Stochastic thermodynamics for classical non-Markov jump processes

Abstract

Stochastic thermodynamics investigates energetic and entropic bounds in small systems. Foundational results, e.g., the first and second laws, predominantly rely on the Markov (memoryless) assumption. Although physicists recognise that the Markov assumption is questionable in real experimental setups, extending stochastic thermodynamics to general non-Markov systems has proven challenging. Fundamentally, it has been elusive how to model memory-dependent non-Gaussian fluctuations consistently with thermodynamic laws. Here we establish a general theory of stochastic thermodynamics for classical non-Markov jump processes. We introduce a key technique, called the Fourier embedding, which converts non-Markov jump processes into Markovian field dynamics of auxiliary Fourier modes. This yields necessary and sufficient conditions for time-reversal symmetry and enables the derivation of the second law for a broad class of strong-memory dynamics that admit the Fourier embedding. We demonstrate the power of our framework by presenting two novel non-Markov models: (i) a history-dependent two-level system and (ii) a history-dependent random walk. Our work accommodates diverse non-Markov dynamics in realistic experimental settings and offers a guiding principle for physics-informed modelling of history-dependent fluctuations.