Dynamics of non-Markovian systems: Markovian embedding vs effective mass approach

TL;DR

This paper compares the Markovian embedding and effective mass methods for modeling non-Markovian dynamics, demonstrating their applicability, accuracy, and efficiency through a Brownian particle example, and providing guidance for future research.

Contribution

It introduces a comparative analysis of two prominent methods for non-Markovian systems, highlighting the effective mass approach's efficiency and accuracy advantages.

Findings

Effective mass approach is faster and sufficiently accurate for short memory times.

Optimal parameters can enhance accuracy and reduce computational costs in Markovian embedding.

The study offers a practical blueprint for analyzing non-Markovian dynamics.

Abstract

Dynamics of non-Markovian systems is a classic problem yet it attracts an everlasting activity in physics and beyond. A powerful tool for modeling such setups is the Generalized Langevin Equation, however, its analysis typically poses a major challenge even for numerical means. For this reason, various approximations have been proposed over the years that simplify the original model. In this work we compare two methods allowing to tackle this great challenge: (i) the well-known and successful Markovian embedding technique and (ii) the recently developed effective mass approach. We discuss their scope of applicability, numerical accuracy, and computational efficiency. In doing so we consider a paradigmatic model of a free Brownian particle subjected to power-law correlated thermal noise. We show that when the memory time is short, the effective mass approach offers satisfying precision and typically is much faster than the Markovian embedding. Moreover, the concept of effective mass can be used to find optimal parameters allowing to reach supreme accuracy and minimal computational cost within the embedding. Our work therefore provides a blueprint for investigating the dynamics of non-Markovian systems.