Ω) → R + b y µ sc (Ω 0 ) = µ s  Ω 0 ∩ (Ω \ B )  . Note that it follo ws from this definition that µ s = µ d + µ sc . It is readily seen that µ d is discrete w.r.t. λ , b ecause B is a coun table set of p oints in Ω, for whic h we ha ve µ d (Ω \ B ) = µ d  (Ω \ B ) ∩ B  = µ d ( ∅ ) = 0 , and (since B ⊂ A 1 ) λ ( B ) = λ ( B ∩ A 2 ) 6 λ ( A 1 ∩ A 2 ) = λ ( ∅ ) = 0 . W e no w show that µ sc is singular contin uous w.r.t. λ . Let x ∈ Ω b e arbitrary . µ sc ( x ) = µ s  { x } ∩ (Ω \ B )  . Th us, if x / ∈ Ω \ B then µ sc ( x ) = µ sc ( ∅ ) = 0. On the other hand, if x ∈ Ω \ B then µ sc ( x ) = µ s ( x ) = 0 b ecause x / ∈ B (the last step has b een shown ab ov e). Th us, µ sc ( x ) = 0 for all x ∈ Ω. Consider the set A 1 . First of all, λ ( A 1 ) = λ ( A 1 ∩ A 2 ) = λ ( ∅ ) = 0. Moreo v er, µ sc (Ω \ A 1 ) = µ s  (Ω \ A 1 ) ∩ (Ω \ B )  = µ s (Ω \ A 1 ), b ecause B ⊂ A 1 . Since µ s is singular, we ha ve µ s (Ω \ A 1 ) = µ s  (Ω \ A 1 ) ∩ A 1  = µ s ( ∅ ) = 0. W e no w hav e that µ sc is singular con tinuous w.r.t. λ . 99 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology 3. T o prov e uniqueness of this decomp osition, assume that we hav e µ s = µ 1 d + µ 1 sc and µ s = µ 2 d + µ 2 sc , where µ i d is discrete w.r.t. λ and µ i sc is singular contin uous w.r.t. λ for i ∈ { 1 , 2 } . F or all Ω 0 ∈ B (Ω) w e thus ha ve ( µ 1 d − µ 2 d )(Ω 0 ) = ( µ 2 sc − µ 1 sc )(Ω 0 ) =: ˜ µ (Ω 0 ) . Assume that ˜ A 1 and ˜ A 2 are the coun table sets, suc h that µ i d (Ω \ ˜ A i ) = λ ( ˜ A i ) = 0 for i ∈ { 1 , 2 } . Define A := ˜ A 1 ∪ ˜ A 2 , which is ob viously coun table. W e hav e for each i ∈ { 1 , 2 } , and all Ω ∈ B (Ω) µ i d (Ω 0 ∩ A ) 6 µ i d (Ω 0 ) = µ i d (Ω 0 ∩ A ) + µ i d (Ω 0 \ A ) 6 µ i d (Ω 0 ∩ A ) + µ i d (Ω \ ˜ A i ) = µ i d (Ω 0 ∩ A ) , and th us µ i d (Ω 0 ) = µ i d (Ω 0 ∩ A ). This implies ˜ µ (Ω 0 ) = ˜ µ (Ω 0 ∩ A ). Note that Ω 0 ∩ A is a subset of A and is thus a countable collection of p oints in Ω, sa y { y 1 , y 2 , . . . } . W e can also write Ω 0 ∩ A = ∪ ∞ k =1 { y k } , which is a disjoin t union. W e thus hav e (for each i ∈ { 1 , 2 } ): µ i sc (Ω 0 ∩ A ) = P ∞ k =1 µ i sc ( y k ) = 0, b ecause eac h term of the sum is zero b y definition of singular con tinuous measures. As a result ˜ µ (Ω 0 ) = ˜ µ (Ω 0 ∩ A ) = µ 2 sc (Ω 0 ∩ A ) − µ 1 sc (Ω 0 ∩ A ) = 0 , for all Ω 0 ∈ B (Ω), b y which w e hav e uniqueness of the decomp osition. 100 Mo delling Cro wd Dynamics App endix C Pro of of Lemma 2.3.1 C.1 Pro of of P art 1 of Lemma 2.3.1 This pro of is based on [26], p. 133. Pr o of. By assumption ν is a p ositiv e measure. Note that the Radon-Nik o dym Theorem pro- vides the existence of dν /dµ . F or simplicit y of notation, let us write: dν /dµ = h . W e no w first pro ve that the positivity of ν implies that h > 0 almost ev erywhere w.r.t. µ . The pro of go es by contradiction. Assume there exists an ˜ Ω ∈ B (Ω) satisfying µ ( ˜ Ω) > 0 and h < 0 µ -almost ev erywhere on ˜ Ω. W e in tro duce the following sets: F 0 := { x ∈ Ω   h ( x ) < 0 } , F n := { x ∈ Ω   h ( x ) 6 − 1 n } , for all n ∈ N + . Consider the disjoin t union ˜ Ω = ( ˜ Ω \ F 0 ) ∪ ( ˜ Ω ∩ F 0 ). F rom the assumption that h < 0 µ -almost ev erywhere in ˜ Ω, it follows that µ ( ˜ Ω \ F 0 ) = 0. Note that since ˜ Ω has strictly p ositiv e measure µ ( ˜ Ω), w e get 0 < µ ( ˜ Ω) = µ ( ˜ Ω \ F 0 ) + µ ( ˜ Ω ∩ F 0 ) = µ ( ˜ Ω ∩ F 0 ) . F urthermore, w e hav e for all n ∈ N + 0 6 ν ( ˜ Ω ∩ F n ) = Z ˜ Ω ∩ F n hdµ 6 Z ˜ Ω ∩ F n − 1 n dµ = − 1 n µ ( ˜ Ω ∩ F n ) 6 0 , whic h implies that µ ( ˜ Ω ∩ F n ) = 0 for an y n ∈ N + . Note that F 0 = S ∞ n =1 F n holds. 1 Since µ is a positive measure, this leads to the follow- ing inequalit y: µ ( ˜ Ω ∩ F 0 ) = µ ˜ Ω ∩  ∞ [ n =1 F n  ! 6 ∞ X n =1 µ ( ˜ Ω ∩ F n ) = 0 , 1 If ˜ x ∈ S ∞ n =1 F n then there exists a specific ˜ n ∈ N + suc h that ˜ x ∈ F ˜ n . The following holds: h ( ˜ x ) 6 − 1 / ˜ n < 0, so ˜ x ∈ F 0 . If ˜ x ∈ F 0 then h ( ˜ x ) is strictly negative, so h ( ˜ x ) 6 − 1 / ˜ n < 0 for ˜ n :=  1 /h ( ˜ x )  . Th us: ˜ x ∈ F ˜ n ⊂ S ∞ n =1 F n . 101 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology where the last step follows from the fact that µ ( ˜ Ω ∩ F n ) = 0 for any n ∈ N + . W e now hav e a contradiction with the statement µ ( ˜ Ω ∩ F 0 ) > 0. Thus, there is no suc h set ˜ Ω of p ositive measure, on which h < 0 almost ev erywhere w.r.t. µ . This implies that h > 0 µ -almost ev erywhere in Ω. No w let h satisfy h ( x ) > 0 for all x ∈ Ω (note that it is p ossible to choose such a repre- sen tant). Ev ery nonnegativ e measurable function is the (p oint wise) limit of an increasing sequence of nonnegativ e simple functions; assume that { h k } ∞ k =1 is such sequence. Simple functions are alw ays measurable, and therefore the follo wing is true: lim k →∞ Z Ω h k dµ = Z Ω hdµ, see [26], p. 112. This limit is also v alid if the domain of in tegration is an y arbitrary Ω 0 ∈ B (Ω), as we will show no w. Let 1 Ω 0 denote the c haracteristic function of Ω 0 . The sequence { h k 1 Ω 0 } is increasing and consists of nonnegative, measurable functions (pro ducts of t wo measurable functions, are again measurable). F urthermore 1 Ω 0 h is (b y definition of { h k } ) the p oint wise limit of the sequence { h k 1 Ω 0 } . Consequen tly (cf. [26], p. 112): lim k →∞ R Ω h k 1 Ω 0 dµ = R Ω h 1 Ω 0 dµ , whic h can also b e written as lim k →∞ Z Ω 0 h k dµ = Z Ω 0 hdµ, for all Ω 0 ∈ B (Ω) . F or con venience, write dµ/dλ = p (the existence of which is guaran teed by the Radon- Nik o dym Theorem). F ollo wing the same arguments as for h , we can conclude that it is p ossible to tak e a representativ e of p , suc h that p ( x ) > 0 for all x ∈ Ω. The Radon-Nik o dym Theorem furthermore ensures the measurabilit y of p . The sequence { h k p } ∞ k =1 consists th us of measurable functions, as eac h elemen t is the pro duct of tw o measurable functions. Both p and h k (for all k ∈ N + ) are nonnegative, so the same holds for the elements of the sequence. The sequence is increasing, b ecause { h k } is increasing. As a result, we can conclude that lim k →∞ R Ω h k pdλ = R Ω hpdλ (cf. [26], p. 112). Multiplication with 1 Ω 0 and some further reasoning yields, lik e b efore, that lim k →∞ Z Ω 0 h k pdλ = Z Ω 0 hpdλ, for all Ω 0 ∈ B (Ω) . The functions h k are simple functions, so for each k ∈ N + there exists a finite collection { A ( k ) i } N ( k ) i =1 of m utually disjoint measurable subsets of Ω, where N ( k ) ∈ N + , suc h that h k ( x ) = N ( k ) X i =1 α ( k ) i 1 A ( k ) i ( x ) , for all x ∈ Ω , for some collection of co efficients { α ( k ) i } N ( k ) i =1 ⊂ R + . F or any measurable set A , the following iden tity holds: Z Ω 0 1 A dµ = µ (Ω 0 ∩ A ) = Z Ω 0 ∩ A pdλ = Z Ω 0 1 A pdλ, for all Ω 0 ∈ B (Ω) . 