Longtime convergence of the Temperature-Accelerated Molecular Dynamics Method

TL;DR

This paper rigorously analyzes the convergence properties of the temperature-accelerated molecular dynamics (TAMD) method, establishing exponential convergence, measure proximity, and error estimation, thereby validating and optimizing TAMD for free energy calculations.

Contribution

The paper provides a mathematical analysis of TAMD, proving exponential convergence, measure closeness, and a Central Limit Theorem, which were previously unestablished.

Findings

Exponential convergence rate close to the effective dynamics.

Invariant measures are close to a reference measure with quantifiable error.

A Central Limit Theorem for error estimation in ergodic averages.

Abstract

The equations of the temperature-accelerated molecular dynamics (TAMD) method for the calculations of free energies and partition functions are analyzed. Specifically, the exponential convergence of the law of these stochastic processes is established, with a convergence rate close to the one of the limiting, effective dynamics at higher temperature obtained with infinite acceleration. It is also shown that the invariant measures of TAMD are close to a known reference measure, with an error that can be quantified precisely. Finally, a Central Limit Theorem is proven, which allows the estimation of errors on properties calculated by ergodic time averages. These results not only demonstrate the usefulness and validity range of the TAMD equations, but they also permit in principle to adjust the parameter in these equations to optimize their efficiency.