Three-dimensional Random Voronoi Tessellations: From Cubic Crystal Lattices to Poisson Point Processes

1 Thr ee-dim ensional Rand om V or onoi T essel la t io ns: Fr om Cubic Crystal L attices to P oi ss on Point P r ocesses Valer io Luca rini Departme n t o f Phy sics, U n ive r sity of B ologna, V iale Be r ti Pic h at 6 /2, 40127 , Bo logna, Italy Istituto Naz ionale di F isica Nucle are – Sezio n e di B olo gn a, Viale Be rti Pic h at 6/2, 40 127, Bo logna, Italy Email: luca rini@adgb .df.unibo .i t Abstr act We pert u r b t he simple cubic (SC), body -center ed cubic (BCC), and face-ce n te r ed cubic (FCC) str uctu r es with a spat ial Gaussian n o ise whos e adim e nsional stre n gth is controlled by the pa r amete r α and analy ze th e st ati s tical prope rti es o f th e ce lls of th e r esulti n g Vo r onoi t ess ellations. We c o n centrate on topo l o gical pr o perties, such as the n umber of faces , an d on metric properties , suc h a s the ar ea and th e vo lume. Th e to po l ogical prope r tie s of th e Vo r o n oi tess ellations of th e SC and FCC c ry stals are u nstable w ith r espe ct t o the intr o ductio n of noise, because th e co rr espo ndin g poly hedra ar e geo metr ically degenerate, whe r eas the tesse lla tio n of th e BCC cr y stal i s topo l ogic al ly stable e v en again s t noise of smal l but finite intensity . Whereas th e av erage vo lum e of the cells is the int e nsity par amete r of th e s y ste m and does not depe n d on the noise, th e average area of th e cells ha s a r ath er int e resting behavio r w ith respec t t o n oise in te n sity . For weak n oise , th e mean ar ea of th e Voro n oi tesse ll atio n s co rrespo ndin g t o pe r turbe d BCC and FCC perturbe d increases quadratic al ly with th e n oise in tensity . In the case of pert urb ed SCC cr y stal s, the r e is an optimal am o unt of n o ise t hat minimizes the m ea n area of th e cells. Already for a mode rate a mou nt of n o ise ( α > 0.5), the statistic al prope rties of th e three pe r turbed tes sellati o n s are indistinguis hable, and f or i ntense n o ise ( α > 2) , r es ult s co nverge to th os e of th e Po isson- Vo r onoi tesse ll atio n. No tably , 2-pa r ameter gam ma di strib utio n s constitute an exc ellent mode l f or t h e empirical pdf of all conside r ed to polo gical a nd met r ic p r ope r ties . By analy zin g jo in tly th e statistica l prope r ties of the ar ea a n d o f the volume of th e cells, we di scover that n o t only th e area and the volume of t he cells f luctuat e, but so does th eir shape too . The shape of a cell can be measur ed wit h the isope r ime tri c quotient, in dica ting h ow sph erical th e cell i s . Th e Voronoi tesse llat io n s of th e BCC and of th e FCC st r uctu r es r esult to be l ocal maxima fo r the isope r imet ri c quo tient among space -filling tess ellations. In t h e BCC case, t hi s suggests a weaker f orm o f th e Kelvin c onjec tur e, w hich h as recently bee n disprov ed. Due to the f luctuations of the shape o f t he c ells, anomalous scali n gs w ith expo nents larger tha n 3/ 2 are obs erved be t wee n the area and the vo lumes of t he cells fo r al l cases co n sidered, and, exc ept fo r the FCC st r uct ure, a lso for infinites imal n oise . I n th e Po isso n -Vo r onoi limit , the exp onent is ~1.67. As t h e numb er of face s is h eavily co rr elated w ith th e sp h e r icity of the cells (c ells w ith more faces a r e bulkie r ), the anom al ous sc aling is h eavily r educe d w hen w e perfo r m po wer law f its se parately on cells with a spe cific n umb er of faces. Keyw ords: Voronoi tess ellat io n, , Nume r ica l Simula ti o ns, Random Ge ometry S y mm etry Break, Poiss on Point P r oc ess , Cubic Cr y stals, Gaussia n Noise , A n omalous Sc aling, Fluc tuations.. 2 1. Intr oducti on A Vo ro noi t essella t i on (Vo ronoi 1907 ; 1908) is a part i t i oning o f a n E uclidean N -d im ens i o nal space Ω defi ned i n ter ms o f a given d iscrete s et o f po in t s Ω ⊂ X . F o r alm ost any po in t Ω ∈ a the re is one specific po in t X x ∈ which i s c l o sest to a . Som e p o in t a m ay b e equa lly dista nt f ro m t wo o r m or e po in t s o f X . If X conta in s only two poi n t s, x 1 a nd x 2 , then t he set of all po in t s w i th t he same dis t ance f ro m x 1 an d x 2 is a hyperplane, which h as co d im ens i o n 1. The hyperplane bisect s perpen d icularly t h e segment f ro m x 1 a nd x 2 . I n ge neral, t h e set o f all po in ts cl o ser t o a po in t X x i ∈ t h an to any ot her poin t i j x x ≠ , X x j ∈ i s t h e in t er i or o f a co nvex ( N -1) -poly to pe usually ca lled the Voro n o i cell f o r x i . T he set o f t h e ( N -1)- poly to pes Π i , eac h co rrespo n d in g to - and co ntaining - o ne point X x i ∈ , is the V o ronoi t essella t i o n co rresponding to X. Extensi o ns t o the case o f n on- Euc li dea n spac e s have a l so been presented (Isokawa 2000; Okabe et al . 2000). S in ce Vo ro noi t essellat i ons bo il up t o bein g o ptimal part iti oning s of the space result i ng fro m a set of gener ating points , t hey have lo n g been cons i d ered f o r appli cat i o ns in se veral research areas, such as t el e co mm u nicat i o ns (S ort ai s et al . 2007), bi o logy ( Finney 1975), astronomy (Icke 1996), f or estry (Barrett 1997) atomi c p hysi c s (Go ede e t al . 1997), m et all urg y ( Weai r e et al . 1986) , polym e r science ( Dot era 1999), m at er i als science (Bennett et al. 1986), an d bi o p hy s ics ( Soye r et al. 2000) I n a geo physic a l co n t ext, Vo ronoi t essella t i o n s hav e b een w idely u sed t o an a lyze spat i a lly d istr ib ut ed observ at ional o r m o de l o utput data (Tsai et al . 2004; L ucar i n i et a l . 2007, Lucari ni et al 2008) . In condense d matt er physics , t h e Vo ronoi cell o f t he latt i ce po in t o f a cr y st al is known as t he Wigne r- Seitz cell , w h erea s t he Voronoi ce ll o f t he rec i pro cal latt i ce po in t i s the Bri l l o uin zo n e (Ba ssani a nd Pastori Parravicini 1975 ; Ashcroft an d Mermin 1976) . V o ronoi t ess e l lat i o ns have bee n used f o r perf o r mi ng st ructure analysis f or cr ys t alline so li ds a nd superco ol ed liquids (T sum ura ya et al . 1993 ; Yu et al . , 2005), f o r de t ecting gl ass tr an sit i o ns (Hen t sch e l et a l . 2007), for em p hasizing the geom et r i cal effects underlying t h e vib r at i ons in t he g l a ss (Luchni ko v et al ., 2000), and for perf o r mi ng det ail ed a n d e fficien t e l ect roni c ca l cu lat i o ns ( Averill a nd Pain t er 1989 ; Rapcew i cz et a l . 1998). F o r a r evi ew o f t he t h eo r y and app licat i o ns of Vo ronoi tesse ll at i o ns, see Aurenhamm er (1991) an d Okabe et al . (2000) . As t h e t h eo reti cal resu l t s on t he st atistical pr opert i es o f general N- dimensi o nal Vo ro n o i t ess e ll at i o n s are st ill re latively limi t ed, d i rect numer i ca l s imulat i o n const i t ut e a cruci al inves t i gat iv e appro ach . At co m putat i o nal lev e l, t he e val uat i o n of t he Vo ronoi t ess e llat i o n o f a g i ve n d is cret e set of po in t s X is n o t a t rivial t ask, an d the de fi n i t i o n of o pt i mal pro cedures is o ngoin g a nd inv o lves va r i o us scient ifi c co mm u ni t i e s (Bow y er 1981; Watso n 1981; Tan e m ur a et a l . 1983; Barber et al . 