102 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology Therefore Z Ω 0 h k dµ = Z Ω 0 N ( k ) X i =1 α ( k ) i 1 A ( k ) i dµ = N ( k ) X i =1 α ( k ) i Z Ω 0 1 A ( k ) i dµ = N ( k ) X i =1 α ( k ) i Z Ω 0 1 A ( k ) i pdλ = Z Ω 0 N ( k ) X i =1 α ( k ) i 1 A ( k ) i pdλ = Z Ω 0 h k pdλ, for all Ω 0 ∈ B (Ω) and for all k ∈ N + . R Ω 0 hdµ and R Ω 0 hpdλ are thus limit v alues of the same sequence, which means they must b e equal: ν (Ω 0 ) = R Ω 0 hdµ = R Ω 0 hpdλ . F rom this, it follows that the integrand hp is in fact the Radon-Nik o dym deriv ativ e dν /dλ , which finishes the pro of. C.2 Pro of of P art 2 of Lemma 2.3.1 Pr o of. Define the p ositive part g + and the negativ e part g − of the function g by g + ( x ) := max { g ( x ) , 0 } , g − ( x ) := − min { g ( x ) , 0 } . These t wo functions are measurable, b ecause g is measurable (see [46], p. 15). Assume for the momen t that neither of g + and g − is iden tically zero. If w e define for all Ω 0 ∈ B (Ω) the follo wing: γ + / − (Ω 0 ) := Z Ω 0 g + / − dµ, then γ + and γ − are measures, e.g. due to [46] (p. 23). Note that b y the sup erscript ‘+ / − ’, we indicate that the statement holds in b oth the ‘+’ case and the ‘ − ’ case. Both g + and g − are nonnegativ e, so γ + and γ − are p ositive measures. g + / − (Ω) > 0 follo ws from the assumption that g + and g − are not identically zero. Since g ∈ L 1 µ (Ω), it follo ws that γ + / − (Ω) < ∞ . The Radon-Nik o dym Theorem can now b e applied to the measures γ + and γ − , and pro- vides the existence of dγ + / − /dµ . The uniqueness of Radon-Nikodym deriv ativ es implies that dγ + / − /dµ = g + / − . No w apply Part 1 of Lemma 2.3.1, by setting ν = γ + and ν = γ − subsequen tly . Note that Mo delling Cro wd Dynamics 103 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology γ + / −  µ is guaranteed by defining γ + / − as an integral with resp ect to µ (cf. [26], p. 104). The result is Z Ω 0 g + / − dµ = γ + / − (Ω 0 ) = Z Ω 0 dγ + / − dλ dλ = Z Ω 0 dγ + / − dµ dµ dλ dλ = Z Ω 0 g + / − dµ dλ dλ, for all Ω 0 ∈ B (Ω), where P art 1 has b een used in the third equalit y . Note that Z Ω 0 g + / − dµ = Z Ω 0 g + / − dµ dλ dλ is automatically satisfied if g + / − ≡ 0. W e can th us pro ceed without distinguishing betw een g + / − ≡ 0 and g + / − 6≡ 0. Finally , w e conclude that for all Ω 0 ∈ B (Ω) Z Ω 0 g dµ = Z Ω 0 ( g + − g − ) dµ = Z Ω 0 g + dµ − Z Ω 0 g − dµ = Z Ω 0 g + dµ dλ dλ − Z Ω 0 g − dµ dλ dλ = Z Ω 0 ( g + − g − ) dµ dλ dλ = Z Ω 0 g dµ dλ dλ, b y which w e hav e the desired statement. C.3 Pro of of P art 3 of Lemma 2.3.1 Pr o of. Set ν = λ and apply P art 1, whic h states dλ dλ = dλ dµ dµ dλ . W e now claim that dλ/dλ ≡ 1, whic h can easily b e sho wn. Obviously λ  λ , thus the Radon-Nik o dym Theorem yields the follo wing: λ (Ω 0 ) = Z Ω 0 dλ dλ dλ, for all Ω 0 ∈ B (Ω) . Ho wev er, also λ (Ω 0 ) = R Ω 1 Ω 0 dλ = R Ω 0 1 dλ holds. The uniqueness of Radon-Nik o dym deriv a- tiv es implies that dλ/dλ ≡ 1. W e thus hav e 1 = dλ dµ dµ dλ , 104 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology and hence dµ dλ =  dλ dµ  − 1 . C.4 Pro of of P art 4 of Lemma 2.3.1 Pr o of. Let Ω 0 ∈ B (Ω 1 × Ω 2 ) b e arbitrary . Define the following set: Ω 0 1 := { x 1 ∈ Ω 1   ( x 1 , x 2 ) ∈ Ω 0 for some x 2 ∈ Ω 2 } . Also define Ω 0 x 1 := { x 2 ∈ Ω 2   ( x 1 , x 2 ) ∈ Ω 0 } , for any choice of x 1 ∈ Ω 1 . The set Ω 0 x 1 is called the Ω 1 - se ction of Ω 0 determined by x 1 (cf. [26], p. 141). Note that Ω 0 x 1 is a subset of Ω 2 . In fact, x 1 can b e regarded as a parameter here. It can be prov ed that every section of a measurable set is a measurable set (see [26], p. 141). W e first pro ve that ν 1 ⊗ ν 2  µ 1 ⊗ µ 2 . Assume that Ω 0 ∈ B (Ω 1 × Ω 2 ) is suc h that ( µ 1 ⊗ µ 2 )(Ω 0 ) = 0. By definition (see [26], p. 144) of the pro duct measure µ 1 ⊗ µ 2 w e hav e ( µ 1 ⊗ µ 2 )(Ω 0 ) = Z Ω 0 1 µ 2 (Ω 0 x 1 ) dµ 1 ( x 1 ) = 0 . As a consequence (cf. [26], p. 147) µ 2 (Ω 0 x 1 ) = 0 , for µ 1 -a.e. x 1 ∈ Ω 0 1 . Define Ω 0 µ 2 := { x 1 ∈ Ω 0 1   µ 2 (Ω 0 x 1 ) 6 = 0 } . Let ˜ x 1 b e such that ˜ x 1 / ∈ Ω 0 µ 2 , then of course µ 2 (Ω 0 ˜ x 1 ) = 0. F rom the hypothesis that ν 2  µ 2 , it follo ws that ν 2 (Ω 0 ˜ x 1 ) = 0. If w e define Ω 0 ν 2 := { x 1 ∈ Ω 0 1   ν 2 (Ω 0 x 1 ) 6 = 0 } , then w e hav e thus prov ed that Ω 0 ν 2 ⊂ Ω 0 µ 2 . Since µ 2 (Ω 0 x 1 ) = 0 for µ 1 -a.e. x 1 ∈ Ω 0 1 , we clearly hav e that µ 1 (Ω 0 µ 2 ) = 0. By hypothesis of the lemma ν 1 is absolutely contin uous w.r.t. µ 1 , and th us µ 1 (Ω 0 µ 2 ) = 0 implies ν 1 (Ω 0 µ 2 ) = 0. W e no w use Ω 0 ν 2 ⊂ Ω 0 µ 2 , to conclude ν 1 (Ω 0 ν 2 ) 6 ν 1 (Ω 0 µ 2 ) = 0 , b y which w e find that ν 1 (Ω 0 ν 2 ) = 0. This statemen t can also b e written as ν 2 (Ω 0 x 1 ) = 0 , for ν 1 -a.e. x 1 ∈ Ω 0 1 . Mo delling Cro wd Dynamics 105 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology It follo ws trivially that ( ν 1 ⊗ ν 2 )(Ω 0 ) = Z Ω 0 1 ν 2 (Ω 0 x 1 ) dν 1 ( x 1 ) = 0 . W e ha ve th us shown that ν 1 ⊗ ν 2  µ 1 ⊗ µ 2 . W e can now apply the Radon-Nik o dym Theorem, whic h pro vides the existence of the unique Radon-Nik o dym deriv ativ e d ( ν 1 ⊗ ν 2 ) d ( µ 1 ⊗ µ 2 ) . Let Ω 0 ∈ B (Ω 1 × Ω 2 ) b e arbitrary , and let 1 A denote the characteristic function of the set A , where A is a set in B (Ω 1 ), B (Ω 2 ) or B (Ω 1 × Ω 2 ). By hypothesis of the lemma the Radon-Nik o dym deriv atives dν 1 /dµ 1 and dν 2 /dµ 2 exist. W e can now p erform the follo wing calculations: ( ν 1 ⊗ ν 2 )(Ω 0 ) = Z Ω 0 1 ν 2 (Ω 0 x 1 ) dν 1 ( x 1 ) = Z Ω 0 1 Z Ω 0 x 1 dν 2 ( x 2 ) ! dν 1 ( x 1 ) = Z Ω 0 1 Z Ω 0 x 1 dν 2 dµ 2 dµ 2 ( x 2 ) ! dν 1 dµ 1 dµ 1 ( x 1 ) = Z Ω 0 1 Z Ω 0 x 1 dν 1 dµ 1 dν 2 dµ 2 ! dµ 2 ( x 2 ) dµ 1 ( x 1 ) = Z Ω 1 Z Ω 2 dν 1 dµ 1 dν 2 dµ 2 1 Ω 0 1 ( x 1 ) 1 Ω 0 x 1 ( x 2 ) ! dµ 2 ( x 2 ) dµ 1 ( x 1 ) = Z Ω 1 × Ω 2 dν 1 dµ 1 dν 2 dµ 2 1 Ω 0 1 ( x 1 ) 1 Ω 0 x 1 ( x 2 ) ! d ( µ 1 ⊗ µ 2 )( x 1 , x 2 ) , where the last step follo ws from [26], p. 147. Let ( ˜ x 1 , ˜ x 2 ) b e an element of Ω 1 × Ω 2 . By definition of Ω 0 ˜ x 1 , w e hav e that 1 Ω 0 ˜ x 1 ( ˜ x 2 ) = 1 if and only if ( ˜ x 1 , ˜ x 2 ) ∈ Ω 0 . If ( ˜ x 1 , ˜ x 2 ) ∈ Ω 0 , then also 1 Ω 0 1 ( ˜ x 1 ) = 1, b y definition of Ω 0 1 . W e th us ha v e 1 Ω 0 1 ( ˜ x 1 ) 1 Ω 0 ˜ x 1 ( ˜ x 2 ) = 1 . On the other hand, if ( ˜ x 1 , ˜ x 2 ) / ∈ Ω 0 then 1 Ω 0 ˜ x 1 ( ˜ x 2 ) = 0, and th us 1 Ω 0 1 ( ˜ x 1 ) 1 Ω 0 ˜ x 1 ( ˜ x 2 ) = 0 . 