title: A perturbative solution to metadynamics ordinary differential equation.
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.
authors.affiliation: Department of Chemistry, Columbia University, New York, New York 10027, USA.
authors.forename: Pratyush
authors.initials: P
authors.lastname: Tiwary
authors.affiliation: Department of Chemistry, The James Franck Institute, Institute for Biophysical Dynamics, and Computation Institute, The University of Chicago, Chicago, Illinois 60637, USA.
authors.forename: James F
authors.initials: JF
authors.lastname: Dama
authors.affiliation: Department of Chemistry and Applied Biosciences, ETH Zurich, Zurich, Switzerland and Facoltà di Informatica, Instituto di Scienze Computationali, Università della Svizzera Italiana (USI), Via Giuseppe Buffi 13, CH-6900 Lugano, Switzerland.
authors.forename: Michele
authors.initials: M
authors.lastname: Parrinello