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

Details Irreversible Thermodynamics Across Scales: Emergent Inter-attractoral Discrete Markov Jump Process and Its Underlying Continuous Nonlinear Stochastic Dynamics Irreversible Thermodynamics Across Scales: Emergent Inter-attractoral Discrete Markov Jump Process and Its Underlying Continuous Nonlinear Stochastic Dynamics

2011-03-16

arxiv.org/abs/1103.3306v2

Physics

Condensed Matter

Statistical Mechanics

21 pages, 1 figure

Scientific paper

The consistency across scales of a recently developed mathematical irreversible-thermodynamic structure, between a continuous stochastic nonlinear dynamical (Langevin-Fokker-Planck) system 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 \emph{thermodynamics} derived from the underlying detailed continuous dynamics is \emph{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 to include thermodynamics: while a mechanical description of a molecule is in terms of continuous bond lengths and angles, chemical reactions are phenomenologically described the law of mass action with rate constants, and Markov thermodynamics.

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