106 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology W e conclude that for all ( x 1 , x 2 ) ∈ Ω 1 × Ω 2 the follo wing identit y holds: 1 Ω 0 1 ( x 1 ) 1 Ω 0 x 1 ( x 2 ) = 1 Ω 0 ( x 1 , x 2 ) . This means that w e can write ( ν 1 ⊗ ν 2 )(Ω 0 ) = Z Ω 1 × Ω 2 dν 1 dµ 1 dν 2 dµ 2 1 Ω 0 ( x 1 , x 2 ) ! d ( µ 1 ⊗ µ 2 )( x 1 , x 2 ) = Z Ω 0 dν 1 dµ 1 dν 2 dµ 2 ! d ( µ 1 ⊗ µ 2 ) . Uniqueness of the Radon-Nik o dym deriv ativ e now makes sure that d ( ν 1 ⊗ ν 2 ) d ( µ 1 ⊗ µ 2 ) = dν 1 dµ 1 dν 2 dµ 2 , whic h finishes the pro of. Mo delling Cro wd Dynamics 107 App endix D Deriv ation of the en trop y densit y for an ideal gas The following deriv ation is due to [50]. More details can b e found in [12] (pp. 16–19 and 96–102). W e consider a gas that is compressible, non-viscous and thermal (i.e. the temp erature in- fluences its b ehaviour). The set of indep endent v ariables is Q := { ρ, T , j } , where ρ is the densit y , T the absolute temp erature and j = ∇ T , the temp erature gradien t. All other de- p enden t v ariables are functions of Q . W e assume the follo wing conserv ation laws: 1. Balance of mass: ∂ ρ ∂ t + ∇ · ( ρv ) = 0 . (D.1) 2. Balance of energy: ρ D ε D t = − p ∇ · v + ∇ · q + ρr. (D.2) 3. Entrop y inequality (Clausius-Duhem): ρ D η D t − ∇ ·  q T  − ρr T > 0 . (D.3) In the balance of energy , ε is the in ternal energy densit y . F urthermore, q denotes the heat flux and r is the heat supply densit y . The entrop y inequality is the lo cal version of the Clausius- Duhem Inequality . It is precisely the inequalit y given by Postulate 3.5.1, for one comp onen t. Moreo ver j η = q /T has b een taken, as w as also suggested b y (1.6.9) in [11] (the results on p. 29 ha ve to be reduced to one comp onen t to see this). Note that w e require T > 0 for the entrop y inequality to make sense. Since (D.1) can also b e written as 0 = ∂ ρ ∂ t + ρ ∇ · v + ∇ ρ · v = D ρ D t + ρ ∇ · v , (D.4) 108 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology it follo ws that ∇ · v = − 1 ρ D ρ D t . Note that ρ D ε D t = ρ  ∂ ε ∂ t + ∇ ε · v  = ρ ∂ ε ∂ t + ρ ∇ ε · v + ε  ∂ ρ ∂ t + ∇ · ( ρv )  | {z } =0 = ρ ∂ ε ∂ t + ε ∂ ρ ∂ t + ∇ ε · ( ρv ) + ε ∇ · ( ρv ) = ∂ ∂ t ( ρε ) + ∇ · ( ρv ε ) , and a similar iden tity holds if w e take η instead of ε . Let F denote the Helmholtz fr e e ener gy , defined b y F := ε − η T . T aking the deriv ativ e D /D t in the definition of F and m ultiplicating afterw ards b y ρ , leads to ρ D ε D t = ρ D F D t + ρη D T D t + ρT D η D t = − p ∇ · v + ∇ · q + ρr. (D.5) The latter equality is due to the balance of energy (D.2). Equation (D.5) can also b e written as ρT D η D t = − ρ D F D t − ρη D T D t − p ∇ · v + ∇ · q + ρr. (D.6) A suitable combination of the entrop y inequalit y (D.3) and (D.6), ∇ · v = − 1 ρ D ρ D t and of the fact that T > 0, yields − ρ D F D t − ρη D T D t + p ρ D ρ D t = ρT D η D t − ∇ · q − ρr = ρT D η D t − ∇ ·  q T T  − ρr = ρT D η D t − T ∇ ·  q T  − q T · ∇ T − ρr = T  ρ D η D t − ∇ ·  q T  − ρr T  | {z } > 0 − q T · ∇ T > − q T · ∇ T , (D.7) or − ρ D F D t − ρη D T D t + p ρ D ρ D t + 1 T q · ∇ T > 0 . (D.8) Since F is a depends only on Q = { ρ, T , j } , we hav e D F D t = ∂ F ∂ ρ D ρ D t + ∂ F ∂ T D T D t + ∂ F ∂ j D j D t , Mo delling Cro wd Dynamics 109 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology b y which (D.8) transforms in to  p ρ − ρ ∂ F ∂ ρ  D ρ D t − ρ  η + ∂ F ∂ T  D T D t − ρ ∂ F ∂ j D j D t + 1 T q · ∇ T > 0 . The co efficien ts of D ρ/D t , D T /D t and D j /D t , and 1 T q · ∇ T are indep endent of D ρ/D t , D T /D t and D j /D t . F urthermore, D ρ/D t , D T /D t and D j /D t can be chosen arbitrarily (p oin t wise), and th us the follo wing c onstitutive e quations must hold p = ρ 2 ∂ F ∂ ρ , η = − ∂ F ∂ T , and ∂ F ∂ j = 0 , (D.9) and since T > 0 also q · ∇ T > 0 . (D.10) Remark D.1. It follo ws even tually from (D.10) that q = λ ( ρ, T , j ) ∇ T , where λ > 0. If line ar thermal c onduction is assumed (that is, q is linear w.r.t. j ), then λ = λ ( ρ, T ) and thus F ourier’s La w is obtained. Ho wev er, this result is not so imp ortant in the sequel. If w e wan t a more explicit form of the constitutive equations in (D.9), we are required to giv e also an explicit choice of F . It is clear from the third constitutive equation in (D.9) that F = F ( ρ, T ) . Definition D.2 (Ideal gas) . A gas is c al le d ide al gas, if F ( ρ, T ) := c ( T − T 0 ) − cT log  T T 0  + RT log  ρ ρ 0  , wher e T 0 and ρ 0 ar e an arbitr ary r efer enc e temp er atur e and density, c is the sp e cific he at, and R is the universal gas c onstant. In fact, c = c V is the sp ecific heat at constant volume, which can only b e a function of T (cf. [52], p. 114). W e hav e made the assumption that c V is constant. In nature, this assump- tion is only v alid ov er wide temp erature ranges for monoatomic gases; see [52], pp. 114–115. F or the sak e of clarity , we will not go in to details for non-constant sp ecific heat c V = c V ( T ). F or an ideal gas, it follo ws from (D.9) that p = R ρT , and η = c log  T T 0  − R log  ρ ρ 0  . In p = R ρT w e recognize the ideal gas la w. F urthermore, note that ε can b e determined explicitly using ε = F + η T . The in ternal energy is a function of the temp erature only: ε = ε ( T ) = c ( T − T 0 ). 110 Mo delling Cro wd Dynamics App endix E Mo difications in the pro ofs of Theorem 4.5.1 and Corollary 4.5.4 W e describ e the details that need to b e adapted in the proofs of Theorem 4.5.1 (global existence of the time-discrete solution) and Corollary 4.5.4 (conserv ation of mass) to make them compatible with the new assumptions made in Section 4.5.3. E.1 The pro of of Theorem 4.5.1 In Part 2 of the pro of of Theorem 4.5.1, the identit y ( χ α n ) − 1 (Ω) = Ω is used. This iden tit y holds if the motion mapping is in v ertible. Using the pre-image, we only hav e that ( χ α n ) − 1 (Ω) ⊂ Ω . W e mo dify the pro of that the absolutely con tin uous part µ α ac ,n +1 is finite accordingly , by inserting an inequalit y: µ α ac ,n +1 (Ω) = µ α ac ,n  ( χ α n ) − 1 (Ω)  6 µ α ac ,n (Ω) < ∞ . Consider the argumen ts to prov e that µ α d ,n +1 is finite in P art 3 of the pro of of Theorem 4.5.1. There, we used that { x i } i ∈J ⊂ Ω implies { χ α n ( x i ) } i ∈J ⊂ Ω due to the fact that the motion mapping is a homeomorphism mapping from Ω to Ω. Note that { x i } i ∈J ⊂ supp µ α n holds. In Section 4.5.3 we require that { χ α n   supp µ α n maps from supp µ α n to Ω. Thus { χ α n ( x i ) } i ∈J ⊂ Ω is still true. Regarding P art 4 of the pro of, note that Ω \ B n =  χ α n  − 1  χ α n  Ω \ B n   no longer holds if pre-images are used. Instead we hav e that Ω \ B n ⊂  χ α n  − 1  χ α n  Ω \ B n   . W e need to pro ve that if µ α sc ,n  Ω \ B n  = 0 111 T ec hnische Univ ersiteit Eindhov en Universit y of T echnology holds, then µ α sc ,n +1  Ω \ χ α n  B n   = 0 . Since B n ⊂ ( χ α n ) − 1  χ α n ( B n )  , w e hav e that Ω \ ( χ α n ) − 1  χ α n ( B n )  ⊂ Ω \ B n . It follo ws that 0 = µ α sc ,n (Ω \ B n ) > µ α sc ,n  Ω \ ( χ α n ) − 1  χ α n ( B n )   , th us µ α sc ,n  Ω \ ( χ α n ) − 1  χ α n ( B n )   = 0 . No w use that ( χ α n ) − 1 (Ω) ⊂ Ω , and ( χ α n ) − 1 (Ω) \ ( χ α n ) − 1  χ α n ( B n )  = ( χ α n ) − 1  Ω \ χ α n ( B n )  , to obtain µ α sc ,n  Ω \ ( χ α n ) − 1  χ α n ( B n )   > µ α sc ,n  ( χ α n ) − 1 (Ω) \ ( χ α n ) − 1  χ α n ( B n )   = µ α sc ,n  ( χ α n ) − 1  Ω \ χ α n ( B n )  = µ α sc ,n +1  Ω \ χ α n  B n   . Because µ α sc ,n  Ω \ ( χ α n ) − 1  χ α n ( B n )   = 0 . w e also hav e that µ α sc ,n +1  Ω \ χ α n  B n   = 0 , and th us we are done. W e also need to show that λ d ( B n ) = 0 implies that λ d  χ α n ( B n )  = 0. This follows directly from the newly made assumption (cf. Section 4.5.3) that there is a constant c n > 0 such that c n λ d  χ α n (Ω 0 )  6 λ d (Ω 0 ) , for all Ω 0 ∈ B (Ω) . When proving finiteness of µ α sc ,n +1 w e again can only use that ( χ α n ) − 1 (Ω) ⊂ Ω instead of ( χ α n ) − 1 (Ω) = Ω. It follo ws that µ α sc ,n +1 (Ω) = µ α sc ,n  ( χ α n ) − 1 (Ω)  6 µ α sc ,n (Ω) < ∞ . 112 Mo delling Cro wd Dynamics T ec hnische Univ ersiteit Eindhov en Universit y of T echnology E.2 The pro of of Corollary 4.5.4 W e redo the pro of of Corollary 4.5.4 completely , b ecause it is slightly more inv olved. Pr o of. F or eac h n ∈ { 0 , 1 , . . . , N T − 1 } and α ∈ { 1 , 2 , . . . , ν } , consider the measure µ α ω ,n , where ω ∈ { ac, d, sc } . By definition of µ α ω ,n +1 (see the constructive pro of of Theorem 4.5.1, P arts 2, 3 and 4): µ α ω ,n +1 := χ α n # µ α ω ,n . F or eac h n we th us hav e µ α ω ,n +1 (Ω) = µ α ω ,n  ( χ α n ) − 1 (Ω)  . Note that supp µ α ω ,n ⊂ supp µ α n ⊂ ( χ α n ) − 1 (Ω). This implies that µ α ω ,n +1 (Ω) = µ α ω ,n  ( χ α n ) − 1 (Ω)  = µ α ω ,n  ( χ α n ) − 1 (Ω) ∩ supp µ α n  + µ α ω ,n  ( χ α n ) − 1 (Ω) \ supp µ α n  = µ α ω ,n  supp µ α n  + µ α ω ,n  ∅  = µ α ω ,n  supp µ α n ∩ supp µ α ω ,n  + µ α ω ,n  supp µ α n \ supp µ α ω ,n  | {z } =0 = µ α ω ,n  supp µ α ω ,n  = µ α ω ,n (Ω) . By an inductiv e argument: µ α ω ,n (Ω) = µ α ω , 0 (Ω) for eac h n . Since µ α n = µ α ac ,n + µ α d ,n + µ α sc ,n for all n ∈ N , the abov e implies trivially that µ α n (Ω) = µ α 0 (Ω) , for all n ∈ N . Mo delling Cro wd Dynamics 113 App endix F P ap er ‘Mo deling micro-macro p edestrian coun terflo w in heterogeneous domains’ The follo wing pages contain the full conten t of [23]: our pap er ‘Mo deling micro-macro p edes- trian counterflo w in heterogeneous domains’, published in Nonline ar Phenomena in Complex Systems . 114 Mo deling micro-macro p edestrian coun terflow in heterogeneous domains Jo ep Evers ∗ Dep artment of Mathematics and Computer Scienc e Eindhoven University of T e chnolo gy PO Box 513, 5600 MB Eindhoven, The Netherlands Adrian Mun tean † Centr e for Analysis, Scientific Computing and Applic ations (CASA) Dep artment of Mathematics and Computer Scienc e Institute for Complex Mole cular Systems (ICMS) Eindhoven University of T e chnolo gy PO Box 513, 5600 MB Eindhoven, The Netherlands (Dated: March 11, 2011) W e presen t a micro-macro strategy able to describ e the dynamics of crowds in heterogeneous spatial domains. Herein we fo cus on the example of pedestrian counterflo w. The main working tools include the use of mass and p orosity measures together with their transp ort as well as suitable application of a v ersion of Radon-Nikodym Theorem formulated for finite measures. Finally , we illustrate numerically our microscopic model and emphasize the effects produced by an implicitly defined so cial v elo city . P ACS num b ers: 89.75.Fb; 02.30.Cj; 02.60.Cb; 47.10.ab; 45.50.Jf; 47.56.+r Keywords: Crowd dynamics; mass measures; p orosity measure; so cial netw orks I. INTRODUCTION One of the most annoying examples of collective b eha vior[1] is p edestrian jams – p eople get clogged up together and cannot reach within the desired time the target destination. Suc h jams are the immediate consequence of the simple exclusion pr o c ess [2, 3], whic h basically says that t wo individuals cannot occupy the same p osition x ∈ Ω ⊂ R d at the same time t ∈ S :=]0 , T [, where T ∈ ]0 , ∞ [ is the final momen t at whic h w e are still observing our social netw ork. Observ ational data (cf. e.g. [4]) clearly indicates that such jams typically take place in certain neigh- b orho o ds of b ottlenecks[5] (narrow corridors, exits, corners, inner obstacles/pillars, ...). The effect of heterogeneities[6] on the ov erall dynamics of the crowd is what motiv ates our work. In this paper w e start off with the assumption that inside a given ro om (e.g. a shopping mall), whic h we denote by Ω, there are a priori known zones with restricted access for p edestrians (e.g. closed ro oms, prohibited access areas, inner concrete structures)[7], whose union we call Ω s . Let us also assume that the remaining region, say Ω p , whic h is defined by Ω p := Ω − Ω s , is connected. Consequen tly , Ω p is accessible to p edestrians. The exits of Ω – target that each p edestrian wan ts to reach – are assumed to belong to the b oundary of Ω p . The wa y we imagine the heterogeneit y of Ω is sk etched in Figure 1. ∗ j.h.m.evers@studen t.tue.nl; Corresponding author. † a.mun tean@tue.nl FIG. 1. Sc hematic representation of the heterogeneous medium Ω. The little black discs represent the pedestrians, while the dark gray zones are the parts where the pedestri- ans cannot p enetrate (i.e. subsets of Ω s ). The p edestrians are considered here to be the microscopic entities, while the ligh t gra yish shado w indicates a macroscopic crowd; see Sec- tion I I A for the precise distinction b etw een micr o and macr o made in terms of supp orts of micro and macro measures. In this framework, we c ho ose for the following w orking plan: Firstly , we extend the multiscale approach devel- op ed by Piccoli et al. [8] (see