2015/10/06 by Pratyush Tiwary, James F. Dama, Michele Parrinello
Chemistry · Mathematics · Physics and Astronomy · #Applied mathematics #Biasing #Chemistry #Convergence (economics) #Differential equation #Lightning and Electromagnetic Phenomena #Limit (mathematics) #Mathematical analysis #Mathematics #Metadynamics #Molecular dynamics #Nonlinear Photonic Systems #Ordinary differential equation #Physics #Quantum mechanics #Riccati equation #Robustness (evolution) #Statistical physics #Theoretical and Computational Physics #Voltage #cond-mat.stat-mech #physics.chem-ph #physics.comp-ph
paper · pdf · doi:10.1063/1.4937945
submitted to J. Chem. Phys
arxiv created 2015/10/06 · openalex publication_date 2015/12/21 · arxiv updated 2016/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.