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arXiv · 1309.2715

The Kac Model Coupled to a Thermostat

Federico Bonetto·Michael Loss·Ranjini Vaidyanathan
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Abstract

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $β$. The system admits the canonical distribution at inverse temperature $β$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.

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Federico Bonetto, Michael Loss, Ranjini Vaidyanathan. 2014-07-18. The Kac Model Coupled to a Thermostat. https://doi.org/10.1007/s10955-014-0999-6

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