On the Convergence of Metadynamics with Gaussian Hills
Abstract
Metadynamics is a class of enhanced-sampling methods that is widely used in molecular modeling. Here, we consider metadynamics with a one-dimensional collective variable and explore the limitations of using Gaussian hills in light of existing convergence results. We reduce the corresponding evolution problem to a replicator-type differential equation and analyze its long-time behavior. For a periodic collective variable, we prove the convergence of metadynamics. However, the situation is fundamentally different when the collective variable is defined on a finite interval. In this case, the stationary solution only exists in the weak sense as a finite atomic measure. The solution to the differential equation weakly converges to it. Consequently, metadynamics is ineffective due to the absence of a clear quasi-stationary state. A quasi-stationary, transient solution can only be captured in the "INTERVAL" framework if the free energy outside the finite interval remains nearly constant across sufficiently large regions compared to σ.
Related
- Critical convergence and Hausdorff measures for generalized Flint Hills series
- Convergence of Kikuchi matrices to Γ-independent and q-Gaussian limits
- Optimal local convergence criteria for integer and Gaussian integer continued fractions
- Convergence Rates for Lasserre's Upper-Bound Hierarchy on ℝd with Gaussian Measure
- Gaussian-Restricted Barycenters for KL-Unbalanced Optimal Transport: Variational Theory and Fixed-Point Convergence
BibTeX
@misc{badanin2026convergence,
title = {{On the Convergence of Metadynamics with Gaussian Hills}},
author = {Andrey Badanin and Olga Rogacheva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01934},
eprint = {2609.01934},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01934},
}
RIS
TY - UNPB
TI - On the Convergence of Metadynamics with Gaussian Hills
AU - Andrey Badanin
AU - Olga Rogacheva
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01934
UR - https://arxiv.org/abs/2609.01934
AB - Metadynamics is a class of enhanced-sampling methods that is widely used in molecular modeling. Here, we consider metadynamics with a one-dimensional collective variable and explore the limitations of using Gaussian hills in light of existing convergence results. We reduce the corresponding evolution problem to a replicator-type differential equation and analyze its long-time behavior. For a periodic collective variable, we prove the convergence of metadynamics. However, the situation is fundamentally different when the collective variable is defined on a finite interval. In this case, the stationary solution only exists in the weak sense as a finite atomic measure. The solution to the differential equation weakly converges to it. Consequently, metadynamics is ineffective due to the absence of a clear quasi-stationary state. A quasi-stationary, transient solution can only be captured in the "INTERVAL" framework if the free energy outside the finite interval remains nearly constant across sufficiently large regions compared to σ.
ER -