1996; Han a nd Bray 2006). T he spec ifi c a nd relev a nt pro bl e m o f co m put ing the geo m et ric ch aract er i st i c s of Po i sso n-Voronoi tessellat i o ns has b een the subject o f intense t heo reti ca l a nd 3 com put at i o nal eff o rt . Pois son-Vo ro n o i t ess e ll at i o n s are obtained start in g fro m f o r a r an do m set o f points X gen erat ed as o ut put of a h o mogeneous ( in t he co nsi d ered space) Poisson po in t pr ocess. This problem ha s a great r el e van c e at pract i cal l e vel because i t co rrespo nds , e.g. , to s t udy ing cr y st a l aggregates w i t h r an do m n uc leati o n s i t es and u nifo rm gr owth rat es. Exact r esul t s concerni ng t he me a n stat i st i c al pr opert i e s o f t he i nter f ace area, i n ner area, n u mber o f ve rt i ce s, etc. o f t he Po iss o n - Voro n o i t essellat i o ns have bee n obta in ed f or E uclidean spaces (Me ij er ing 1953; Christ et al ., 1982; Dro uff e and It zy k so n 1984; Cal ka 2003; H il hor st 2006) ; an especially impress ive account i s g iven by F inch (2003). Several co mputat i o nal stud i e s on 2D an d 3D s pa ces hav e f o und r esul t s in agreem e nt wi t h t he t heoreti cal fi ndings, and, m or eover, h ave s h o wn t ha t b ot h 2-param et er (K u m ar et al . 1992) and 3- parame t er (H in de a n d M ile s 19 80) ga mm a distribut i o ns fi t up to a hi g h degree o f accuracy t he em p irical pd fs o f se veral geo m et r ical c h aract er i st i c s o f t h e ce ll s (Z h u et a l . 2001; Tanemura 2003) . T he ab- initio der iv at i o n o f t he pd f o f t he geo metri ca l pr oper tie s of Po i sso n - Voro n o i t ess e lla t i o n s have not b een ye t o btaine d , except i n asympt otic reg im es (H ilh o rst 2005) , which ar e, surprisingly , not com pat ibl e w i t h the gamma d is t rib ut ions family. In a previ o u s paper (Lucar ini 2008), we h ave a naly zed in a rat her gen era l framewo rk t h e statist i ca l pr opert ie s o f Vo ro noi tessella t i on o f t he Euc lide a n 2D p lane. I n par t i cu la r, we h a ve f o ll o wed t h e t ransi t i o n f ro m r egul ar t ri a ngular, square an d hexagonal honey co mb V o rono i t ess e ll at i o n s t o t hose o f t he Po i sson-Vor onoi case, t hus analyzing in a co mm o n f r a me wo rk symm et r y - break pro cess es and the approach to unif o r mly ra ndom d istr ib ut ions o f t ess e lla t i o n - gen erat in g po in t s, which is b a sically realized when the typic a l displacem e nt beco m e s l arger t han the la t t i ce u ni t v ect or. Thi s a nalysis h as been acco m p lish ed by a Monte Carl o analys i s o f t essellat i ons gen erat ed by po in t s w hose regu l ar pos i t i o ns are pe rt urb ed t hrough a Gaussian no ise. T he symm et r y brea k induced by t h e in t ro duc t i o n o f no i se de stro y s t he tr i a ngular and square t essell at i o n, which are structur al ly u ns t able, w he reas t he honeyco mb hexagonal t essella t i on is stable also f o r sm all but fini t e nois e. Moreo ve r, in dependent ly o f t he un pert urbed struct ure, for all no i se in t ensi t ies ( in c ludin g in fini t esimal), h e xagons co nsti t ut e the m o st comm o n c l a ss o f cells a n d t he ensemble me a n of t he cells area and pe r im et er res t ricted to the he xagonal c ell s coinci d es wi t h the ful l ensemble m ea n , which re inf o rces t he ide a t h at h e x ago ns ca n b e t aken a s generic po ly go ns in 2D Voro n o i t essell at i o ns . The r easons why the regular hex ago nal t essellat i o n has such pecu li ar pro pert i es o f ro bus t ne ss re lies o n the fac t that i t i s opt im a l b o t h i n t erms of peri m et er-to- area rati o and in t er m s o f cost (see New man 1982 ; Du and Wang 2005) . The extremal pro per t i es o f such a t essellat i o n are cl ear ly hi g hlighted by Karc h et al . (2006) , wh ere i t is noted t h at a Gi bbs s y st em o f repulsi ve c harges in 2D arra n ges spo n t ane o usly for l o w t empe ra t ures (f reezes) a s a regu la r hexagonal cr y st al . 4 Moreover, a regu l ar hexagonal str uc t ure has been f o u nd f o r t he Vo ro noi t ess e lla t i on b u il t f ro m t he spontaneously arr anged latt i ce o f h o t spots (stronges t upward m ot i o n o f h o t fl u i d) o f t h e R ayl e igh- Bènard co nv e ct ive cell s, wi t h the co m pe nsat in g downward m ot i on o f cool ed fluid co nc e ntrated o n t h e s i de s o f t he Voro n o i cells ( Rapaport 2006). In this paper we wa nt to ex t end the a naly s is perfor m ed in Lucarini (2008) to the 3D case, which is probably o f w ider appl i cat ive in t erest . W e co nside r t hree cu bic cr ys t als covering t h e 3D Euc li dea n space, n a mely t he sim p le cu bic (SC), t he f ac e-centered cubi c (FCC) a nd t he b o d y - cen t ered cu bic (BCC) latt i ces (Bassani a nd Pa sto ri Parravicini 1975). T he co rrespondin g space - filli ng Vo ronoi ce lls o f suc h cr ysta ls ar e the cube, t he rho m bic dodecah edro n, a nd t he t runc at ed octahedron. T he cubic cr ys t al s ys t em is one o f the m o st co mm o n cr ys t al s ys t em s f o u n d in elementa l me t a l s, a n d na t ura lly occurr in g cr ystals a nd minerals . T hese cr y st al s featur e extraor di nar y geom et r i cal pr oper ti es: • t h e cube is t h e o nly space-filling regu l ar so li d ; • t h e FC C (to ge t her w i t h t he Hex ago nal Cl o se Packed st ructure) f eat ures t h e la rgest po ssible packin g f ract i on - t h e 1611 K ep l er’s co nje ct ure has b een rece n t ly pro ved by Hales (2005) ; • t h e Voro n o i ce ll o f BCC has been conjectured by Kel v in in 1887 as being t he space-filling cell wi t h t h e smallest sur f ace to volume rat i o , and o nly recent ly a very cu mberso m e co unter- ex a mple has bee n g iven by Wearie a nd P helan (1 994); moreo v er, t h e t runcated o c t ahe dro n is conj ect ured to b e hav e t he l o west cost am o ng a ll 3D space- filli ng cells (see Du a n d Wang 2005). Because o f its l o w dens i t y , basically due t o the l o w p acking fact i on, t he SC syste m has a high en erg y s t ruct ure and i s rare in nature, and i t i s f o un d o nly in t h e alp h a- f o r m o f Po . The B CC is a m o re co m pact sys t em and have a l o w energy structur e , i s therefore m o re c o mm o n in n at ure. Examples o f BCC struct ures in c lude Fe, Cr, W , and Nb . Finally , t hanks to its e x t re m a l pro pert i es in t erm s o f pack ing fract i o n a nd t he resu l t in g hi g h de nsi t y , F CC cr ysta l s are f a i r ly co mm o n a nd specifi c e xamples include P b, Al , Cu, Au a n d Ag. The e xtrem a l pr operties o f t he BCC st ruct ure can bas i ca lly be in t erpret ed as t he fact t h at t he correspondin g Vor o n o i ce ll defines a natur al discrete ma t hemat i c a l m easure, a nd imply t h at t runc at ed octahedra const i t ut e an optimal too l f o r achieving d ata co m press i on (E ntezar i et a l . 2008). An o t he r outstanding pro perty is that the Vo ro n o i cell o f t he BCC str ucture, as opposed to SC and FC C, i s to p o l og i cal ly stable w i t h r espect t o i nfini t esimal pert urbati o ns t o t he po si t i on o f t he lattice points (T roadec e t al . 