Intrinsic convergence properties of entropic sampling algorithms

Abstract

We study the convergence of the density of states and thermodynamic properties in three flat-histogram simulation methods, the Wang–Landau (WL) algorithm, the 1/t algorithm, and tomographic sampling (TS). 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 → ∞, so that the standard deviation σg of g over energy values is a measure of the overall sampling error. Neither the WL algorithm nor TS converge: in both cases σg saturates at long times. In the 1/t algorithm, by contrast, σg decays $\propto 1/\sqrt{t}$ . Modified TS and 1/t procedures, in which f ∝ 1/tα, converge for α values between 0 < α ≤ 1. There are two essential facets to convergence of flat-histogram 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 with 0 < α ≤ 1. Convergence is optimal for α = 1. For α ≤ 0 the sampling noise never decays, while for α > 1 the initial error is never completely eliminated.