Estimation of the infinitesimal generator by square-root approximation

TL;DR

This paper introduces a novel method for estimating transition rates in molecular processes by discretizing the infinitesimal generator using Voronoi tessellations and approximating it through geometric averages of Boltzmann weights, linking energy surfaces to transition rates.

Contribution

The paper proposes a new approach for estimating infinitesimal generators in molecular dynamics using Voronoi tessellations and geometric averaging, providing a direct connection to energy surfaces.

Findings

Effective approximation of the infinitesimal generator for molecular conformations.

Successful application to 2D diffusion and Alanine dipeptide.

Demonstrates correlation between potential energy and transition rates.

Abstract

For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approxi-mated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correla-tion between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.