On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas

Abstract

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a `piston', i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.

23 pages, 6 figures

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• Heat Transport in low-dimensional systems
• Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs
• Heat conduction and Fourier's law by consecutive local mixing and thermalization
• Heat conduction and Fourier's law in a class of many particle dispersing billiards
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Cited by in corpus (5)

• Equilibration of energy in slow-fast systems
• Role of conserved quantities in Fourier's law for diffusive mechanical systems
• Equidistribution for standard pairs in planar dispersing billiard flows
• Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases
• Assortative Exchange Processes