Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process

TL;DR

This paper establishes the mathematical foundations for a particle approximation method for the Adaptive Biasing Force process, ensuring its existence, uniqueness, and convergence in molecular dynamics simulations.

Contribution

It introduces a novel interacting particle system to approximate a nonlinear stochastic differential equation with a conditional expectation term.

Findings

Proved existence and uniqueness of the nonlinear SDE.

Developed a discretization scheme using interacting particles.

Provided convergence analysis for the particle approximation.

Abstract

We prove existence and uniqueness for some nonlinear stochastic differential equation used in molecular dynamics, whose nonlinearity comes from a conditional expectation term. We also introduce an interacting particle system in order to approximate this conditional expectation, providing a discretization scheme for this equation.