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arXiv · 2607.18202

Entropy production in Knudsen thermodynamics of compartmented systems

Timothy Chumley·Renato Feres·Zhenyi Zhang
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Abstract

We investigate entropy production and nonequilibrium transport in a class of random dynamical systems modeling a Knudsen gas confined to a compartmented container. The system consists of a single particle undergoing random billiard motion, with collisions leading to random reflection or transmission through semi-reflecting, compartment-separating walls. The stationary entropy production rate is first expressed as the relative entropy between forward and time-reversed path measures. Under a reciprocity assumption, used to introduce temperature into the general random billiard system, it is shown how this information-theoretic definition reduces to the classical thermodynamic formula: the mean energy transferred to the walls divided by their local temperatures, a stochastic Clausius relation. We then develop a modular analysis of compartmented systems. Each open compartment is characterized by a compartment scattering operator and its sojourn statistics. The sequence of compartment entrance states defines a Markov chain whose stationary distribution is used in a renewal-reward theorem to assemble the compartment contributions into the global entropy production rate. The framework is illustrated with a series of examples of increasing complexity governed by a generalized Maxwell-Smoluchowski scattering operator and amenable to detailed and explicit analysis. Such operators are defined by a few parameters: temperature, a partial thermal accommodation, the height of potential barriers, and a porosity coefficient. The central example is a cyclic three-compartment system consisting of two thermal walls at different temperatures and a potential barrier. For full thermal accommodation we obtain closed-form expressions for the entropy production rate and net probability circulation around the cycle, revealing a thermal ratchet effect analogous to thermal transpiration.

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Timothy Chumley, Renato Feres, Zhenyi Zhang. 2026-07-20. Entropy production in Knudsen thermodynamics of compartmented systems. https://arxiv.org/abs/2607.18202

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