also the con text describ ed in [9] and [10]) to the case of coun terflow[11] of pedestri- ans; then we allo w the p edestrian dynamics to take place in the heterogeneous domain Ω, and finally , w e include an implicit v elo city la w for the p edestrians motion. The main reason wh y w e c ho ose the counterflo w scenario [also Mo delling Cro wd Dynamics 115 2 called bidirectional flow [12]] out of the man y other w ell- studied cro wd dynamics scenarios is at least threefold: (i) Pedestrians counterflo w is often encountered in the ev eryda y life: at p edestrian traffic lights, or just observ e next week-end, when you go shopping, the dynamics of people coming against y our w alking di- rection [esp ecially if you are positioned inside nar- ro w corridors]. (ii) The walk ers trying to mov e faster by a voiding local in teractions with the oncoming p edestrians facili- tate the o ccurrence of a well-kno wn self-organized macroscopic pattern – lane formation; see, for in- stance [13]. (iii) W e expect the solution to microscopic mo dels p osed in narro w corridors to b e computationally c heap. Consequen tly , extensiv e sensitivity analyses can b e p erformed and the corresp onding sim ulation results can b e in principle tested against existing experi- men tal observ ations [4, 14]. The presence of heterogeneities is quite natural. Pedes- trians t ypically follow existing streets, walking paths, they trust building maps, etc. They tak e into account the local environmen t of the place where they are lo cated. If the n umber of p edestrians is relatively high compared to the a v ailable walking space, then the cr owd-structur e inter action b ecomes of vital importance; see e.g. [15] for preliminary results in this direction. As long term plan, w e wish to understand what are the microscopic mechanisms b ehind the formation of lanes in heterogeneous environmen ts. In other words, w e aim at identifying links b et ween so cial for c e -type microscopic mo dels (see [16, 17], e.g.) and macroscopic mo dels for lanes (see [13, 18], e.g.) in the presence of heterogeneities. Here w e follo w a measure-theoretical approac h to describe the dynamics of crowds[19]. Our w orking strategy is very muc h inspired by the works b y M. B¨ ohm [20] and Piccoli et al. [8]. The paper is organized as follo ws: In Section I I w e in tro duce basic mo deling concepts defining the mass and p orosit y measures needed here, as w ell as a coupled system of transp ort e quations for measures. In Section I I I we presen t our concept of so cial velo city . Section IV contains the main result of our paper – the weak form ulation of a micro-macro system for p edestrians mo ving in heterogeneous domains. W e close the pap er with a numerical illustration of our microscopic mo del (Section V) exhibiting effects induced by an implicitly defined v elo city . I I. MODELING WITH MASS MEASURES. THE POR OSITY MEASURE F or basic concepts of measure theory and their inter- pla y with mo deling in materials and life science, we re- fer the reader, for instance, to [21] and resp ectively to [9, 20, 22]. A. Mass measure Let Ω ⊂ R d b e a domain (read: ob ject, bo dy) with mass. Since we hav e in mind physically relev ant situations only , w e consider d ∈ { 1 , 2 , 3 } . How ever, most of the considerations rep orted here do not dep end on the c hoice of the space dimension d . Let µ m (Ω 0 ) be defined as the mass in Ω 0 ⊂ Ω. Note that whenever we write Ω 0 ⊂ Ω, we actually mean that Ω 0 is suc h that Ω 0 ∈ B (Ω), where B (Ω) the σ -algebra of the Borel subsets of Ω. As a rule, w e assume µ m to be defined on all elemen ts of B (Ω). In Sections I I A 1 and I I A 2, we consider tw o spe- cific interpretations of this mass measure that w e need to describe the b ehavior of p edestrians at tw o separated spatial scales. 1. Microsc opic mass me asur e Supp ose that Ω con tains a collection of N p oint masses (eac h of them of mass scaled to 1), and denote their po- sitions b y { p k } N k =1 ⊂ Ω, for N ∈ N . W e wan t µ m to be a counting measure (see Section 1.2.4 in [23], e.g.) with resp ect to these point masses, i.e. for all Ω 0 ∈ B (Ω): µ m (Ω 0 ) = # { p k ∈ Ω 0 } . (1) This can b e achiev ed by representing µ m as the sum of Dirac measures, with their singularities located at the p k , k ∈ { 1 , 2 , . . . , N } , namely: µ m = N X k =1 δ p k . (2) W e refer to the measure µ m defined b y (2) as micr osc opic mass me asur e . 2. Macrosc opic mass me asur e Let us no w consider another example of mass measure µ m . T o do this, we assume that the following p ostulate applies to µ m : P ostulate I I.1 (Assumptions on µ m ) (i) µ m > 0 . (ii) µ m is σ -additive. 116 Mo delling Cro wd Dynamics 3 (iii) µ m  λ d , wher e λ d is the L eb esgue-measur e in R d . By P ostulate II.1 (i) and (ii), we hav e that µ m is a p osi- tiv e measure on Ω, whereas (iii) implies that there is no mass present in a set that has no volume (w.r.t. λ d ). A mass measure satisfying Postulate I I.1 is in this con- text referred to as a macr osc opic mass me asur e . Radon- Nik o dym Theorem[24] (see [21] for more details on this sub ject) guaran tees the existence of a real, non-negativ e densit y ˆ ρ ∈ L 1 λ d (Ω) suc h that: µ m (Ω 0 ) = Z Ω 0 ˆ ρ ( x ) dλ d ( x ) for all Ω 0 ∈ B (Ω) . (3) Similarly , we introduce time-dep endent mass measures µ t , where the time slice t ∈ S enters as a parameter. B. Porosit y measure Let Ω ⊂ R d b e a heterogeneous domain comp osed of t w o distinct regions: fr e e sp ac e for p edestrian motion and a matrix (obstacles) such that Ω = Ω s ∪ Ω p (disjoin t union), where Ω s is the matrix (solid part) of Ω and Ω p is the free space (p ores). This notation is very muc h inspired b y the mo deling of transp ort and chemical reactions in porous media; see [25], e.g. Let µ p (Ω 0 ) be the volume of p ores in Ω 0 ⊂ Ω. P ostulate I I.2 (Assumptions on µ p ) (i) µ p > 0 . (ii) µ p is σ -additive. (iii) µ p  λ d . By P ostulate I I.2 (i) and (ii), w e ha ve that µ p is a measure on Ω. W e refer to µ p as a p or osity me asur e (cf. [20]). The absolute con tinuit y statement in (iii) formulates mathe- matically that there cannot be a non-zero v olume of pores included in a set that has zero volume (w.r.t. λ d ). As- sume that Ω is such that λ d (Ω) < ∞ . Then the Radon- Nik o dym Theorem ensures the existence of a function φ ∈ L 1 + (Ω) suc h that: µ p (Ω 0 ) = Z Ω 0 φdλ d for all Ω 0 ∈ B (Ω) . (4) Note that µ p (Ω 0 ) measures