Stochastic thermodynamics across scales: Emergent inter-attractoral discrete Markov jump process and its underlying continuous diffusion

We investigate the internal consistency of a recently developed mathematical thermodynamic structure across scales, between a continuous stochastic nonlinear dynamical system, i.e., a diffusion process with Langevin and Fokker–Planck equations, and its emergent discrete, inter-attractoral Markov jump process. We analyze how the system’s thermodynamic state functions, e.g. free energy F , entropy S , entropy production e p , free energy dissipation F ̇ , etc., are related when the continuous system is described with coarse-grained discrete variables. It is shown that the thermodynamics derived from the underlying, detailed continuous dynamics gives rise to exactly the free-energy representation of Gibbs and Helmholtz. That is, the system’s 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 a large system, with 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 incorporating thermodynamics: while a mechanical description of a molecule is in terms of continuous bond lengths and angles, chemical reactions are phenomenologically described by a discrete representation, in terms of exponential rate laws and a stochastic thermodynamics.