1998). Usin g a n e ns e mble - based appro ach , we st udy t he break-up o f t he s ymmetr y o f t h e SC, BC C an d FCC syste ms a nd o f t heir cor respondin g Vo rono i t essellat i o ns by st o ch astically pert urbin g w i t h 5 a space-ho m ogeneous Gauss ian n o is e o f para metr i ca lly contro l led st rength the pos i t i o ns o f t he la t t i ce po in t s x i , and qua nt i t at iv ely evaluat in g ho w t h e s t at i st i cal pr oper tie s o f t h e geo m et ri ca l ch aract er i st i c s o f t he result i ng 3D Vo ro n o i ce lls ch ange. The st ren gt h o f cons i d ered perturbation range s up t o t he po in t where t y p ic a l d is p lacements beco m e larger t ha n t he l at t i ce u ni t vector , which basically leads to t h e limi t ing case o f the Po is son- Voro n o i pro cess. Therefore, o ur w o rk joins o n t h e an a lysis o f Vo ronoi cel l s result i ng fro m infini t esimal ( n a mely , small ) per tur b at i o ns t o regul ar cu bic la t t i ces to f ull y rando m t essellat i ons. Our paper i s o rganized as f o ll o ws. In sect i o n 2 we d is cuss so m e g e n era l pro perti es o f t he Voro n o i t ess e lla t i o n s cons ide red, describe the methodo l o g y o f w o rk and the set o f numerical ex per iments per for m ed. In sec t i on 3 we sh o w our resu l t s. I n sect i o n 4 we pre sent our concl usions an d per spect i ves f o r future work. 2. The or etical and C o m putational Is sues General Properties of Voronoi tesse llations W e consider a rando m po in t pr ocess ch aract erized by a spat i a lly h o m o gen eo us co arse-grained in t e nsi t y 0 ρ , such t hat t he expe ct at i o n value o f t h e number o f po in ts x i in a ge ne r ic reg i o n Γ œ  3 i s Γ 0 ρ , whe re Γ is the L ebesgue m easure o f Γ , whereas t he fl u ct uati o ns i n t h e n u m ber o f po in t s are Γ ≈ 0 ρ . I f 1 0 >> Γ ρ , we are in t h e t hermodynamic limit a nd boundary e ff ect s are n eg ligibl e , so that t he number o f ce lls o f t he Vo ro noi t essellat i o n resul t ing f ro m t he set o f po i nts x i a n d contain ed inside Γ is Γ ≈ 0 ρ V N . In t his pap er we co n sider per fe ct crys t als, cr ysta ls with ran do m d is l o cat ions , a nd sparse points r esul t in g f ro m a spat ially ho m o geneous Po isson pr ocess. P erf ect cry sta l s are obta in ed w hen t h e probabili t y d istr ib ut i o n fu n ct i o n (pd f ) o f t he r an do m po in t pro cess can be e xpressed as a sum o f Dirac m asse s o beyi ng a d iscret e transl at i onal symmetr y . Cr y st als wi t h r ando m d isl o cat ions are per i o d i cal is a stat ist i ca l sense since the pdf o f the po in t - process i s a n o n -s ingular L e besgue me a surable f u nct i on o beying d iscrete t ranslati o nal s ymm etr y . Finall y , t he pd f o f the ho m o geneo us Po i s son po i nt process i s co ns t ant in space. Using sc ali ng argu ments ( Luca r ini 2008) , one o b t a in s t hat in a ll ca ses co nsi dered t he statist i ca l pr o pertie s o f t he Vo ronoi t essell at i o n a re intensive, so t h at t h e poin t densi t y 0 ρ can be scaled t o unit y , o r, al t erna t iv ely, t he do m a i n m a y can be sc aled t o t he Cart esi a n cu be [ ] [ ] [ ] 1 , 0 1 , 0 1 , 0 1 ⊗ ⊗ = Γ . W e w ill st i ck t o t h e s econd approach. W e t hen define ( ) Y µ ( ( ) Y σ ) as the 6 me a n va lu e (st an dard de vi at i o n) o f t he variable Y o v er t he V N cells f o r t he single r ealizat i o n o f t he rand o m pr ocess, wherea s the e x pres si on E ( [ ] E δ ), in dicates the ensemble me a n (standard devi at i o n) o f the ran do m variable E . W e have t hat ( ) ( ) 1 0 , − ∝ ρ σ µ V V , wh ere V i s the vo l u me o f t h e V o ronoi c e ll , ( ) ( ) 3 2 0 , − ∝ ρ σ µ A A , wh er e A i s the sur fac e a rea of t h e V o ronoi c e ll , an d ( ) ( ) 3 1 0 , − ∝ ρ σ µ P P , where P i s t h e tot al per i meter o f t he ce ll . T he proport i o nali t y co nstants depen d o f t he specific rando m po in t process consi d ered. The re f o re, by mul t iplying t he e nsemble me a n est im ator s o f the m ean and standa rd deviati o n o f t he v ar i o us geo m et r i ca l properties o f the Voronoi cell s t im e s the appr o pri ate power of 0 ρ , we ob t a i n uni ver sal funct i o ns. Goin g t o the to pol o g i cal pro pert i es o f t he ce l ls, we re m ind that, sin ce each Voro noi cell is conv ex, its v ert ices ( v ), edges ( e ) , and fac e s ( f ) are connected by t he s implifi ed Eu l er-Po in care f or m u la for 3D po ly hedra 2 = + − f e v . Moreov er, in a gener i c soli d vert i ces are tr iv a l e n t ( i . e. giv e n by t he in t ersecti o n o f t h ree edges), so t h at v e 2 3 = , which implies t h at 2 2 1 + = v f , so that the know l edge o f t h e num ber o f vert i ces of a cell pro vi des a rat her com p le t e inf o r m at i on a bout the poly hedron. An o t h er gen era l result is t ha t in eac h c e ll t he average num ber ( n ) o f si d es o f ea ch f ace is ( ) 6 12 6 2 6 < − = + = f v v n , which m arks a c lear d i ffe re nce w i t h respe c t to t he p l ane ca se, wh ere t h e E ul er t heorem appli es (Lucarini 2008). Exact R esults W e firs t c o nsi der the per fec t SC , BC C, an d F CC cubic cr ys t al s having a tot al o f 0 ρ l at t i ce po in ts per uni t vol u me a n d 0 ρ c o rresp o n d i ng Vo ronoi cell s i n 1 Γ . T h ere f o re, t he leng t h o f t he side o f t h e cub es o f t he SC, BCC, a n d FCC cr ys t als are 3 1 0 − ρ , 3 1 0 3 1 2 − ρ , and 3 1 0 3 1 4 − ρ , respec t iv e ly. Bas i c Eucli d ean geo m et r y all o ws us t o f u lly a naly ze these st ructures. The cell s o f t h e Vo ro n o i t es sellat i on of t h e SC c rystal are cub es (h aving 12 e dges, 6 f a ces, 8 t riv alent v ert i ces) of si de l e n gt h 3 1 0 − ρ and total sur fa ce area 3 2 0 6 − = ρ A . The ce l ls o f the Vo ron o i t es s ellati on o f t he BCC cr y sta l are t runca t ed octahe dra (havin g 36 edges , 14 face s, 24 tr i valen t v ert i ces) of si de length 3 1 0 3 1 0 6 7 4454 . 0 2 − − − ≈ ρ ρ an d t otal surface area ( ) 3 2 0 3 2 0 3 4 3147 . 5 2 3 2 1 3 − − − ≈ + = ρ ρ A . The c e l ls of the Voronoi t ess e ll at ion of t h e FCC cr ys t al are rho m bic dodecahedra ( havi ng 24 edges, 12 fac e s, 6 t riv a len t verti ces, 8 tetraval ent vert i ces) o f side len gt h 3 1 0 3 1 0 3 4 6874 . 0 3 2 − − − ≈ ρ ρ an d tot al s ur f ace area 3 2 0 3 2 0 6 5 3454 . 5 2 3 − − ≈ ⋅ = ρ ρ A . Th e stan dard is o perim etr i c qu o t i e nt 3 2 36 S V Q π = , wh ich 7 m easures t h e in non-dimen sional u ni t s the sur f ace-to -vol u m e rati o o f a so li d ( Q = 1 f or a sphere), is 0.5236, 0.7534 , an d 0.7405 f o r t he SC, BC C, and F CC struct ures, r esp ectiv e ly . On the ot he r en d of t he “s pectru m o f ra ndomnes s”, exact resul t s have b een obta in ed o n Poiss o n-Voron o i t ess ellat i o n s u sing r ather cumbersom e analy t ical too l s. W e r eport som e o f t he resul ts d i scussed by F i nch (2003): • the average n umber o f ve rt ice s i s ( ) 0709 . 27 35 96 2 ≈ = π µ v an d i t s standard devi at i o n is ( ) 6708 . 6 ≈ v σ ; expl o i t ing t h e E ul er-Po inca re r elati o n p lus t he ge ne r ici t y pro per t y , we o btain ( ) ( ) v e µ µ 2 3 = , ( ) ( ) 2 2 1 + = v f µ µ , ( ) ( ) v e σ σ 2 3 = , and ( ) ( ) v f σ σ 2 1 = ; • the av erage s ur f ace area is ( ) 3 2 0 3 2 0 3 1 8209 . 5 3 5 3 256 − − ≈       Γ       = ρ ρ π µ A (w i th ( ) • Γ h ere indi cat in g t he usu a l Gamma f u ncti o n), and i t s standard devia t i on is ( ) 3 2 0 04804 − ≈ ρ σ A ; • the av erage vol ume is, by def ini t i o n , ( ) 1 0 − = ρ µ V , whe reas its stan dard devi at i o n is ( ) 1 0 4231 . 