Brazilian Journal of Probability and Statistics

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

Eric A. Carlen, Joel L. Lebowitz, and Clément Mouhot

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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

Keywords
Non-equilibrium steady state Boltzmann equation

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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References

  • Arkeryd, L. and Nourri, A. (2000). $L^{1}$ solutions to the stationary Boltzmann equation in a slab. Ann. Fac. Sci. Toulouse Math. (6) 9, 375–413.
  • Bobylev, A. V. (1975). Fourier transform method in the theory of the Boltzmann equation for Maxwellian molecules. Dokl. Akad. Nauk USSR 225, 1041–1044.
  • Bobylev, A. V. (1988). The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C. Math. Phys. 7, 111–233.
  • Bonetto, F., Lebowitz, J. L. and Rey-Bellet, L. (2000). Fourier’s law: A challenge to theorists. In Mathematical Physics 2000 (A. Fokas et al., eds.) 128–150. London: Imperial College Press.
  • Bonetto, F., Loss, M. and Vaidyanathan, R. (2014). The Kac model coupled to a thermostat. J. Stat. Phys. 156, 647–667.
  • Carlen, E. A., Esposito, R., Lebowitz, J. L., Marra, R. and Rokhlenko, A. (1995). Nonunique stationary states in driven kinetic systems with applications to plasmas. Phys. Rev. E (3) 52, 40–43.
  • Carlen, E. A., Esposito, R., Lebowitz, J. L., Marra, R. and Rokhlenko, A. (1998). Hydrodynamic limit of a driven kinetic system with non-unique stationary states. Arch. Rational Mech. Anal. 142, 193–218.
  • Carlen, E. A., Gabetta, E. and Toscani, G. (1999). Propagation of smoothness and the rate of exponential decay to equilibrium for a spatially homogeneous Maxwellian gas. Comm. Math. Phys. 305, 521–546.
  • Carlen, E. A., Carvalho, M. C. and Gabetta, E. (2000). Central limit theorem for Maxwellian molecules and truncation of the wild expansion. Comm. Pure Appl. Math. 53, 370–397.
  • Dhar, A. (2008). Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537.
  • Esposito, R., Lebowitz, J. L. and Marra, R. (1994). Hydrodynamic limit of the stationary Boltzmann equation in a slab. Comm. Math. Phys. 160, 49–80.
  • Gabetta, G., Toscani, G. and Wennberg, B. (1985). Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901–934.
  • Gallagher, I., Saint Raymond, L. and Texier, B. (2014). From Newton to Boltzmann: Hard Spheres and Short-Range Potentials. Zurich Lectures in Advanced Mathematics. Zürich: EMS.
  • Goldstein, S. and Lebowitz, J. L. (2003). On the (Boltzmann) entropy of non-equilibrium systems. Phys. D 193, 33–66.
  • Goldstein, S., Lebowitz, J. L. and Presutti, E. (1978). Mechanical systems with stochastic boundaries. Coll. Math. Soc. J. Bolyai 27, 401–419.
  • Kipnis, C., Goldstein, S. and Ianiro, N. (1985). Stationary states for a mechanics system with stochastic boundary conditions. J. Stat. Phys. 41, 915–939.
  • Lanford, O. (1976). On a derivation of the Boltzmann equation. In International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975). Asterisque 40, 117–137. Paris: Soc. Math. France.
  • Lepri, S., Livi, R. and Politi, A. (2003). Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80.
  • Otto, F. (2001). The geometry of dissipative evolution equations: The porous medium equation. Comm. Partial Differential Equations 26, 101–174.
  • Pulvirenti, M., Saffirio, C. and Simonella, S. (2014). On the validity of the Boltzmann equation for short range potentials. Rev. Math. Phys. 26, 145001.
  • Tanaka, H. (1973). An inequality for a functional of probability distributions and its applications to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrsch. Verw. Gebiete 27, 47–52.
  • Tanaka, H. (1978/1979). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46, 67–105.
  • Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Providence, RI: Amer. Math. Soc.