Geodesic
A limit theorem for the position of a particle in the Lorentz model
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 42-68 
V. V. Vysotsky. A limit theorem for the position of a particle in the Lorentz model. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 9, Tome 328 (2005), pp. 42-68. http://geodesic.mathdoc.fr/item/ZNSL_2005_328_a3/
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     title = {A limit theorem for the position of a~particle in the {Lorentz} model},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Consider a particle moving through a random medium. The medium consists of spherical obstacles of equal radii, randomly distributed in $\mathbb R^3$. The particle is accelerated by a constant external field. When colliding with an obstacle, the particle inelastically reflects. We study asymptotics of $X(t)$, which denotes the position of the particle at time $t$, as $t\to\infty$. The result is a limit theorem for $X(t)$. Our proof is based on functional CLT for Markov chains.

[1] Dzh. Dub, Veroyatnostnye protsessy, IL., M., 1956

[2] M. Kats, Neskolko veroyatnostnykh zadach fiziki i matematiki, Nauka, M.

[3] J. Banasiak, G. Frosali, G. Spiga, “Inelastic scattering models in transport theory and their small mean free path analysis”, Mathematical Methods in the Applied Sciences, 23 (2000), 121–145 | 3.0.CO;2-5 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[4] C. Buet, S. Cordier, B. Lucquin-Desreux, S. Mancini, “Diffusion limit of the Lorentz model: asymptotic preserving schemes”, Mathematical Modelling and Numerical Analysis, 36:4 (2002), 631–655 | DOI | MR | Zbl

[5] Ph. A. Martin, J. Piasecki, Lorentz's model with dissipative collisions, Preprint, 1998; arXiv: /cond-mat/9810070

[6] S. P. Meyn, R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, London, 1993 | MR | Zbl

[7] K. Ravishankar, L. Triolo, “Diffusive limit of the Lorentz model with a uniform field starting from the Markov approximation”, Markov Processes and Related Fields, 5:4 (1999), 385–421 | MR | Zbl

[8] D. P. Sanders, Fine structure of distributions and central limit theorem in diffusive billiards, Preprint, 2004 ; arXiv: /nlin.CD/0411012 | MR

[9] D. R. Wilkinson, S. F. Edwards, “Spontaneous interparticle percolation”, Proceedings of the Royal Society of London, Series A, 381, 33–51 | MR

[10] K. I. Wysokinski, W. Park, D. Belitz, T. R. Kirkpatrick, “Density expansion for the mobility in a quantum Lorentz model”, Phys. Rev. E, 52:1 (1995), 612–622 | DOI