Analysis of an Adaptive Biasing Force method based on self-interacting dynamics

TL;DR

This paper provides a rigorous mathematical analysis of an adaptive biasing method in molecular dynamics, demonstrating its convergence and effectiveness in approximating free energy using self-interacting dynamics.

Contribution

It proves the almost sure convergence and near-accuracy of an adaptive biasing algorithm based on self-interacting dynamics, filling a gap in theoretical understanding.

Findings

Proves convergence of the bias to the ideal free energy gradient

Shows the bias approximates the free energy as an auxiliary parameter approaches zero

Establishes the method's consistency and efficiency in molecular dynamics simulations

Abstract

This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to $0$. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.