Stochastic Thermodynamics Across Scales: Emergent Inter-attractoral Discrete Markov Jump Process and Its Underlying Continuous Diffusion

TL;DR

This paper demonstrates that the thermodynamic structure of a continuous stochastic system is exactly preserved when coarse-grained into a discrete Markov jump process, bridging microscopic dynamics and emergent thermodynamics across scales.

Contribution

It establishes the exact thermodynamic equivalence between continuous stochastic dynamics and their emergent discrete Markov models in the thermodynamic limit.

Findings

Thermodynamics from continuous dynamics matches discrete models in the large system limit.

The approach generalizes chemical reaction thermodynamics to stochastic nonlinear systems.

The results support the consistency of thermodynamic descriptions across different scales.

Abstract

The consistency across scales of a recently developed mathematical thermodynamic structure, between a continuous stochastic nonlinear dynamical system (diffusion process with Langevin or Fokker-Planck equations) and its emergent discrete, inter-attractoral Markov jump process, is investigated. We analyze how the system's thermodynamic state functions, e.g. free energy $F$, entropy $S$, entropy production $e_p$, and free energy dissipation $\dot{F}$, etc., are related when the continuous system is describe with a coarse-grained discrete variable. We show that the thermodynamics derived from the underlying detailed continuous dynamics is exact in the Helmholtz free-energy representation. That is, the system thermodynamic structure is the same as if one only takes a middle-road and starts with the "natural" discrete description, with the corresponding transition rates empirically determined. By "natural", we mean in the thermodynamic limit of large systems in which there is an inherent separation of time scales between inter- and intra-attractoral dynamics. This result generalizes a fundamental idea from chemistry and the theory of Kramers by including thermodynamics: while a mechanical description of a molecule is in terms of continuous bond lengths and angles, chemical reactions are phenomenologically described by the Law of Mass Action with rate constants, and a stochastic thermodynamics.