Determining extended Markov parameterizations for vector-valued generalized Langevin Equations

TL;DR

This paper extends algorithms for data-driven Markov modeling of generalized Langevin equations from scalar to vector-valued processes, demonstrated on a two-dimensional particle velocity example, achieving accurate autocorrelation fits.

Contribution

The paper introduces an extension of existing algorithms to handle vector-valued processes in Markov approximations of generalized Langevin equations.

Findings

Achieved accurate autocorrelation data fitting over relevant time intervals.

Developed Markov models with 10-20 auxiliary variables.

Validated approach on a two-dimensional particle velocity process.

Abstract

The generalized Langevin equation is used as a model for various coarse-grained physical processes, e.g., the time evolution of the velocity of a given larger particle in an implicitly represented solvent, when the relevant time scales of the dynamics of the larger particle and the solvent particles are not strictly separated. Since this equation involves an integrated history of past velocities, considerable efforts have been made to approximate this dynamics by data-driven Markov models, where auxiliary variables are used to compensate for the memory term. In recent works we have developed two algorithms which can be used for this purpose, provided the dynamics in question are scalar processes. Here we extend these algorithms to vector-valued processes. As a physical test bed we consider an S-shaped particle sliding on a planar substrate, which gives rise to a truly two-dimensional velocity process. The two algorithms provide Markov approximations of this process with 10-20 auxiliary variables and a very accurate fit of the given autocorrelation data over the entire time interval where these data are non-negligible.