Dynamics of Micro and Nanoscale Systems in the Weak-Memory Regime

TL;DR

This paper develops a rigorous framework for approximating non-local in time evolution equations in small-scale systems, extending the Markov approximation to a weak-memory regime with explicit error bounds.

Contribution

It introduces a systematic method to extend the Markov approximation to systems with comparable time scales, providing explicit error bounds and a convergent perturbation scheme.

Findings

Derived explicit bounds on local approximation errors

Established a convergent perturbation scheme for non-local equations

Unified framework for describing memory effects in small-scale systems

Abstract

Memory effects are ubiquitous in small-scale systems. They emerge from interactions between accessible and inaccessible degrees of freedom and give rise to evolution equations that are non-local in time. If the characteristic time scales of accessible and inaccessible degrees of freedom are sharply separated, locality can be restored through the standard Markov approximation. Here, we show that this approach can be rigorously extended to a well-defined weak-memory regime, where the relevant time scales can be of comparable order of magnitude. We derive explicit bounds on the error of the local approximation and a convergent perturbation scheme for its construction. Being applicable to any non-local time evolution equation that is autonomous and linear in the variables of interest, our theory provides a unifying framework for the systematic description of memory effects.