0 − ≈ ρ σ V . Simulations For the SC , BCC, a nd FCC latti c es, we in t roduce a symm etr y - brea k ing 3D-ho m ogeneous ε - Gaus s i a n n o ise , whi c h rand o mi z es the pos i t i o n of each o f t he po in t s x i about i t s determi n is t ic posi t i o n w i t h a spat i a l varianc e 2 ε . By defining 2 2 0 2 2 Q l α ρ α ε = = , th us e xpressing t he mea n squared displa c ement a s a f r act i o n 2 α o f the i nverse of t h e de nsi t y o f po in t s, whi ch is t h e natural squared l ength scale 2 Q l . The para m et er 2 α can be loose l y in t erpreted as a n or m a liz ed t em perature of t he l att i ce. Note that in a l l cases , when ensembles are consi d ered, t he d is t rib ut i o n o f t h e x i is st ill periodi c. The st a t is t i cal analysis is per f o r m ed over 100- m e mb ers e n sembles o f Voro n o i tessell at i o ns gen erated f o r all val ue s of α ran g in g f ro m 0 t o 2 w i th step 0.01, p l us additional values aime d at ch eck in g t h e weak- and high-no i se limi t s. An o t h er set of s i mulat i o ns is per f or m ed by co m put in g an ensem b le o f 100 Po isson -Vo ronoi t ess ellat i o n s generated startin g f ro m a s et of u nif o r mly ra ndom l y dis t r i buted 0 ρ po in t s per uni t vol u m e. The actua l s im u la t i ons are per f o r m ed by app lyin g, w i t hin a cust omi zed rout in e, the MATLA B7.0  func t i o ns v o ro n o i n . m and conv hulln . m, whi ch implemen t the a l go ri t hm in t ro duced by B ar ber et al . (1996), to a se t o f po in t s x i havi ng c o arse gra in ed d e n si t y 100000 0 = ρ an d gen erated accordin g t o the co nsi dered rando m process . T h e f unct i on voronoin . m a ssoc i at es to each 8 poin t t he vert i ce s o f t h e correspondin g Voro noi cell a n d i t s vo l u me, whereas t he f u nct i o n convh ulln.m is use d t o gen erate the conv ex hull of t he cell . Note t hat t he conv e x h u ll is given in t er m s of 2-s im p li c es, i. e. t ri angles. Wh erea s t his inf or m at ion is sufficient f o r co m put in g the t otal sur f a ce area o f the ce l l, a n add i t i onal step i s nee ded in o rder t o def ine t he to pol o g i ca l pro perti es o f the cell . I n f act, in o rder to dete rmi ne the actual n u mbe r of faces o f t he ce ll a nd define exact l y wh at po l ygon eac h face is, we ne ed t o expl or e wh et h er nei g hb o ur in g simpli c es are cop l a na r, and t h us constitut e hi gher order po ly gons. Thi s is accom p lished by co m put in g t he u ni t vecto r k i ˆ ort h o gon a l to ea ch s implex k s and co mp ut in g the ma t r i x o f the scalar pro duc t s k j i i ˆ , ˆ for al l t he simplices o f t he cell . Wh e n the scalar pro duc t s q p i i ˆ , ˆ , 1 , − + ≤ ≤ n m q p m of u ni t vecto rs ort h o gonal t o n n ei g hbour i ng s im p lices k s 1 − + ≤ ≤ n m k m are close t o 1 - wi t hin a specifie d to l erance ξ , c orrespondin g t o a tole ra nc e o f ab o ut ξ 2 in t he angle between t h e u ni t vector - we hav e t hat U 1 − + = n m m k k s is a po ly go n w i t h n+ 2 s i d es. W e have consis t ent ly v er ifi ed t ha t choo sin g a ny t oleranc e smaller t h a n 8 10 − = ξ we o b t ain bas i call y t h e sam e results. Other val ue s o f 0 ρ - s m a ller a nd larger t h a n 100000 0 = ρ - h a ve been used in o rder to check t h e pr evi o usly de scribed scaling laws, which are f o un d to be preci sely ve r ifi ed in a ll numerica l ex per im e n t s. T h e ben e fi t o f usin g such a large value o f 0 ρ re li es in the fac6t t hat fl uct uati o n s in t he ensem b le s are qu i t e s m a ll . Tessellati o n h as b een p erf or m ed start in g fro m po in t s x i bel o nging to t he square [ ] [ ] [ ] [ ] [ ] [ ] 1 , 0 1 , 0 1 , 0 1 . 1 , 1 . 0 1 . 1 , 1 . 0 1 . 1 , 1 . 0 1 ⊗ ⊗ = Γ ⊃ − ⊗ − ⊗ − , but o nly t h e cells b e l o nging t o 1 Γ h a v e been consi dered for ev aluat in g t he stat i st ic a l pro perti es, in o rder to basic a lly avo i d 0 ρ depl et i o n in t he case of la rge val ue s o f α due t o o n e-step Browni a n d iffusi o n o f t h e po in t s ne ar b y t h e bounda r i es. 3. Fr om Regular to Rand o m Lattice s By de fini t i o n , if α = 0 we are i n the determi n is t ic c ase of S C, BCC , and FCC latti ces. We study h o w t h e geo m et ri cal pro perti es o f t he Vo ronoi cells c h a n ge with α , covering the w ho l e r ange go in g fro m t h e s ymm et r y break, o ccurrin g when α bec o m e s po si t ive , up to t he pro gressi vely m o re and m o re unif o r m d is t r ib ut i o n o f x i , o bta i ned when α is large wi t h res pect to 1 and the pdf s o f ne ar by po in t s x i overl ap m or e an d m or e si g ni fica ntly . 9 Faces , E dges, Vertices Wh en sp atial no i se is present in t he system , t he resul t in g Vor onoi cell s are gen er i c po ly hedra, so t h at degene rat e quadri valent v ert i ce s, suc h as those presen t in the rho mb ic do deca hedron (Tro adec et al . 1998) a re re m o ve d wi t h pro babili t y 1. Theref o re, we e xpec t that ( ) ( ) v e µ µ 2 3 = , ( ) ( ) 2 2 1 + = v f µ µ , ( ) ( ) v e σ σ 2 3 = , an d ( ) ( ) v f σ σ 2 1 = . Thes e re l at ion s have b e en v er ifi ed up t o a v ery high degree o f accuracy in o ur sim u la t i ons, so t hat, i n o rder to descri be t he topol o gy of the cell , i t is su ffi cient to presen t t he stat i st i c al properti e s of j u st one am o n g e , f , an d v . In Fi gure 1 we prese nt our results rel at iv e t o the number o f f aces of the Voronoi ce ll s. In t he case o f t he SC a nd FCC cr ystals, the in t ro ducti o n o f a minimal a m o u nt o f s ymm et r y - b reak in g no i se in duc es a t ransi t i o n t he stat is t ic s o f ( ) f µ and ( ) f σ , s ince ( ) f µ and ( ) f σ are discontin uo us in 0 = α . In the SC case, t h e average numb er o f fa ces j umps f r o m 8 to o v er 16, wh ereas, as di scussed by Tro adec e t al . (1998), t he di sappearance of t he quadriv alent v ert i ces in the rho m bic do decahe dro n case causes an in cre ase of t wo uni t s (up to exac t ly 14) i n the average n u mbe r o f f aces . Ne ar 0 = α , f o r both SC and FCC per t urbed cr y st als ( ) f µ depen ds li nearly o n α as ( ) ( ) γα µ µ α + ≈ + = 0 f f , wh ere by + = • 0 α we m ean t he li mi t f or i nfini t es imal no is e o f the quan t i t y • . The pr oporti o nali t y co ns t an t has o pposi t e sign in t he t wo cas es, w i t h 5 . 1 − ≈ γ f or t h e SC an d 1 ≈ γ f o r t he F CC cas e. Moreove r, the in t roducti o n o f no i se ge ne rat es t he sudden appearance o f a fi n i t e standa r d devi at i o n in the number o f f a ces in eac h ce ll ( ) + = 0 α σ f , which is larger for SC cr y sta l s. I n the case of FCC cr ystal s, Tr oadec et a l . (1998) pro p o se a theoret i ca l va l u e ( ) 3 4 0 = + = α σ f , whereas o ur n u me r ical est i mate is a bout 10% lower, ac t ually in go od ag reem e n t wi t h t he numerical r es u l t s presen t ed by Tr oadec et al . i n t h e sam e p aper. Som ewhat surpr i s ingly, the ( ) f σ is a lm o st constan t f or a fini te ra nge nea r 0 = α ran g ing up t o about 3 . 0 ≈ α f or t he SC cr y stal and 2 . 0 ≈ α f or t h e FCC cr ys t a l , thus de fini ng a n in t r i nsic w id t h o f t he d is t rib ut i o n o f fac es for a – well-defined - “ weakly perturbed” s t ate. T h us, we f ind, in the case of the pert urbe d FCC crystals, the ran ge o f appli cabili t y o f the weak n o i se li near pert urbati on an a ly s is by Tr oadec et al . ( 1998). Wh en co nside r in g t he B CC crysta l , t he im pact of in t roduc in g n o i se i n t he pos i t i o n o f t he poin t s x i i s rather d i ffe r e n t . Res ul ts are also sh o wn in Fig. 1. I nfi n i t esimal no i se does not eff ect a t all t h e t essel lat i o n, in t he sens e t ha t a ll Vo ronoi cells are 14- f aceted (as in t h e u nperturbed state). Moreove r, ev en fi ni t e- size n o ise b a sically does not dis t ort cells in su ch a way t hat ot her po ly hedra are created,. We h ave not obse r ved – also go in g to highe r densi t ies - a ny non-14 fa c eted poly hedro n 10 f or up to 1 . 