Micro-reversibility and thermalization with collisional baths

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Micro-reversibility plays a central role in thermodynamics and statistical mechanics. It is used to prove that systems in contact with a thermal bath relax to canonical ensembles. However, a problem arises when trying to reproduce this proof for classical and quantum collisional baths, i.e. particles at equilibrium interacting with a localized system via collisions. In particular, micro-reversibility appears to be broken and some models do not thermalize when interacting with Maxwellian particles. We clarify these issues by showing that micro-reversibility needs the invariance of evolution equations under time reversal plus the conservation of phase space volume in classical and semiclassical scenarios. Consequently, all canonical variables must be considered to ensure thermalization. This includes the position of the incident particles which maps their Maxwellian distribution to the effusion distribution. Finally, we show an example of seemingly plausible collision rules that do not conserve phase-space volume, and consequently violate the second law. (C) 2019 Elsevier B.V. All rights reserved.
We acknowledge fruitful discussions with Carlos Mejia-Monasterio. JE wishes to thank Andreas Engel for valuable discussions and for enabling a research stay for JE in Madrid. ME is supported by the European Research Council project NanoThermo (ERC-2015-CoG Agreement No. 681456). FB acknowledges the financial support of FONDECYT grant 1191441 and of the Millennium Nucleus "Physics of active matter'' of the Millennium Scientific Initiative of the Ministry of Economy, Development and Tourism (Chile). JMRP acknowledges financial support from the Spanish Government (Grant Contract, FIS-2017-83706-R).
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