## Brazilian Journal of Probability and Statistics

### Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas

#### Abstract

We investigate a kinetic model of a system in contact with several thermal reservoirs at different temperatures $T_\alpha$. Our system is a spatially uniform dilute gas whose internal dynamics is described by the non-linear Boltzmann equation with Maxwellian collisions. Similarly, the interaction with reservoir $\alpha$ is represented by a Markovian process that has the Maxwellian $M_{T_\alpha}$ as its stationary state. We prove existence and uniqueness of a non-equilibrium steady state (NESS) and show exponential convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wennberg (GTW). This shows that the GTW distance between the current velocity distribution to the steady-state velocity distribution is a Lyapunov functional for the system. We also derive expressions for the entropy production in the system plus the reservoirs which is always positive.

#### Article information

Source
Braz. J. Probab. Stat., Volume 29, Number 2 (2015), 372-386.

Dates
First available in Project Euclid: 15 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1429105593

Digital Object Identifier
doi:10.1214/14-BJPS263

Mathematical Reviews number (MathSciNet)
MR3336871

Zentralblatt MATH identifier
1318.82035

#### Citation

Carlen, Eric A.; Lebowitz, Joel L.; Mouhot, Clément. Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas. Braz. J. Probab. Stat. 29 (2015), no. 2, 372--386. doi:10.1214/14-BJPS263. https://projecteuclid.org/euclid.bjps/1429105593

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