Generalized Ensemble and Tempering Simulations: A Unified View

TL;DR

This paper derives a unified mathematical framework for generalized ensemble and tempering simulations using stochastic processes, providing insights into optimizing flow and understanding limitations in ergodic systems.

Contribution

It introduces a unified view based on Master equations, deriving Fokker-Planck and hopping process models for these simulations, and discusses optimization and ergodicity limitations.

Findings

Unified stochastic process representation for simulations

Flow optimization linked to first passage time minimization

Limitations identified under broken ergodicity conditions

Abstract

From the underlying Master equations we derive one-dimensional stochastic processes that describe generalized ensemble simulations as well as tempering (simulated and parallel) simulations. The representations obtained are either in the form of a one-dimensional Fokker-Planck equation or a hopping process on a one-dimensional chain. In particular, we discuss the conditions under which these representations are valid approximate Markovian descriptions of the random walk in order parameter or control parameter space. They allow a unified discussion of the stationary distribution on, as well as of the stationary flow across each space. We demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space, and discuss the consequences of our results for optimizing simulations. Finally, we point out the limitations of these representations under conditions of broken ergodicity.