Intrinsic convergence properties of entropic sampling algorithms

RONALD DICKMAN; ROLANDO ELIO BELARDINELLI; VICTOR DANIEL PEREYRA; BRUNO JEFERSON LOURENÇO

Boston

Conferencia; CCP2014 XXVI IUPAP Conference on Computational Physics; 2014

IUPAP

We study the convergence of the density of states and thermodynamic properties in three flat-histogram simulation methods, the Wang-Landau (WL) algorithm [1], the 1/t algorithm [2], and tomographic sampling (TS) [3]. In the first case the refinement parameter f is rescaled (f -> f/2) each time the flat-histogram condition is satisfied, in the second f ~ 1/t after a suitable initial phase, while in the third f is constant (t corresponds to Monte Carlo time). To examine the intrinsic convergence properties of these methods, free of any complications associated with a specific model, we study a featureless entropy landscape, such that for each allowed energy E = 1,...,L, there is exactly one state, that is, g(E)=1 for all E. Convergence of sampling corresponds to g(E,t) -> const. as t -> infinity, so that the standard deviation σ of g over energy values is a measure of the overall sampling error. Neither the WL algorithm nor TS converge: in both cases σ saturates at long times. In the 1/t algorithm, by contrast, σ decays ~ 1/t^1/2. Modified TS and 1/t procedures, in which f ~ 1/t^α, converge for α in the interval (0,1]. There are two essential facets to convergence of flathistogram methods: elimination of initial errors in g(E), and correction of the sampling noise accumulated during the process. For a simple example, we demonstrate analytically, using a Langevin equation, that both kinds of errors can be eliminated, asymptotically, if f ~ 1/t^α for α in (0,1]. Convergence is optimal for α = 1. For α less than or equal to zero the sampling noise never decays, while for α > 1 the initial error is never completely eliminated.