the volume of a subset of Ω 0 (namely of Ω 0 ∩ Ω p ). So, w e get that µ p (Ω 0 ) = λ d (Ω 0 ∩ Ω p ) 6 λ d (Ω 0 ) for all Ω 0 ∈ B (Ω) . (5) W e thus hav e R Ω 0 φdλ d 6 R Ω 0 dλ d , or R Ω 0 (1 − φ ) dλ d > 0. Since the latter inequality holds for an y choice of Ω 0 , it follo ws that φ 6 1 almost ev erywhere in Ω. C. T ransport of a measure F or the sequel, we wish to restrict the presen tation to the case d = 2. F or our time interv al S and for eac h i ∈ { 1 , 2 } , we denote the velocity field of the corresp onding measure b y v i ( t, x ) with ( t, x ) ∈ S × Ω. Let also µ 1 t , and µ 2 t b e t wo time-dep endent mass measures. Note that for eac h c hoice of i , the dep endence on t of v i is comprised in the functional dep endence of v i on both measures µ 1 t , and µ 2 t . This is clearly indicated in (9). The fact that here w e deal with tw o mass measures µ 1 t and µ 2 t , transp orted with corresp onding velocities v 1 and v 2 , translates in to:        ∂ µ 1 t ∂ t + ∇ · ( µ 1 t v 1 ) = 0 , ∂ µ 2 t ∂ t + ∇ · ( µ 2 t v 2 ) = 0 , for all ( t, x ) ∈ S × Ω . (6) These equations are accompanied by the following set of initial conditions: µ i t = µ i 0 as t = 0 for i ∈ { 1 , 2 } . (7) It is worth noting that (6) is the measure-theoretical coun terpart of the Reynolds Theorem in con tinuum me- c hanics. T o b e able to interpret what a partial differ- en tial equation in terms of measures means, we giv e a w eak formulation of (6). Essentially , for all test func- tions ψ 1 , ψ 2 ∈ C 1 0 ( ¯ Ω) and for almost every t ∈ S , the follo wing iden tity holds: d dt Z Ω ψ i ( x ) dµ i t ( x ) = Z Ω v i ( t, x ) · ∇ ψ i ( x ) dµ i t ( x ) (8) for all i ∈ { 1 , 2 } . Definition I I.1 (W eak solution of (6)) The p air ( { µ 1 t } t > 0 , { µ 2 t } t > 0 ) is c al le d a weak solution of (6), if for al l i ∈ { 1 , 2 } the fol lowing pr op erties hold: 1. the mappings t 7→ R Ω ψ i ( x ) dµ i t ( x ) ar e absolutely c ontinuous for al l ψ i ∈ C 1 0 ( ¯ Ω) ; 2. v i ∈ L 2  S ; L 1 µ i t (Ω)  ; 3. Equation (8) is fulfil le d. W e refer the reader to [26] for an example where the exis- tence of weak solutions to a similar (but easier) transport equation for measures has b een rigorously shown. I I I. SOCIAL VELOCITIES W e follow v ery muc h the philosophy dev elop ed by Hel- bing, Vicsek and coauthors (see, e.g. [13] and references cited therein) whic h defends the idea that the p edes- trian’s motion is driven b y a so cial force. Is worth noting that similar though ts w ere giv en in this direction (motion Mo delling Cro wd Dynamics 117 4 of so cial masses/net w orks) m uch earlier, for instance, b y Spiru Haret [27] and Antonio Portuondo y Barcel´ o [28]. Moreo v er, other authors (for instance, Ho ogendo orn and Bo vy [29]) prefer to account also for the Zipfian principle of least effort for the h uman b ehavior. W e do not attempt to capture the least effort principle in this study . A. Sp ecification of the velocity fields v i Un til no w, we hav e not explicitly defined the velocity fields v i ( i ∈ { 1 , 2 } ). V ery muc h inspired b y the so cial for c e mo del by Dirk Helbing et al. [16], the velocity of a p edestrian is mo deled as a desir e d velo city v i des p er- turb ed by a comp onent v i [ µ 1 t ,µ 2 t ] . The latter component is due to the presence of other individuals, b oth from the p edestrian’s own subpopulation and from the other subp opulation. The desired velocity is indep endent of the measures µ 1 t and µ 2 t , and represents the velocity that an agent w ould hav e had in absence of other p edestrians. F or each i ∈ { 1 , 2 } , the velocity v i is defined by sup erp osing the tw o velocities v i des and v i [ µ 1 t ,µ 2 t ] as follo ws: v i ( t, x ) := v i des ( x ) + v i [ µ 1 t ,µ 2 t ] ( x ) , (9) for all t ∈ (0 , T ) and x ∈ Ω. F or a c ounterflow scenario, the desired velocities of the t w o subp opulations follow opp osite directions. W e th us tak e v i des ( x ) = v i des ∈ R 2 fixed (for i ∈ { 1 , 2 } ) and v 1 des = − v 2 des . The comp onent v i [ µ 1 t ,µ 2 t ] mo dels the effect of in teractions with other p edestrians on the current velocity[30]. Since the in teractions b etw een mem b ers of the same subp opu- lation differ (in general) from the interactions b etw een mem b ers of opp osite subp opulations, we assume that v i [ µ 1 t ,µ 2 t ] consists of t wo parts: v i [ µ 1 t ,µ 2 t ] ( x ) := Z Ω \{ x } f own ( | y − x | ) g ( α i xy ) y − x | y − x | dµ i t ( y ) + Z Ω \{ x } f opp ( | y − x | ) g ( α i xy ) y − x | y − x | dµ j t ( y ) , (10) for i ∈ { 1 , 2 } , where j = 1 if i = 2 and vice versa. In (10) w e ha ve used the following: • f own and f opp are contin uous functions from R + to R , describing the effect of the mutual distance b et ween individuals on their in teraction. Compare the concept of distanc e inter actions defined in [22]. f own incorp orates the influence by mem b ers of the same subp opulation, whereas f opp accoun ts for the in teraction b etw een mem b ers of opp osite subpopu- lations. f own is a comp osition of t wo effects: on the one hand individuals are repelled, since they wan t to a void collisions and congestion, on the other hand they are attracted to other group mates, in order not to get separated from the group. f opp only contains a repulsiv e part, since we assume that p edestrians do not w an t to stick to the other sub- p opulation. • α i xy denotes the angle b etw een y − x and v i des ( x ): the angle under which x sees y if it were moving in the direction of v i des ( x ). • g is a function from [ − π , π ] to [0 , 1] that enco des the fact that an individual’s vision is not equal in all directions. Regarding the sp ecific choice of f own , f opp and g w e are very m uch inspired by [16] and [8], e.g. Ho wev er w e do not use exactly their w ay of mo deling pedestrians’ in- teraction forces. W e list here the following forms for the functions f own , f opp and g that matc h the giv en c harac- terization: f opp ( s ) :=    − F opp  1 s 2 − 1 ( R opp r ) 2  , if s 6 R opp r ; 0 , if s > R opp r , (11) f own ( s ) :=      − F own  1 s − 1 R own r  1 s − 1 R own a  , if s 6 R own a ; 0 , if s > R own a , (12) g ( α ) := σ + (1 − σ ) 1 + cos( α ) 2 , for α ∈ [ − π .