Markovian thermodynamics of non-Markovian Langevin equations

TL;DR

This paper develops a thermodynamic framework for non-Markovian Langevin equations by embedding them into a high-dimensional Markovian system, enabling consistent entropy production analysis.

Contribution

It introduces a method to embed non-Markovian dynamics into Markovian systems, ensuring unique thermodynamic quantities and monotonic entropy production.

Findings

Explicit embedding construction for linear memory kernels.

Unique thermodynamic quantities from the Markovian embedding.

Monotonic entropy production in the embedded system.

Abstract

We develop the thermodynamics of non-Markovian generalized Langevin equations by embedding them in a high-dimensional Markovian representation involving auxiliary degrees of freedom. If the memory is linear and satisfies detailed balance with the noise, we provide an explicit construction of the embedding for non-Markovian dynamics with many degrees of freedom and hydrodynamic interactions. Moreover, while the embedding is generally not unique, we show that it results in unique values of thermodynamic quantities of the Markovian system. This allows us to define the Markovian entropy production of a non-Markovian system, which, in contrast to the definition based directly on the non-Markovian dynamics, is guaranteed to increase monotonically with time. Moreover, the Markovian representation allows us to identify the apparent decrease in the non-Markovian entropy with heat and information exchange between the system and the auxiliary degrees of freedom.