Journal of Statistical Physics

, Volume 172, Issue 3, pp 762–780 | Cite as

Equilibration and Diffusion for a Dynamical Lorentz Gas

  • Émilie SoretEmail author


We consider a model of a dynamical Lorentz gaz: a single particle is moving in \({\mathbb {R}}^d\) through an array of fixed and soft scatterers each possessing an internal degree of freedom coupled to the particle. Assuming the initial velocity is sufficiently high and modelling the parameters of the scatterers as random variables, we describe the evolution of the kinetic energy of the particle by a Markov chain for which each step corresponds to a collision. We show that the momentum distribution of the particle approaches a Maxwell–Boltzmann distribution with effective temperature T such that \(k_BT\) corresponds to an average of the scatterers’ kinetic energy.


Mathematical physics Probability Lorentz gas Diffusion processes Equilibration Stationary distribution 



This work is in part supported by IRCICA, USR CNRS 3380 and the Labex CEMPI (ANR-11- LABX-0007-01). The author thanks S. De Bièvre and P.E. Parris for their helpful discussions.


  1. 1.
    Aguer, B., De Bièvre, S., Lafitte, P., Parris, P.E.: Classical motion in force fields with short range correlations. J. Stat. Phys. 138(4–5), 780–814 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aguer, B.: Comportements asymptotiques dans des gaz de Lorentz inèlastiques. PhD thesis, 2010. Thèse de doctorat dirigée par De Bièvre, Stephan and Lafitte-Godillon, Pauline Mathématiques appliquées Lille 1 (2010)Google Scholar
  3. 3.
    Chandrasekhar, S.: Dynamical friction. I. General considerations: the coefficient of dynamical friction. Astrophys. J. 97, 255 (1943)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chandrasekhar, S.: Dynamical friction. II. The rate of escape of stars from clusters and the evidence for the operation of dynamical friction. Astrophys. J. 97, 263 (1943)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chandrasekhar, S.: Dynamical friction. III. A more exact theory of the rate of escape of stars from clusters. Astrophys. J. 98, 54 (1943)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    De Bièvre, S., Parris, P.E.: Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases. J. Stat. Phys. 142(2), 356–385 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ethier, S.N., Kurtz, T.G.: Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)Google Scholar
  8. 8.
    Mandl, P.: Analytical treatment of one-dimensional Markov processes. Die Grundlehren der mathematischen Wissenschaften, Band 151. Academia Publishing House of the Czechoslovak Academy of Sciences, Prague (1968)Google Scholar
  9. 9.
    Pavliotis, Grigorios A.: Stochastic Processes and Applications. Texts in Applied Mathematics. Springer, Berlin (2014)Google Scholar
  10. 10.
    Soret, E., De Bièvre, S.: Stochastic acceleration in a random time-dependent potential. Stoch. Process. Appl. 125(7), 2752–2785 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Silvius, A.A., Parris, P.E., De Bièvre, S.: Adiabatic-nonadiabatic transition in the diffusive hamiltonian dynamics of a classical holstein polaron. Phys. Rev. B 73(1), 014304 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 233. Springer, Berlin (1979)Google Scholar

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Authors and Affiliations

  1. 1.IEMN UMR CNRS 8520Villeneuve-d’AscqFrance
  2. 2.IRCICA CNRS 3024Villeneuve-d’AscqFrance

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