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Partially thermostated Kac model. (English) Zbl 1330.82036

The departure point for the present paper is the paper of F. Bonetto et al. [J. Stat. Phys. 156, No. 4, 647–667 (2014; Zbl 1302.35290)], where the Kac \(N\)-particle system has been coupled to an infinite reservoir (gas at thermal equilibrium). Presently, the Kac model is partially thermostated, by assuming that only \(m<N\) particles remain actually in contact with the Maxwellian thermostat. One motivation is that studying partially thermostated systems allows to introduce spatial inhomogeneities to the Kac model. Another one is associated with an attempt to better understand the role of the interparticle interactions while approaching the equilibrium Gaussian state. For infinite thermostats, the convergence to equilibrium is known to persist even in the absence of those interactions. For partially thermostated systems, interparticle interactions can never be neglected. Two indicators of the approach to equilibrium are investigated. First, the spectral gap of the evolution operator is proved to behave like \(\sim m/N\) for large \(N\). Second, the relative entropy is shown to decay at an exponential rate \(\sim m/N^2\) for large \(N\). The relationship between the Bonetto et al. thermostat [loc. cit.] and the presently employed Maxwellian thermostat is established through the Van Hove limit.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
70F35 Collision of rigid or pseudo-rigid bodies
28D20 Entropy and other invariants
47D06 One-parameter semigroups and linear evolution equations

Citations:

Zbl 1302.35290
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References:

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