A perturbative solution to metadynamics ordinary differential equation

TL;DR

This paper presents a perturbative solution to the differential equation governing metadynamics, demonstrating the method's robustness and convergence properties in exploring complex free energy landscapes.

Contribution

It introduces a perturbative approach to solve the metadynamics differential equation in Riccati-like form, enhancing understanding of its convergence behavior.

Findings

The solution confirms robustness of metadynamics to biasing parameters.

Demonstrates convergence to true free energy surface.

Applicable to any generic biasing kernel.

Abstract

Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.