## The Boltzmann-Grad limit of the periodic Lorentz gas.(English)Zbl 1237.37014

Consider the $$d$$-dimensional Lorentz gas defined by infinitely many spherical scatterers of radius $$\rho$$, whose centers are placed in the points of a lattice $$\mathcal{L} \subset \mathbb{R}^d$$. For this configuration space, the authors study the statistical properties of the billiard dynamics (point particle traveling freely between scatterers and experiencing Fresnel reflections at each collision with a scatterer), in the Boltzmann-Grad limit. The Boltzmann-Grad limit is defined as the limit $$\rho \to 0^+$$, together with a rescaling of the spatial coordinates by a factor $$\rho^{d-1}$$, so that the average cross-section and the mean free path between collisions remain constant.
Building on the very strong results the same authors had achieved in [“The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems”, Ann. Math. (2) 172, No. 3, 1949–2033 (2010; Zbl 1211.82011)], they prove various forms of the following limit theorem, which we state in a non-rigorous way for the sake of clarity.
Denoting by $$(q(t), v(t))$$ a trajectory in the Lorentz gas, which depends on the initial conditions $$(q_0, v_0)$$ and on the choice of $$\rho$$, define the stochastic process $$\{ Q(T), V(T) \}_{T \geq 0}$$ via the relation $$(Q(T), V(T)) = (\rho^{d-1} q(\rho^{-(d-1)}T), v(\rho^{-(d-1)}T))$$ and initial conditions randomly chosen according to a certain probability distribution. This process depends on $$\rho$$. As $$\rho \to 0^+,$$ the process tends (in the sense of the finite-dimensional distributions for finite sets of times) to a persistent Markov process. The discrete-time version of this process, using the collision times, is a Markov chain with memory two.

### MSC:

 37A60 Dynamical aspects of statistical mechanics 37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010) 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 60J25 Continuous-time Markov processes on general state spaces

Zbl 1211.82011
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### References:

