Multicanonical entropy like-solution of statistical temperature weighted histogram analysis method (ST-WHAM)

The multiple histogram method [8,11], also known as weighted histogram analysis method (WHAM), has greatly improved the efficiency of Monte Carlo simulations [12,13], mainly when it comes to dealing with data obtained from replica-exchange method (REM) at different temperatures [14,15]. The REM simulation has been particularly important to sample conformations in protein folding simulations. Generalized ensemble algorithms, in which REM, the multicanonical algorithm (MUCA) and their extensions [16][17][18][19][20] are included, allow for the simulation to overcome the free-energy barriers frequently encountered in the energy landscapes.

The WHAM equations combine data from an arbitrary number M of independent Monte Carlo or molecular dynamics simulations to enhance the sampling, thereby producing thermal averages as a continuous function of the coupling parameter, being the temperature the most often parameter. The solutions of the WHAM equations, obtained from energy data stored in histograms H α (E), α = 1, • • • , M yield free-energy differences via an iterative numerical process. The success of the iterative process depends on the number of different histograms and their overlaps [11]. More recently [21], an iteration-free approach to solve the WHAM equations in terms of intensive variables has been developed. This numerical approach, namely the statistical temperature weighted histogram analysis method (ST-WHAM), yields the inverse temperature β S = ∂S/∂E directly from a new form of WHAM equations. Thermodynamic quantities like entropy, can be evaluated upon numerical integration of this statistical temperature.

The formulation of the new WHAM equations now starts from a weighted average of the numerical estimates for the density of states Ω α ,

with the normalization condition α fα (E) = 1. The density of states are estimated from the energy histograms

of the WHAM formulation [21]. The quantities N α , W α , and Z α are the number of energy entries in the histograms H α , the sampling weights, and the (unknown) partition functions, respectively. The final weighted temperature is given by

where

, and fα = Π α / α Π α . The ST-WHAM approach estimates the thermodynamic temperature from the first term in Eq. ( 3) only. This is because the second term, which amounts to the difference between ST-WHAM and WHAM, can be neglected for large samplings, N α » 1. This remark is what makes the ST-WHAM an iteration-free method to estimate the inverse effective temperature βS . Now, entropy estimates follow from a careful integration of βS (E) when one disregards the second term in Eq. (3),

Microcanonical analysis and aims -Two sampling algorithms are successfully used to estimate the density of states Ω(E): the Wang-Landau algorithm [22,23] and MUCA [24,25]. This success is intimately related to their performances in obtaining a reasonable number of round-trips between two extremal energy values [26]. Thus, depending on whether the system presents strong free-energy barrier, or not, we may face some failure. Of course this is not a feature of these algorithms only, but in fact it is an overall behavior even for other generalized-ensemble algorithms [19]. Algorithms like MUCA facilitate the microcanonical analysis, which is important for characterization of the thermodynamic aspects of phase transition in small systems. Among the MUCA applications and microcanonical analysis are the studies of heteropolymer aggregations [27][28][29][30].

The microcanonical analysis contrasts with the usual data analysis obtained from simulations in the canonical ensemble. For example, it is possible to observe thermodynamic features like temperature discontinuity and negative specific heat in the microcanonical ensemble [31][32][33][34][35][36], which appear at first-order phase transitions. A negative specific heat is a consequence of the nonconcave behavior of the microcanonical entropy S micro (E) as a function of the energy and leads to the so-called convex intruder in S micro (E) [37], a feature that is prohibited in the canonical ensemble. Systems where one finds convex intruders in the entropy present ensemble inequivalence [32,38,39]. It is noteworthy that the functions entropy S(E) and free-energy F (β) are related by the Legendre-Fenchel (LF) transform,

This relation is always true, independent of the shape of S(E), even with a nonconcave piece in its domain. The LF transform always produces a concave function.

On the other hand, if a function is not concave in its domain, then it cannot be obtained as the LF transform of another function [39]. Thus, if S micro (E) has a convex intruder, it cannot be calculated from free energies in that energy domain and, as a consequence one has thermodynamically nonequivalent systems