Abstract
We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in Carlen et al. (Braz J Probab Stat 29:372–386, 2015), we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilistic techniques, and is quite different form the analysis in Carlen et al. (Braz J Probab Stat 29:372–386, 2015) that was based on contraction mapping methods.
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Acknowledgements
E.A. Carlen is partially supported by NSF Grant Number DMS 1501007. J.L. Lebowitz is partially supported by AFOSR Grant FA9550-16-1-0037. JLL thanks IAS Systems Biology for its hospitality. C. Mouhot is partially supported by the ERC consolidator grant MAFRAN.
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Dedicated to Juerg Froehlich, Thomas Spencer and Herbert Spohn.
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Carlen, E.A., Esposito, R., Lebowitz, J.L. et al. Approach to the Steady State in Kinetic Models with Thermal Reservoirs at Different Temperatures. J Stat Phys 172, 522–543 (2018). https://doi.org/10.1007/s10955-018-2074-1
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DOI: https://doi.org/10.1007/s10955-018-2074-1