π ] . (13) Here F opp and F own are fixed, p ositive constan ts. The constan ts R opp r , R own r ( r adii of r epulsion ) and R own a ( r adius of attr action ) are fixed and should be chosen suc h that 0 < R own r < R own a and 0 < R opp r . F urthermore, the restriction max { R own a , R opp r }  L has to be fulfilled. The interaction f opp is designed such that individuals “feel” repulsion (i.e. f opp < 0) from another p edestrian if they are placed within a distance R opp r from one another. The corresp onding statemen t holds for f own if individuals are within the distance R own r . Additionally , an individual is attracted to a second individual if their m utual distance ranges betw een R own r and R own a . The function g ensures that an individual exp eriences the strongest influence from someone straight ahead, since g (0) = 1 for any σ ∈ [0 , 1]. The constan t σ is a tuning parameter called p otential of anisotr opy . It determines how strongly a p edestrian is fo cussed on what happ ens in front of him, and ho w large the influence is of p eople at his sides or b ehind him. In the remainder of this section, we suggest four differen t alternatives for the definition of v i [ µ 1 t ,µ 2 t ] b y 118 Mo delling Cro wd Dynamics 5 indicating v arious sp ecial choices of distance inter- actions and visibility angles (conceptually similar to α i xy ) as they arise in (10). All of them b oil down to including an implicit dependency of the actual velocity v i = v i des + v i [ µ 1 t ,µ 2 t ] . Note that this effect increases the degree of realism of the mo del, but on the other hand it makes the mathematical justification of the corresp onding models muc h harder to get. 1. Modific ation of the angle α i xy W e defined the angle α i xy as the angle b etw een the v ector y − x and v i des ( x ). How ever this is not a go o d definition if the pedestrian in p osition x is not moving in the direction of v i des ( x ) (or, in a broader sense, if the ac- tual speed cannot b e approximated sufficiently w ell by the desired velocity). Therefore w e suggest to define α i xy = α i xy ( t ) as the angle b etw een y − x and v i ( t, x ). 2. Pre diction of mutual distanc e in (near) futur e Up to now the functions f own and f opp dep ended on the actual distance b etw een x and y at time t . How ever p edestrians are likely to an ticipate on the distance they exp ect to ha ve after a certain (small) time-step (say , some fixed ∆ t ∈ R ). In practice, this means that at a time t ∈ S a p erson will mo dify his velocity (either in direction, or in magnitude, or b oth) if he foresees a collision at time t + ∆ t ∈ S . T o predict the m utual distance b etw een x and y at time t + ∆ t , the curren t velocities at x and y are used for extrap olation. The predicted distance is: | ( y + v ( y , t )∆ t ) − ( x + v ( x, t )∆ t ) | . Consequen tly , sticking to the notation in (10), the in teraction p otential f own and f opp should depend on | ( y + v i ( t, y )∆ t ) − ( x + v i ( t, x )∆ t ) | and on | ( y + v j ( t, y )∆ t ) − ( x + v i ( t, x )∆ t ) | resp ectively (where j = 1 if i = 2 and vice v ersa). 3. Pre diction of mutual distanc e within a time interval in the (ne ar) futur e The disadv antage of using | ( y + v ( y , t )∆ t ) − ( x + v ( x, t )∆ t ) | is that ∆ t is fixed. A p edestrian can thus only predict the distance at an a priori sp ecified p oint in time in the future. How ever, p eople are able to an- ticipate also if they expect a collision to o ccur at a time that is not equal to t + ∆ t . W e assume no w that we are giv en a fixed ∆ t max ∈ R + suc h that an individual can predict mutual distances by extrap olation for an y time τ ∈ ( t, t + ∆ t max ). Th us, ∆ t max imp oses a b ound on how far can an individual lo ok ahead in to the future. T o cap- ture this effect, we suggest to replace f own ( | y − x | ) and f opp ( | y − x | ) b y: 1 ∆ t max Z ∆ t max 0 f own (   ( y + v i ( t, y ) τ ) − ( x + v i ( t, x ) τ )   ) dτ , (14) and 1 ∆ t max Z ∆ t max 0 f opp (   ( y + v j ( t, y ) τ ) − ( x + v i ( t, x ) τ )   ) dτ , (15) resp ectiv ely . 4. Weighted pr e diction Since an individual probably attaches more v alue to his predictions for p oin ts in time that are nearer b y than others, one additional modification comes to our mind. Let h : [ t, t + ∆ t max ] → [0 , 1] b e a w eight function. Then instead of (14) and (15), we prop ose 1 ∆ t max Z ∆ t max 0 f own (   ( y + v i ( t, y ) τ ) − ( x + v i ( t, x ) τ )   ) h ( τ ) dτ , (16) and 1 ∆ t max Z ∆ t max 0 f opp (   ( y + v j ( t, y ) τ ) − ( x + v i ( t, x ) τ )   ) h ( τ ) dτ . (17) If h is decreasing, then the influence of t 1 is larger than the influence of t 2 , if t 1 < t 2 (whic h matches our intu- ition). B. Two-scale measures W e now consider the explicit decomp osition of the measures µ 1 t and µ 2 t . Let the pair ( θ 1 , θ 2 ) b e in [0 , 1] 2 , and consider the following decomposition of µ 1 t and µ 2 t : µ i t = θ i m i t + (1 − θ i ) M i t , i ∈ { 1 , 2 } . (18) Here, m i t is a microscopic measure. W e consider { p i k ( t ) } N i k =1 ⊂ Ω to b e the p ositions at time t of N i c ho- sen p edestrians, that are members of subp opulation i . W e w ant m i t to b e a counting measure with resp ect to these pedestrians, i.e. for all Ω 0 ∈ B (Ω): m i t (Ω 0 ) = # { p i k ( t ) ∈ Ω 0 } , i ∈ { 1 , 2 } . (19) W e thus define m i t as the sum of Dirac masses (cf. Section I I A 1), cen tered at the p i k , k = 1 , 2 , . . . , N i : m i t = N i X k =1 δ p i k ( t ) , i ∈ { 1 , 2 } . (20) M i t is the macroscopic part of the measure, which tak es in to accoun t the part of the crowd that is considered Mo delling Cro wd Dynamics 119 6 con tin uous. W e consequen tly hav e M i t  λ 2 , since a set of zero volume cannot contain an y mass. Note that we are th us in the setting of Section I I A 2. Now, Radon- Nik o dym Theorem guaran tees the existence of a real, non-negativ e densit y ˆ ρ i ( t, · ) ∈ L 1 λ 2 (Ω) suc h that: M i t (Ω 0 ) = Z Ω 0 ˆ ρ i ( t, x ) dλ 2 ( x ) (21) for all Ω 0 ∈ B (Ω) and all i ∈ { 1 , 2 } . IV. MICRO-MA CRO MODELING OF PEDESTRIANS’ MOTION IN HETER OGENEOUS DOMAINS W e ha ve already made clear that w e w ant to mo del the heterogeneit y of the in terior of the corridor. In practice this means that pedestrians cannot en ter all parts of the domain. As describ ed in Section I I B, we hav e a measure µ p corresp onding to the porosity of the domain (whic h is fixed in time). How ever, w e note that the concept of p orosit y (cf. Section I I B) is a macroscopic one. F or this reason only the macroscopic part of the mass measure in (18) needs some mo dification with respect to the p oros- it y . In this context, one should b e a ware of the analogy with