Violation of the method of images in non-Markovian processes and its connection to stochastic thermodynamics

TL;DR

This paper reveals the failure of the method of images in non-Markovian diffusive processes and connects this breakdown to principles in stochastic thermodynamics, providing new insights into the thermodynamic interpretation of such processes.

Contribution

It demonstrates why the method of images fails for non-Markovian processes and introduces a thermodynamic framework to understand this breakdown.

Findings

The method of images does not accurately describe non-Markovian diffusions.

A conjugate path concept obeys a fluctuation theorem-like relation.

Thermodynamic interpretation clarifies the breakdown mechanism.

Abstract

We discuss a failure of the wide-spread method of images solution to describe the time evolution of probability distribution in diffusive processes with memory. For a path that touches a target during stochastic evolution, we define its conjugate twin of reflected path, and show that their path probability ratio obeys a relation analogous to the fluctuation theorem. With a key quantity properly identified as the heat, the resultant thermodynamic interpretation of the processes provides a quantitative basis as well as an intuitive physical picture on how and why the method of images breaks down for non-Markovian processes.