The Adaptive Biasing Force algorithm with non-conservative forces and related topics

TL;DR

This paper analyzes the robustness of the Adaptive Biasing Force algorithm under non-conservative forces, establishing conditions for convergence and stationary states using entropy techniques.

Contribution

It introduces a fixed point framework for the existence of stationary states and proves exponential convergence for both Adaptive Biasing Force and its projected variant.

Findings

Ensures flat histogram property under generic forces

Establishes existence of stationary states via fixed point analysis

Proves exponential convergence of biasing force and distribution

Abstract

We propose a study of the Adaptive Biasing Force method's robustness under generic (possibly non-conservative) forces. We first ensure the flat histogram property is satisfied in all cases. We then introduce a fixed point problem yielding the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms, relying on generic bounds on the invariant probability measures of homogeneous diffusions. Using classical entropy techniques, we prove the exponential convergence of both biasing force and law as time goes to infinity, for both the Adaptive Biasing Force and the Projected Adaptive Biasing Force methods.