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2024 | OriginalPaper | Chapter

Approach to Equilibrium for the Kac Model

Authors : Federico Bonetto, Eric A. Carlen, Lukas Hauger, Michael Loss

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

This is a review on the Kac Master Equation. Various issues will be presented such as the resolution of Kac’s conjecture about the gap for the three dimensional hard sphere gas, entropic propagation of chaos and other topics such as systems coupled to reservoirs and thermostats. The discussion is informal with few proofs and those who are presented are only sketched.

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previous chapter Results About the Free Kawasaki Dynamics of Continuous Particle Systems in Infinite Volume: Long-time Asymptotics and Hydrodynamic Limit

next chapter Hydrodynamic Limit from the Boltzmann Equation in a Slightly Compressible Regime

Note that the collision mechanism (24) is linear, i.e it can be represented by a matrix

$$\begin{aligned} \begin{pmatrix} v_i^*(\sigma ) \\ v_j^*(\sigma ) \end{pmatrix} = M_{(i,j)}^\sigma \begin{pmatrix} v_i \\ v_j \end{pmatrix} \,, \quad M_{(i,j)}^\sigma :=\begin{pmatrix} I - \sigma \oplus \sigma & \sigma \oplus \sigma \\ \sigma \oplus \sigma & I - \sigma \oplus \sigma \end{pmatrix} \,. \end{aligned}$$

(30)

Extending $M_{(i,j)}^\sigma $ such that it leaves all other particles invariant, we obtain a collision matrix $\underline{M}_{(i,j)}^\sigma $ acting on ${\mathord {\mathbb R}}^{3M+3N}$ that satisfies $F(u_1,\dots ,u_i^*(\sigma ),\dots ,u_j^*(\sigma ),\dots ,u_{M+N}) = F(\underline{M}_{(i,j)}^\sigma \boldsymbol{u})$.

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Title: Approach to Equilibrium for the Kac Model
Authors: Federico Bonetto
Eric A. Carlen
Lukas Hauger
Michael Loss
Publisher: Springer International Publishing
Book: From Particle Systems to Partial Differential Equations
Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-65195-3_8

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