mathematical homo genization . This technique dis- tinguishes b etw een microscopic and macroscopic scales, where we also see that some (av eraged) characteristics are only defined on the macroscopic scale. F or more de- tails, the reader is referred to [25] or [31]. In R 2 , we ha v e µ p  λ 2 . F urthermore M i t  µ p for i ∈ { 1 , 2 } and a.e. t ∈ S . This is obvious, since no p edestrians can b e present in a set that has no p ore space (i.e. zero p orosit y measure). A basic prop erty of Radon-Nikodym deriv ativ es no w gives us: dM i t dλ 2 = dM i t dµ p dµ p dλ 2 (22) for eac h i ∈ { 1 , 2 } and for almost every t ∈ S . W e hav e already defined ˆ ρ i ( t, · ) := dM i t dλ 2 and φ := dµ p dλ 2 . If w e no w denote by ρ i ( t, · ) the Radon-Nik o dym deriv ativ e dM i t dµ p , the following relation holds: ˆ ρ i ( t, · ) ≡ ρ i ( t, · ) φ ( · ) for all i ∈ { 1 , 2 } . A. W eak formulation for micro-macro mass measures W e now hav e the follo wing measure: µ i t = θ i m i t + (1 − θ i ) M i t , i ∈ { 1 , 2 } , (23) as w as given in (18), where now: m i t = N i X k =1 δ p i k ( t ) , dM i t ( x ) = ρ i ( t, x ) φ ( x ) dλ 2 ( x ) . (24) This sp ecific form of the measure will now b e included in the weak form ulation (8), with velocity field (9)-(10). The real p ositive num b ers θ i ( i ∈ { 1 , 2 } ) are intrinsic scaling parameters depending on N i . The transport equation (8) tak es the following form: d dt  θ i N i X k =1 ψ i  p i k ( t )  + (1 − θ i ) Z Ω ψ i ( x ) ρ i ( t, x ) φ ( x ) dλ 2 ( x )  = θ i N i X k =1 v i  t, p i k ( t )  · ∇ ψ i  p i k ( t )  + (1 − θ i ) Z Ω v i ( t, x ) · ∇ ψ i ( x ) ρ i ( t, x ) φ ( x ) dλ 2 ( x ) , (25) for all i ∈ { 1 , 2 } . Here w e hav e used the sifting property of the Dirac distribution. In the same manner, w e specify v i [ µ 1 t ,µ 2 t ] from (10) as 120 Mo delling Cro wd Dynamics 7 v i [ µ 1 t ,µ 2 t ] ( x ) = θ i N i X k =1 p i k ( t ) 6 = x f own ( | p i k ( t ) − x | ) g ( α i xp i k ( t ) ) p i k ( t ) − x | p i k ( t ) − x | +(1 − θ i ) Z Ω f own ( | y − x | ) g ( α i xy ) y − x | y − x | ρ i ( t, y ) φ ( y ) dλ 2 ( y ) + θ j N j X k =1 p j k ( t ) 6 = x f opp ( | p j k ( t ) − x | ) g ( α i xp j k ( t ) ) p j k ( t ) − x | p j k ( t ) − x | +(1 − θ j ) Z Ω f opp ( | y − x | ) g ( α i xy ) y − x | y − x | ρ j ( t, y ) φ ( y ) dλ 2 ( y ) , (26) for i ∈ { 1 , 2 } , and j as b efore ( j = 1 if i = 2 and vice v ersa). W e hav e omitted the exclusion of { x } from the domain of integration (in the macroscopic part), since { x } is a nullset and th us negligible w.r.t. λ 2 . Note that the sums ma y b e ev aluated in any p oint x ∈ Ω (not necessarily x = p i k ( t ) for some i and k ); the integral parts ma y also b e ev aluated in all x , including x = p i k ( t ) for some i and k . V. NUMERICAL ILLUSTRA TION W e wish to illustrate no w the microscale description of a counterflo w scenario (i.e. for θ 1 = θ 2 = 1) by presen ting plots of the configuration of all individuals situated in a given corridor at sp ecific moments in time. W e consider a sp ecific instance in which there are in total 40 individuals (20 in each subpopulation). The dimensions of the corridor are d = 4 and L = 20. The v elo city is taken as defined in (10)-(13). F urthermore, the following mo del parameters are used: v 1 des = 1 . 34 e 1 , v 2 des = − 1 . 34 e 1 , F opp = 0 . 3, F own = 0 . 3 , R opp r = 2 , R own a = 2 , R own r = 0 . 5, F w = 0 . 5, R w = 0 . 5 , σ = 0 . 5 . In Figure 2, we show the configuration in the cor- ridor at times t = 0, t = 7 . 5, and t = 15. The individuals of the subp opulation 1 are colored blue, while the individuals of the subp opulation 2 are colored red. Clearly , self-organization can be observ ed in the system: P edestrians that desire to mov e in the same direction form lanes (in this case, three of them). This feature is observ ed and des crib ed extensively in literature, cf. e.g. [13]. Another feature, pointed out b y Figure 2, is the follo wing: Within the three already formed lanes, small clusters of p eople are formed. This flo cking is a result of the typical choice for f own in (12). Members of the same subp opulation are repe lled if their mutual distance is in the range (0 , R own r ); they are attracted if their mutual distance is in the range ( R own r , R own a ). No in teraction tak es place if individuals are more than a distance R own a apart. The attraction part of the interaction causes indi- viduals that are already relativ ely close to get even closer, un til they are at a distance R own r . F or distances around R own r , there is an interpla y b etw een repulsion and attraction, even tually leading to some equilibrium in the mutual distances b et ween neighboring individuals in one cluster. In Figure 2, we observe self-organized patterns ev en within clusters. W e should men tion here that further sim ulation is needed, in particular on multiply-connected domains. Note how ever, that as the mo del is now, moving individ- uals are not a priori hindered to penetrate any obstacle. Moreo v er we wish to p oin t out that the n umber of particles should b e sufficiently large, in order to b e statistically acceptable. A CKNOWLEDGMENTS W e ackno wledge fruitful discussions within the ”Par- ticle Systems Seminar” of ICMS (Institute for Complex Molecular Systems, TU Eindhov en, The Netherlands), esp ecially with H. ten Eik elder, B. Markv o ort, F. Nardi, M. Peletier, M. Renger, and F. T oschi. F urthermore w e w ould like to thank F ons v an de V en (Eindhov en) for sharing his ideas. A.M. is indebted to Michael B¨ ohm (Bremen) for introducing him to the fascinating world of mo deling with measures. J.E. wishes to express his grat- itude to the Department of Mathematics and Computer Science, TU Eindho ven, for giving him the opp ortunity to do an Honors Program in Industrial and Applied Math- ematics. [1] See the question of scale of Vicsek [32]. [2] C. Kipnis and C. Landim, Sc aling Limits of Inter acting Particle Systems (Springer V erlag, 1998). Mo delling Cro wd Dynamics 121 8 −10 −8 −6 −4 −2 0 2 4 6 8 10 −4 −3 −2 −1 0 1 2 3 4 −10 −8 −6 −4 −2 0 2 4 6 8 10 −4 −3 −2 −1 0 1 2 3 4 −10 −8 −6 −4 −2 0 2 4 6 8 10 −4 −3 −2 −1 0 1 2 3 4 FIG. 2. The sim ulation of a crowd’s motion in a corridor of length L = 20 and width d = 4. Each of the tw o sub-populations consists of 20 individuals. The images w ere tak en at t = 0 (left), t = 7 . 5 (middle), t = 15 (right). [3] A. 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