0 ≈ α in any memb er o f the e n semble, so that ( ) 14 = f µ and ( ) 0 = f σ in a fini t e ran ge. Ho wev er, s ince t he Gaussian no i s e we are usin g ind uces f o r each po in t x i a d istr ib ut i o n wi t h – an unreali st i c- n on-com pa ct supp o rt, in pr in c iple i t is po ssi ble t o h a v e o utlie rs t h at , at l o ca l level, can d is t o rt h eavily the t essella t i on, so t hat we s houl d i nterpret thi s result as t ha t findin g non 14- fac et ed cell s i s highly – in so m e s ense, ex po nentially - unli ke ly . In Fi gure 1 the Po i sso n -Vo ronoi limi t in g case is indic at ed; o ur s im u lat i o ns provide resul t s in perfe ct a gree m e n t wi t h t he analy t ica l by Fi nc h (2003). For 1 > α t he value o f ( ) f µ and ( ) f σ of t h e Vo ronoi t essella t i o ns of the t hree pertur b ed cr ys t als asymptot i cally converge to wh at resul t in g f ro m t he Po i sso n -Vo ronoi t ess e lla t i o n, as expected. We should note, t h o ugh , tha t i n the 2D cas e t he a sym ptot i c convergence h as been shown t o be much s l o wer (Lu carini 2008), so t hat spati a l n o i s e in 3D seems t o mi x t hings up much m or e e fficient ly . S imila r ly t o t h e 2D ca se, t he pert urbe d t es sell at i o ns are s t ati st i call y undis t in gu ishable – e speci a lly those result i ng f ro m t he B CC a n d FCC dis t ort ed la t t i ces – well bef or e conv erging t o t h e Po i s son-Voronoi case, t hus po in t in g a t som e gen eral beh a vi o r. A dd i t i o nal s t at i st i c al pro pert i es o f t he d i st rib ut i o n o f the numbe r o f fa c es in t he Vo ronoi t es sell at i o n need to be menti oned. The m o de o f t he d is t rib ut i o n is qui t e interest in g s inc e t he number of f aces is , o bvi ou s ly , in t eger. In the FC and B CC case s, up to 3 . 0 ≈ α t he m o de i s 14, w h ereas f o r la rger values o f α 15-f acet ed poly hedra are the m o st com m o n ones . Also t h e Vo ron o i t ess ellat i o ns of m ed i u m - to – hi g hly pertur b ed S C cr y st als are do m inated by 15- face t ed po ly hedra, w heras 16- fac et ed poly hedra do m inate up to 25 . 0 ≈ α . W e also o bs erve that for all non -s ing u la r cas e s - 0 > α for S C and FCC cr ys t als, 1 . 0 > α f or BCC cr y sta ls -, an d a fortiori for t h e Po i sson-Vo ronoi case, the dis t rib ut i o n of faces can be represen t ed to a v er y high degree o f pr ecisi o n wi t h a 2-param eter gam ma d istr ib ut i o n : ( ) [ ] ( ) k x x N k x g k k V Γ − = − θ θ θ ex p , 1 (1), wh ere ( ) k Γ i s t he usua l g amm a funct i on a nd 0 ρ ≈ V N i s, by de fi n i t i o n , t he n or m a lizati o n facto r. W e re mi nd t h at ( ) ( ) ( ) f f k f θ µ = and ( ) ( ) ( ) 2 2 f f k f θ σ = , so t hat a ll infor ma t i on r egard in g ( ) f k an d ( ) f θ can b e deduced f ro m F i gure 1. I n part ic u la r, in t he Po i sson-Voro noi case, our r esul t s are in e xcell e n t agreement wi t h those o f Tan e mu ra ( 2003). As a side not e, we m e n t i o n that in su ch a limi t cas e, we obse rve cells w i t h numb er of f a ces r an g i ng f ro m 6 to 36. 11 S i nce , as previ o usly discussed, t he inv er se of numbe r of faces y =1/ f is re l ated to t h e a v erag e n u mbe r n o f si des per face in eac h Vo ronoi cell as y f n 12 6 12 6 − = − = , ex press i o n (1) can in pr i nciple b e u sed a ls o for studyi ng the stat i st i cal pro per t i e s o f n a nd f o r fi nding explici t ly , w i t h a sim p l e change o f v ar iabl e, t he n d i st rib ut i o n (o r at l east a stat i st i ca l m o de l f o r i t ) , whi c h i s not a 2 - param et er gam ma distr ib ut i o n . Usin g expressi o n ( 1), we have t hat: ( ) ( ) ( ) ( ) ( ) f k k f y n µ θ θ µ µ µ 12 6 12 6 1 12 6 1 12 6 12 6 − = − < − − = − = − = , (3) In the Poi sson-Voro noi limi t, Fin ch (2003) pr o poses : ( ) ( ) ( ) ( ) 22 . 5 35 24 144 12 6 1 12 6 2 2 ≈ + = − = − = π π µ µ µ f f n . (4) This r esul t s seems to be wron g, as di rect num er ical simula t i o ns give ( ) ( ) 19 . 5 1 12 6 ≈ − = f n µ µ . The mi st ake seems to deriv e from the fac t t h at the mean o f the inve rse o f the numbe r o f fa c es is diff erent f ro m the inverse o f t he mean. I f , ins t ead, we use expressi o n (3) f o r com put in g ( ) n µ , thus usin g t h e 2-para m et er gamm a m o de l in the P o i sson-Voronoi c ase, we o btain a n a lm ost ex act resul t . The same app li es als o f o r al l con s i dered values of α an d f or the t hree pe rt urb ed crysta l li n e structures consi d ered in this study . Thi s f urt h er rei n f o rces t he i dea t ha t 2-param eter gam ma f a mily pdfs shoul d be t h o ugh t as e xc e llen t st a t is t ic a l mo del s f o r the dis t r i but i ons o f numbe r of faces in gen eral 3D Vo ronoi t ess ella t i o n. Area and Volume of the cells The stat i st i ca l pro perti es o f t he area and of t he vo l u m e o f t he Vo ronoi t essellat i o ns o f t h e pert urbe d cubi c cr ystals have a less pat h o l o g i cal b e h a vi o ur wi t h respec t to what previ o usly descr i bed whe n n o i s e is tur n ed on, as all pro perti es are cont in uous an d differen t i able in 0 = α . Still , so m e rat her in t erest in g f eat ures can be obse r ved. As menti o ned above, the ens e mbl e mean of t he mean v o l u me o f t h e c ell s is set to 1 0 − ρ in a l l cas es, s o that we d i s cuss t he pro pertie s o f t he e nsembl e mean ( ) V σ , sh o wn in Figure 2. We f irst observ e t hat f or all cubi c s t ruct ures t he s t an dard dev iat i o n co nv erges t o zero wi t h vanishi ng n o is e, t h us m eaning t hat sm a l l v ar i at i o ns in t he po si t ion i n t he lattice po in t s d o n ot crea t e dram at ic rearran ge m e n t s in t h e ce ll s w h e n t hei r vo l u mes are consi dered. Moreover, f o r 3 . 0 < α , a well - defin ed linear b e h a vi o ur ( ) Χ ≈ α σ V is observed f or all S C, BC C, and FC C struct ures. T h e 12 pro p orti o n a li t y co n st an t X i s n o t di st in gu ishable between t h e BCC a n d t he F CC pert urbe d cr y st als, an d actua l ly ( ) PV V σ 2 . 1 ≈ Χ , where t h e p edix r efers t o t he a symptot i c Po i sso n -Vo ronoi val ue ; t he SC curve is so m ewhat st eepe r n ear t he o ri g in. The t hree curves beco m e u n d is t in gu ishable f o r 4 . 0 > α , so when t he no i se is m o derate ly in t e n se a n d r etic u les are st ill re lat i vely o rganiz ed. As previ o usly o bserved, t his s eems t o be a rat her general and ro bus t fe at ure. It is a l so i nterest in g to n o t e that t he attainm ent o f t he Po i sso n-Voronoi limi t i s qu i t e s l o w, as co m pared t o the case o f the statis t i cal pro pert i es o f t he number of fa c es of t he ce l l, and a co m para ble agree men t is obtain ed o nl y f or 3 > α (not sh o wn ). Wh en co nsi der i ng t he area o f t h e cells (s ee Figure 3), f urt h er in t erest in g pro per t i es can be highli g hted. Fi rst, the behavi o ur of t he ensemble mean ( ) A σ is rat he r simil ar to wh at j ust discus sed f o r ( ) V σ . Neverthele ss, in this case the agreemen t between t he three perturbed struct ure is m o re precisely ver ifi ed – t he t hree curves are b are l y dis t in gu ishable f o r all values o f α . Moreove r, we again o bs er ve a li near beh a vi o ur l ike ( ) ( ) PV A A σ α σ ≈ , whi c h suggests t hat t he sys t ems cl o sely “a li g n” t o wards Poiss o n - like ran do mn e ss.f o r small val ues o f α . This cl o sel y mi r ro r what observ ed in the 2 D case f o r t he expecta t i on value o f the standa rd deviat i o n of t he perim et er and of t he area o f t he Vo ronoi cell s (Lucar ini 2008). The pro per t i es of ( ) A µ for the pert urb ed c r y sta l st ructures a re als o sh own i n Fi gure 3. A stri k ing f eature i s that, sim ilar ly to what n ot ed i n t he ca se of tr i a n gu l ar and s quare t essella t i o ns o f t h e plane, t here is a specific a m o u n t o f n o i se t hat o p t imi zes the mea n s ur f ace f o r t he pert urbed S C cry st als. We see t hat t he m ea n area o f the ce lls decreases by a b o ut 8% wh en α is inc reased fro m 0 to about 0. 3 , wh ere a ( quadra t i c) mi ni m u m is attain ed. For stron ger no i se, t he m e an area o f t he Voronoi cells o f t he pert urbed SC cr ystals decreases, a nd , f or 5 . 