Determining extended Markov parameterizations for vector-valued generalized Langevin Equations

TL;DR

This work extends data-driven Markovian embeddings of generalized Langevin dynamics from scalar to vector-valued processes, enabling direct construction of memory-approximate Markov models from trajectories. It introduces two complementary algorithms: Method A, a block Prony-based approach that interpolates time-sampled autocorrelation data with a finite Prony series, and Method B, a generating-function approach that uses a matrix-valued AAA rational approximation to extract Markovian exponents and coefficients. Both methods aim to produce a stable, physically admissible embedding by enforcing constraints on the autocorrelation (e.g., $\varphi(0)=I$) and solving associated Lyapunov or Lur'e equations to obtain $A,B,C,K,L$ and $\Sigma$. Application to a two-dimensional S-shaped probe in a Langevin bath demonstrates that 10–20 auxiliary variables suffice for accurate reproduction of the velocity autocorrelation, with Method A offering higher accuracy at the expense of more variables, and Method B achieving greater parsimony while satisfying the side constraints. The work provides a practical framework for high-dimensional memory-embedded stochastic modeling with potential broad impact on coarse-grained simulations and memory-kernel identification.

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.