 [1] P. Bálint and I. P. Tóth, ”Exponential decay of correlations in multi-dimensional dispersing billiards,” Ann. Henri Poincaré, vol. 9, iss. 7, pp. 1309-1369, 2008. · Zbl 1162.37018 · doi:10.1007/s00023-008-0389-1 [2] P. M. Bleher, ”Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon,” J. Statist. Phys., vol. 66, iss. 1-2, pp. 315-373, 1992. · Zbl 0925.82147 · doi:10.1007/BF01060071 [3] F. P. Boca and A. Zaharescu, ”The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit,” Comm. Math. Phys., vol. 269, iss. 2, pp. 425-471, 2007. · Zbl 1143.37002 · doi:10.1007/s00220-006-0137-7 [4] C. Boldrighini, L. A. Bunimovich, and Y. G. Sinai, ”On the Boltzmann equation for the Lorentz gas,” J. Statist. Phys., vol. 32, iss. 3, pp. 477-501, 1983. · Zbl 0583.76092 · doi:10.1007/BF01008951 [5] J. Bourgain, F. Golse, and B. Wennberg, ”On the distribution of free path lengths for the periodic Lorentz gas,” Comm. Math. Phys., vol. 190, iss. 3, pp. 491-508, 1998. · Zbl 0910.60082 · doi:10.1007/s002200050249 [6] L. A. Bunimovich and Y. G. Sinai, ”Statistical properties of Lorentz gas with periodic configuration of scatterers,” Comm. Math. Phys., vol. 78, iss. 4, pp. 479-497, 1980/81. · Zbl 0459.60099 · doi:10.1007/BF02046760 [7] E. Caglioti and F. Golse, ”On the distribution of free path lengths for the periodic Lorentz gas. III,” Comm. Math. Phys., vol. 236, iss. 2, pp. 199-221, 2003. · Zbl 1041.82016 · doi:10.1007/s00220-003-0825-5 [8] E. Caglioti and F. Golse, ”The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions,” C. R. Math. Acad. Sci. Paris, vol. 346, iss. 7-8, pp. 477-482, 2008. · Zbl 1145.82019 · doi:10.1016/j.crma.2008.01.016 [9] N. I. Chernov, ”Statistical properties of the periodic Lorentz gas. Multidimensional case,” J. Statist. Phys., vol. 74, iss. 1-2, pp. 11-53, 1994. · Zbl 0946.37500 · doi:10.1007/BF02186805 [10] P. Dahlqvist, ”The Lyapunov exponent in the Sinai billiard in the small scatterer limit,” Nonlinearity, vol. 10, iss. 1, pp. 159-173, 1997. · Zbl 0907.58038 · doi:10.1088/0951-7715/10/1/011 [11] E. B. Davies, One-parameter Semigroups, London: Academic Press [Harcourt Brace Jovanovich Publishers], 1980, vol. 15. · Zbl 0457.47030 [12] D. Dolgopyat, D. Szász, and T. Varjú, ”Recurrence properties of planar Lorentz process,” Duke Math. J., vol. 142, iss. 2, pp. 241-281, 2008. · Zbl 1136.37022 · doi:10.1215/00127094-2008-006 [13] G. Gallavotti, ”Divergences and approach to equilibrium in the Lorentz and the wind-tree-models,” Phys. Rev., vol. 185, pp. 308-322, 1969. · doi:10.1103/PhysRev.185.308 [14] F. Golse, ”The periodic Lorentz gas in the Boltzmann-Grad limit,” in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 183-201. · Zbl 1109.82020 [15] F. Golse, ”On the periodic Lorentz gas and the Lorentz kinetic equation,” Ann. Fac. Sci. Toulouse Math., vol. 17, iss. 4, pp. 735-749, 2008. · Zbl 1166.82304 · doi:10.5802/afst.1200 [16] F. Golse and B. Wennberg, ”On the distribution of free path lengths for the periodic Lorentz gas. II,” M2AN Math. Model. Numer. Anal., vol. 34, iss. 6, pp. 1151-1163, 2000. · Zbl 1006.82025 · doi:10.1051/m2an:2000121 [17] M. W. Hirsch, Differential Topology, New York: Springer-Verlag, 1994, vol. 33. · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5 [18] H. Lorentz, ”Le mouvement des électrons dans les métaux,” Arch. Néerl., vol. 10, pp. 336-371, 1905. · JFM 36.0922.02 [19] J. Marklof and A. Strömbergsson, ”The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems,” Ann. of Math., vol. 172, iss. 3, pp. 1949-2033, 2010. · Zbl 1211.82011 · doi:10.4007/annals.2010.172.1949 [20] J. Marklof and A. Strömbergsson, ”Kinetic transport in the two-dimensional periodic Lorentz gas,” Nonlinearity, vol. 21, iss. 7, pp. 1413-1422, 2008. · Zbl 1153.82020 · doi:10.1088/0951-7715/21/7/001 [21] J. Marklof and A. Strömbergsson, The periodic Lorentz gas in the Boltzmann-Grad limit: Asymptotic estimates. · Zbl 1242.82036 · doi:10.1007/s00039-011-0116-9 [22] I. Melbourne and M. Nicol, ”Almost sure invariance principle for nonuniformly hyperbolic systems,” Comm. Math. Phys., vol. 260, iss. 1, pp. 131-146, 2005. · Zbl 1084.37024 · doi:10.1007/s00220-005-1407-5 [23] I. Melbourne and M. Nicol, ”A vector-valued almost sure invariance principle for hyperbolic dynamical systems,” Ann. Probab., vol. 37, iss. 2, pp. 478-505, 2009. · Zbl 1176.37006 · doi:10.1214/08-AOP410 [24] R. G. Newton, Scattering Theory of Waves and Particles, Second ed., New York: Springer-Verlag, 1982. · Zbl 0496.47011 [25] W. Rudin, Real and Complex Analysis, Third ed., New York: McGraw-Hill Book Co., 1987. · Zbl 0925.00005 [26] D. Szász and T. Varjú, ”Limit laws and recurrence for the planar Lorentz process with infinite horizon,” J. Stat. Phys., vol. 129, iss. 1, pp. 59-80, 2007. · Zbl 1128.82011 · doi:10.1007/s10955-007-9367-0 [27] H. Spohn, ”The Lorentz process converges to a random flight process,” Comm. Math. Phys., vol. 60, iss. 3, pp. 277-290, 1978. · Zbl 0381.60099 · doi:10.1007/BF01612893
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