0 > α , t he asymptot i c v alue o f the Poisson -Vo ronoi t essella t i o n is reached. I n ter m s o f cell sur fa ce minimi zat i on, the unpe rt urbed cubi c t ess e lla t i o n is ab o ut 3 % w orse than t he “m o st r an dom” t essella t i o n. The depende n ce o f ( ) A µ with respect to α is ve r y in t erest in g a ls o f or t h e pert urbe d BC C an d FCC cubic cr y sta ls . In bot h case s, the m ean area inc reases quadrat i c a l ly (w i th very similar coeffici e n t ) for s mall values o f α , which s h o ws that t he Voronoi t es sellat i ons o f t h e B CC a n d FCC cubi c crysta l s are l o ca l mi n ima f o r t he mean sur fac e i n t he set o f spac e- filli ng t essella t i o ns. We kn o w t h at n e i t her t he t runca t ed octahed ro n nor the rhombi c do decah edro n are g l o bal minima, since (at le ast) the We a i r e-Phelan s t ructur e h as a small er sur f a ce (Weai r e and P h e l a n 1994). It i s reasonabl e to expec t t hat a si mila r quadrat i c inc rease of t h e average sur fac e shou l d be o bs er ved wh e n pert urbin g w i t h spat i al gaussian n o i se t h e cr y sta lli ne struct ure c o rres po n d in g t o the W eaire- 13 Phel a n ce ll. F o r 3 . 0 > α , the v a l ues o f ( ) A µ f o r perturbed BCC a nd FCC cr ys t als basi c ally coin cide, a n d f o r 5 . 0 > α the Poi sson-Voro n o i limi t i s reach ed w i t hin a high acc urac y . Al so in t h e case of c ell s area and vo l ume, for 0 > α , th e empirical pdfs can be fi t t ed v er y effi cient ly u sin g 2-parameter gam ma distr ib ut i o ns, thus confirmi ng w ha t observ ed i n the 2D case (Lucarini 2008). Fluctuations, Shape an d Anom alous Scali ng The analysi s o f the pro per t ies o f the j o in t area - volume pdf f o r the Voro n o i ce lls o f the co nsidered t es sell at i o ns s heds li g h t o n t he st a t i st i cs o f fl uct uati o ns of these quan t i t i e s. As d i scussed ab o ve , the area and the vo l u me of the Vo ronoi cells res u l t in g f r o m a r ando m t es sell at i o ns cells are highly variable. See Fi gure 4 f o r t he j o in t ce ll s area- vo l u me dis t r ib uti o n in the cas e o f P o i sson-Voronoi t ess e lla t i on. All co nsi d ered perturb ed crysta l st ruc t ures gi ve quali t at iv el y simil ar res u l t s, b ut fe at ure, as obvi o us fro m the p revi o us d iscussi o n, m o re pea ked dis t r ib ut i ons. I n partic u la r, by i ntegrat in g the 2D dis t r ib ut i o n a l o ng ei t her d irec t i on, we obtain a 1D -pd f whi c h , as previ o usly d i scussed, i s c l o sely appro xima t ed wi t h a 2-param et er gamm a distr ib ut i o n . A fi r st in t erestin g st atistical property w here the j o in t c e l ls area-vo l ume pd f h a s t o be conside red is t he is o perim et r i c quo t i e nt 3 2 36 S V Q π = . When e val uat in g i t s e x pectat i on value, we h a v e: ( ) ( ) ( ) ( ) 3 2 3 2 36 36 A V A V Q µ µ π µ π µ ≠ = . (5) This implies t h at t est i ng t h e a verage “spherici t y” – which is basically what Q mea sure s - o f a random t essell at i o n is a slight ly dif fe re nt pro bl e m fro m t estin g t he av erage sur face for a g i ven av erage volume , whi c h is w hat Fi gur e 3 refers to , wh ereas in regu la r t essell at i o ns the t wo problem s are equiv alent. In Figure 5 we present o ur resul t s f o r t h e t h ree pert urb ed crysta l st ruct u res. The observ ed be h a vi o ur are quali t at i ve ly simila r t o wh at co ul d be deduced f ro m Figure 3 using t he quan t i t y ( ) ( ) 3 2 36 A V µ µ π as an es t im ate f o r ( ) Q µ - e.g. ( ) Q µ decreases w i t h α f or t he BCC a nd FCC pert ur b ed cr ys t als - but si g nifican t add i t i o n a l inf o r m at i on e merges . We not e t hat t he discrepancy be t ween the e st im at e an d the exact resu l t f o r ( ) Q µ is posi t iv e a nd m o noto ni ca ll y inc rea sin g wit h α , wi t h a re l at iv e err or o f t h e o rde r o f 10% when t he Po i sso n -Vo ronoi limi t is attain ed. Simila r ly t o what o bs er ve d whe n analyzin g the stat i st i cal pro perti e s o f t he area of the cell s, in t his case we have t hat by opt im a lly t unin g t h e intensi t y o f t he no i se ( 3 . 0 ≈ α ) perturbin g t he S C cr y st al , ( ) Q µ reaches a maxim u m (w i t h a 20% increase fro m bot h the P oiss o n-Voro noi limi t and a 25% in crea se f ro m the regular cry st al limi ts), so that the cor resp ondin g cel l s are on the a verage “m o re s p he r ical”. It shou l d also be e mphasized t hat, f o r all pert urbed cr y st als a n d f o r all 0 > α , Q 14 fea t ures a notable v ar iabili t y ( in t he Poisson-Vorono i limi t Q values range fro m 0.15 to 0.77) , whi c h implies t hat the sh ape of the cell s can v ar y w ildly . An a n o m a l o us sca li ng is o b served whe n fi t t ing a po wer l aw be t ween the value o f t h e area an d o f and the value o f t he vo l ume o f t he cells. Wh e n att emptin g a linear fit be t ween the l o gar i t h m of t he vo l u me a nd o f t he area o f cells, we f i nd t hat t he best fi t o f t he e xpon e nt η such t hat η A V ∝ is in all cas es la rger t h a n 3/2. T he values o f t h e best fi t for η f or t h e pert urbed SC, BCC , a nd FCC cr y st als are shown i n F i gure 6 as a f u n ct i o n o f α ; we w ish t o r em ark t hat t h e qual i t y o f t he fi t s is v er y hi gh, wi t h r elative u n cert aint i es o f the order o f at m o st 2 10 − in all ca ses. In pa rt i cular, in the Po i s son-Voronoi l imi t 67 . 1 ≈ η - bla ck line in Figure 4 - which suggests the o ccurrenc e of a 5/ 3 ex po nen t . It is a ls o re m arka ble t hat, as soon as no is e is t urned o n, an o m a l ou s scaling is o b ser v ed in f or t h e SC an d BCC cu bi c cr ys t al s. In t he SC case, ( ) 9 . 1 0 ≈ = + α η , t he expon e nt i s m o noto ni call y decreasi ng f o r all val ues o f α , and bec o m e s un d is t in gu ishable from t h e Po is son-Vor onoi l imi t for 2 > α . In t he B CC case, as opposed to wh at one co ul d expe ct given t he st ruc t ural stabili t y of the cr y st al , ( ) 2 3 57 . 1 0 > ≈ = + α η , whi ch implies tha t a ( m o dest) an o m a l ou s scaling is o bserved als o f or infi n i t es im a l no i se. The exponent in creases fo r sm al l values o f α , oversh oot s the Po is son- Voro n o i limi t , an d f o r 6 . 0 > α i t s val ue basi c ally co incides wi t what obtain ed in t he SC case. Wh e n conside r ing the FCC pert urbe d cr ys t a l , an a no m a l o us sca li ng is o sb ere vd f o r al l finite values of no i s e, b ut , qui t e not ably , ( ) 2 3 0 = = + α η , which means t h at fo r infi ni t esi m a l n o is e ano mal ous scaling i s not o b serve d. So, i n this regard, the FCC cr ystal seems t o b e m o re robust t h an t he BCC one. The presence o f fl uct uati o ns i n t he val ue s o f t he a rea an d o f t h e v o l u me o f t h e cells m a y be inv o ked as a n e x p lana t i on for t he o bs er v ed a n o mal o us callings. Not e t hat t hi s link is not anobvi o us one: i f the space i s t essell at ed w i t h cu b es f eat urin g a spec t ru m o f s ide lengths, we woul d in al l case s observe a 3/2 sca lin g e xponen t be t ween areas and vol u mes o f t h e cells. Theref o re, t he rea son f or t he a no m al o u s scali ng s hou l d be r elated to the fac t that cel ls o f vari o us s iz es are n o t geom et r i ca lly simi l ar ( in a s t atis t ic a l sense), which agrees w i t h t he f act t hat t he i soper im etr i c rat i o Q ha s a dist in ct v ar i abilit y . Nevertheless, the stati st i cal pro per t i es o f Q do not expl ain why t he FCC cr y st al – a s o pposed to t he BCC o ne- do es n o t featur e ano m a l ous s ca lin g i n t he l o w n o ise l imi t , whi c h im p lies that t h e shape o f i t s Vo ronoi cells are m o re stable t han in t he BCC case In o rder to cla r ify t he i mpact o f “s h ape” fl uct uations i n t h e est im at e o f t he expone nt η , we t ake advantage of a strat egy o f invest i gat i o n co mm o nly ado pt ed for st udy ing Vo ronoi t essell at ions, i.e. t he str a t ifi cat i o n o f t h e ex pe ctati o n v a lue s of t he geo m et ri cal pr opert i es w i t h re spect to cl asses define d by t he numbe r o f s i de s o f t he ce l ls (Z h o u et al . 2001, Hi lh o rst 2005, Lucari ni 2008). I n t he 15 presen t case, i t wou l d be pro fi t able t o s t udy quant i t i es suc h as ( ) f V µ a nd ( ) f A µ , wh ere t h e pedix indicates t h at the average s are per for m ed only o n ce lls w i t h f f aces. It i s readi ly o bse r ved that, in a ll ca ses , b ot h ( ) f V µ a nd ( ) f A µ increase w i t h f (not shown), f o r t he b as ic and i ntu i t ive reason t hat cel l s with a larger numbe r of faces are t y pica lly l arger in v o l ume a nd ha v e larger areas. Moreover, an d t hi s is a m o re in t erest in g po int, l ar ger cell s are also t y p ica lly bu lkier, so t ha t t h e i r is o perime t r i c Q is l arger. An e x a ct resu l t in this sense is that in 2D Po i sso n -Vo ronoi t ess e lla t i ons cell s w i t h a la rge number of sides asymptot i cally conv erge t o ci rc les (H il h o rst 2005). I n Fi gures 7(a), 7(b), and 7(c) we prese n t t he expectat i o n value o f t h e is o per i metr i c quo t i e nt ( ) f Q µ as a funct i o n o f α f or t he t hree pert urbed cu bic cr ystals. We obse r ve t hat in a ll ca ses, for a g iven value of α , ( ) f Q µ inc reases si g nifi ca nt ly wi t h f , t hus confi r ming the geome t ri cal in t u i t i o n. Mo reover, wh erea s in t he BCC a nd FCC cas e ( ) f Q µ f o r a gi ven f de creases m o noto ni cally wi t h α , in the SC pert u rbed cr ystal we ha v e an i ncrease o f ( ) f Q µ with α f o r al l values o f f up to 3 . 0 ≈ α . Th eref o re, t he i soperim etric q uoti e n t op t im ization due to n oi se o bse rved i n Fi gure 3 f o r SC perturb ed crys t al s o ccu rs not only o n the overall av erage , b ut a l so sepa rat ely i n each class o f cells . F o r 1 > α t h e resul t s of the thre e perturb ed cry st als t en d to conv erge to the Pois son-V o ron oi limi t. A t any rate, the m ain inf or ma t i o n c o n t ai n ed i n Figure s 7 is t ha t f or al l perturbe d cry sta l structures: a) cel ls are really v er y va r iabl e in t erm s of s h ape ; b ) the num ber o f fa ces ac t s as a good pr oxy v ar i able f or t he is o perim et ri c quot i ent, an d f or the s ha pe o f t he cell s. It em b ) su ggests us a stra t egy f o r cross-c he cking th e re lev anc e o f the fl uctuati ons in t he s h ape o f the cell s in determi n in g t h e pres ence o f t he a n omal o us ex ponen t 2 3 > η depi ct ed in Figure 6. W e t h en attem pt a power-l aw fi t η A V ∝ sep arate ly f o r each c las s of cells , so t h at f o r each val ue o f noi se we obtai n vi a a b est f i t t h e o ptim al f η for t he subset of c e l ls havi ng f f ace s. It is amazin g t o fi nd t ha t f o r al l c ases consid ered f η resul ts t o be l o wer that t he c o rresp o n din g “ gl o bal” η , and f or m ost v alues o f α and f w e hav e th at 6 . 1 2 3 < < f η . Su ch beh avi o ur is also i ndepen den t of t he t y p ic al popul at i o n o f t h e c l a sses o f ce ll s. W e do not pl ot f η as a fun ct i on o f α f o r the t hree pert urb ed cry stals, si nce wh at i s obtain ed is j u st a pl ateau wi t h n o struc t ure a l ong t he α or f di r ec t i o ns . W e then hav e t ha t the f -cl a ssifi cat i o n re m o ves m o st of t h e s h ape f l uctuati o n s w i t hi n each class, so t h at an o m a l o us scalin g is greatly sup pressed . We beli eve t ha t 16 4. Summ a ry and C oncl us ions In t hi s wor k we hav e st udi ed t h e st atis t i cal pro per ti es o f the 3D Voro noi t ess ellati o n s generated by poin t s s i t uated on per t urb ed cubi c crystal st ructur es , nam ely the SC, BC C, an d FCC latt i ces. The perturb at i o ns t o th e perfec t crys tal s t ructures are ob t ain ed by applying Gaussi an n o i se t o the posi t i o ns o f each po in t .. The vari a nce o f t he pos i t ion of t h e po in t s in duce d by t he Gaussian n o i se is ex presse d a s 3 2 0 2 2 ρ α ε = , wh ere α (whi ch is a sort o f square ro ot o f a norm alize d t em perature) i s t he co n t rol p aram eter, 0 ρ i s t he num ber o f po in t s – an d o f Voro n oi cells – per u ni t v o l u me , an d 3 1 0 − ρ i s t he res ul t ing n at ural l e n gth scal e. By i ncrea sing α , w e expl o re t he tran si t i o n f r o m pe r f ect cry stals t o l ess and le ss regula r st ruc t ures, un til t h e l imi t o f unif or mly r and o m distr ib ut i o n o f po in ts i s att ai n ed. Note t h at, f or all valu e s of α , th e pdf of t he l at ti ce poi nts is in all cases peri od ic . For each v alue of α , we h a ve pe r f or m ed a s et of s im ulati o ns , in order to crea t e an e n sem b l e o f V o ronoi tessel la t i o n s i n the u ni t cub e, a n d have c o m puted the stat i st i c al properti es of the cell s. Th e fi r st n o tabl e resul t is t h at t he Voronoi cells correspon d i ng t o SC an d FCC la t ti ces (cube an d r h o mbi c do dec ah edron, res pect iv e ly ) are not st abl e in terms o f t opol o gi cal st ruc ture. A s so o n as n o i se is t urne d on , t h eir degen eraci es – t o b e i nten ded as non-g e n eric properti es o f t hei r ve rt i ces – are rem oved vi a a s y mm etr y - b reak. W i t h t h e in t roduc t i o n o f a n infini t esim a l am ount of noi se, t he pdf o f t h e d is tr ib ut i on of the num ber o f fac es per cell bec o m es sm oo t h , wi t h a fi ni t e stan dard devi at i o n and a bi ased me an w i t h respe ct to t h e zero noi se case. Ins t ea d, t he zero-noi se limi t of t h e Voron o i tess ellat i o n of the pert urb ed BC C s t ru cture i s co in ciden t w i t h the u n perturbed c ase. Surpri singly , t he to pol o gy o f the perfec t BC C t ess ellat i o n is ro bu st a l so again st sm all but fi n i t e n o i se. For s t r o n g n o i se, the stati stical pro perti es of t h e tess ellat i o n s o f the t hree per t urb ed c r y st als conv erge to those of t h e Poi sson -Voro noi limi t, but, qui t e n o tably , t h e m em or y o f the speci fic ini t ial un perturbe d stat e i s l ost al ready f o r m o derate noi se, sin ce t h e stat is t i cal proper t i es o f the t hre e perturb ed t es sella t i o n s are in d i st ing u i shabl e a l r eady f or 5 . 0 > α . A s opposed to t h e 2D case, whe re h exag o n is so o v erwh elmin g ly t he “ generic p oly gon”, sin ce h exagons consti t ute the st abl e tessel la t i o n , are the m ost com m on po l yg o ns of b as i cally any t ess ellat i o n , and t he a v erage num ber of si des o f t h e cells is six ( Luca r ini 2008), in the 3D case w e do not h ave such a do m inan t topol o g i c a l struc t ure . Ra t h er in t eres t in g fe at ures emerg e l o ok in g at the me t ri c propertie s o f the cells , n a m ely t heir surf ace area and thei r vo l u m e. Giv en t h e i ntens i ve na t ure o f t he Voronoi t ess ellat i o n , t he ens emble av erag e of t h e mean and o f t he st an dard devia t i o n o f t h e area and o f t h e vo l u me o f t h e cell scale a s 3 2 0 − ρ an d 1 0 − ρ , respec t i ve l y, so t h at we ca n ea sily o b t ain unive rsal res u l t s by perf or mi ng sim u la t i o ns wi t h a s pecifi c given densi t y 0 ρ . 17 F o r al l valu es o f α , th e st an dard dev iat i o n of the area o f t he V o ronoi cells is b as i cally the sam e f o r t h e t h ree perturb ed cubi c str uc t ures: i t f e atures a l in ear i n crease f or sm a ll noi se, an d th e n approac he s asym ptot i cally t he Po i sson-V o ronoi limi t for l arge valu es o f α . Practi cally the same appl ies f o r the st an dard devi ati on of t he vol u m e o f t he cell s. Wh ereas t he en sem b l e a v erage o f t h e mean v o l u m e is set to 1 0 − ρ by de fi n i t i o n, t he analy s is of t he pro per t i es o f t h e ensem ble av erage of the me an area o f t h e cells is qu i t e i nsigh t f ul . I n t he c ase of perturbed BCC a n d F CC structures , t he m ean ar ea in creases o nl y quadra t ic ally w i t h α f o r weak n o i se, t h us s ugges t in g tha t th e un perturb ed Voron oi t es sella t i o n are l ocal mi n im a f o r the in t erf ace area . In stead, t h e m ea n a rea o f the pe rt urb ed SC st ruc ture has a m o re i n t eres t i ng depend e nce o n n o i se i ntensi t y : i t decreas es up to 3 . 0 ≈ α , whe re a l oc al mi n im um is attain ed, whe ras f or la rger v alues of α , i t in creases un t i l reachi ng t h e Po i sso n -Voron oi limi t (as f o r FC C and BCC ). S o , coun t er-i ntu i t iv e ly , no i s e ca n ac t as an “optim i z er” i n t he case o f t h e per t urb ed SC struc t ure. In all cases analy zed, 2-param et er gamm a distr i buti o n s are very eff ectiv e i n describi ng t he em pirical pdfs both for di screte va r i a bl e s, such as t h e n umbe r o f fa ces, and f or co n t i nuous vari a bl e, as in the case o f t h e area and vol u m e of t h e cells . In parti cular, su c h an approach h as be en usef ul in i dent if yi ng an d ex plaini ng a mi stake conta i ne d i n Finch (20 03) r ega rd i ng the ens e m ble avera ge o f the m ean num ber of si des per f ace i n ea ch Voron o i cell . Th e pres ence of stro n g fl uct uati o ns i n th e area an d vol u m e o f t he Voron o i cell s h a s sug gested to ch eck so m e j o in t a rea-vo l u m e st ati stic a l properti es. In pa rt i cula r, we hav e consi d ered the i soper i me t ri c qu ot i ent Q , whi ch m easure s the sphe r i c i t y o f t he c ells a n d i s a goo d quan ti t ative m easure o f t he shap e o f t he cells . The e nsem ble a v erage o f t h e m ean val ue of Q is quad rat i c m aximum f o r t h e F CC an d BCC cr y sta l s f o r 0 = α , and, f o r any v a l ue o f α , t h e pe rt urb ed BCC struc t ure h as the la rgest m ean isoperim et r i c quot i ent.. The o b serva t i o n t ha t t h e trun cated octah edron i s a “ larg e ” l ocal maxim u m for Q f o r space - filli ng t es sell at i o ns sugg ests a weak re- f o r m u l at i o n o f the Kel vin conje cture o f gl obal o ptim ali t y o f t h e trun cated o c t ah edron , whi c h h as been pro v ed fals e (W eaire a n d Phelan 1994). More over, wh e n n o is e in incl uded in t he sys t em , Q fe atures in all cases a rat h er la rg e v ar i abili t y , whi ch sugg e sts t h at t he s h ape o f the cell s ca n v ar y a l ot w i thin a g iv e n tes sella t i o n .. I n parti cula r, i t i s obs erved t h at the n u m ber of fa ces of a c e l l is a goo d p roxy f o r i t s i soperi m etr ic quoti e n t : ce l ls wi t h a la rger numb er of fac es are bul k ie r. Th e m ain effect o f th e fl uctuat i o n s i n the sha pe of t he cell – and not m erely o f t hei r area a n d v o l u m e - i s tha t wh e n att em pt in g a power l aw fi t η A V ∝ b etween the a reas and t h e v o l umes of th e Voron o i cel ls , w e obtain f o r al l pert urb ed tessel la t i o n s a n d f or any i n t ensi t y o f n o i se an a n o m al ous scalin g, i .e. 2 3 > η , wi t h 67 . 1 ≈ η in t h e Poi sson-Vor o n o i limi t. Such an an o m al o us s caling is o b serve d also f or i nfini t esimal n o is e, exce pt in 18 the cas e o f t h e FCC cr y stal , w h o se t es sella t i o n, in t his se n se, is t h e m o st stabl e. A stro n g evi dence of t he re l evan ce o f the s hape fl uctuati ons in det er mi n i ng t h e an o m al ou s sc a li ng l ies in t h e f act that i t i s a lm o st sup press ed w hen we c l assify t h e cell s acco rdi ng t o t he num ber o f t h ei r f aces and attem pt the power l aw fi t class by cla ss. Thi s work cl ear ly de fi nes a way of conne ct in g, wi t h a simpl e param etr i c con t rol of spat ial n o i se, cry stal st ructure to unif or mly random d is tr ib uti o n o f poin t s. Such a procedure can be in prin ciple appli ed f o r desc r ibi ng t he “di sso l ut i on” of a ny crystalli ne structure . Our resul ts em p hasi z e the im por tanc e of a naly z in g i n a co mm o n f r amework t h e topol o gi cal and th e metr i c propert i es o f the Voronoi t ess ellati o ns. In part i cula r, i t proves that, i n or der to grasp a bett er un derstan din g, i t is n ecessary t o con side r t h e j o in t st ati st i cal pro per ti e s o f t h e area a n d o f t h e v o l u m e o f t h e cells, t h u s goin g b e y o nd 1D pd f . . T hi s has pro v ed t o b e especi ally us e f u l f o r f r ami ng in all gen erali t y the prope rti es o f o p t i mali t y o f the F CC a n d BCC structures. Sev era l open que st i ons call f or ans wers. The rel ationshi p b etween fl uctuati ons in the shape of t he cells an d t h e ano m a l o us scali ng, as well a s the co nn ect i o n be t ween t h e n u m ber of fac es o f t h e cell s and t h e i r a v erage is oper im et ri c rat i o, su re ly dese rve further analys es. It w oul d be im portant to an alyze t h e im pact o f spat i a l n o i se o n o ther relev ant cr y sta ll ine struct u res, such as t he la t ti ce w h o se Voron o i cell is t he We a i re-Phelan str uctu re, or the He xagon al Cl os e Pac ked crys t al , w hi ch seem s to h ave cl o se correspon den ces t o th e FCC structure, al so w hen i nfini t esim al perturbati o n to the posi t i o n o f the po i nts are consi dered (Tro adec et al . 199 8). Finally , t h e impa ct o f no i se on hi gher order s t a ti st i cal p ro per t i e s, s uch th o se of n eighbourin g cell s (Sen t hi l K u m ar an d K uma ra n 2005) , sh o uld be seri o usly addres sed. 19 Figure 1: Ensemble mean of t he mean and of the standard deviation of t he number of faces (f) of the Voronoi cells f or perturbed SC , BCC and FCC cubic cr ystal s . The error bars, whos e half-w idt h is t w ic e t h e standar d deviation comput ed over the ensemb l e, are too small to be plotted. Th e Poiss on-V oronoi limit is indi cated. Figure 2: Ens emble mean of the standard deviation of the volume (V) of t h e Voronoi c ells f or perturbed SC, BCC and FCC c ubic c r ystal. The ensemble mean of the mean is set to t he inverse of t h e density. The mean valu es i s Values are multiplied times th e appropriat e pow e r of t h e densit y in orde r t o obtain uni ve rsal functions. The err or bars, w hose half- w i dth is tw ice the stand ard devi ation comput ed over t he en semble, ar e t o o small to be plott ed. The Poisson -Voronoi limit is indi cated. 20 Figure 3: Ens emble mean of the mean and of t h e standard deviation of t h e area (A) of the Vorono i cells f or perturb ed SC, BCC and FCC c ubic c r ystals. Values are multiplied t im es the appr opriate pow e r of the de nsi ty in order to obtain universal functions . The error bars, w hose half -w i dth is tw ice the sta ndard deviat ion c omp uted over t h e ensemble , are too small t o b e p lotted. The Poisson-Vor onoi limit is indi cated. Figure 4: J oint distributi on of the area and of the volume of the Voronoi cel ls i n the Poisson-V oronoi tessellation limit. The black lin e indicates t h e best log le as t squar es fit . D et ail s in the text . 21 Figure 5: E nsemb le mean o f the isoperim etric ra ti o 3 2 36 A V Q π = of the Voronoi cells for p erturbed SC , BCC an d FCC c ub ic crystals . Th e impact of c ross -correlations betw een A and V i s given by ( ) ( ) ( ) 3 2 36 A V Q µ µ π µ − , w hich is basically the same for t h e three perturb ed lattices and i s plotted w ith a dash ed-dotted li n e. Details in the text. Figure 6: Scaling exponent η fitting t h e power-law relati on η A V ∝ for the Voronoi cells of perturb e d SC, B CC and FCC c ub ic crystals . The presenc e of an anoma lous scal ing ( 2 3 ≠ η ) due to the f luc tuations in the shape of the cells i s apparent. 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