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 Uspekhi Mat. Nauk, 2016, Volume 71, Issue 2(428), Pages 81–120 (Mi umn9707)

Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas

V. V. Kozlov

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The problem of conditions ensuring the existence of first integrals that are polynomials in the momenta (velocities) is considered for certain multidimensional billiard systems which play an important role in non-equilibrium statistical mechanics. These are the Lorentz gas, a particle in a Euclidean space with (not necessarily convex) scattering domains, and the Boltzmann–Gibbs gas, a system of small identical balls in a rectangular box which collide elastically with one another and the walls of the box. The ergodic properties of such systems are only partially understood: some problems are still waiting for solution, and in certain cases (for instance, when the scatterers are non-convex) the system is known not to be ergodic. An approach to showing the absence of a non-trivial polynomial first integral with continuously differentiable coefficients is developed. The known first integrals for integrable problems in dynamics are mostly polynomials in the momenta (or functions of polynomials). The investigation of multidimensional billiards with non-compact configuration space, when there is no hope for ergodic behaviour, is of particular interest. Applications of the general results on the absence of non-trivial polynomial integrals to problems in statistical mechanics are discussed.
Bibliography: 62 titles.

Keywords: Birkhoff billiards, Lorentz gas, Boltzmann–Gibbs gas, polynomial integral, topological obstructions to integrability, elastic reflection, KAM theory.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work was supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.4213/rm9707

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English version:
Russian Mathematical Surveys, 2016, 71:2, 253–290

Bibliographic databases:

UDC: 514.755+530.1:51+536
MSC: Primary 37D50, 70F35, 70H33; Secondary 70H08

Citation: V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Uspekhi Mat. Nauk, 71:2(428) (2016), 81–120; Russian Math. Surveys, 71:2 (2016), 253–290

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9707
• https://doi.org/10.4213/rm9707
• http://mi.mathnet.ru/eng/umn/v71/i2/p81

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Kozlov, D. V. Treschev, “Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics”, Sb. Math., 207:10 (2016), 1435–1449
2. S. V. Bolotin, “Degenerate billiards”, Proc. Steklov Inst. Math., 295 (2016), 45–62
3. Ivan A. Bizyaev, Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups”, Regul. Chaotic Dyn., 21:6 (2016), 759–774
4. S. V. Bolotin, V. V. Kozlov, “Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom”, Izv. Math., 81:4 (2017), 671–687
5. D. Treschev, “A locally integrable multi-dimensional billiard system”, Discrete Contin. Dyn. Syst. Ser. A, 37:10 (2017), 5271–5284
6. I. A. Bizyaev, A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Integriruemost i neintegriruemost subrimanovykh geodezicheskikh potokov na gruppakh Karno”, Nelineinaya dinam., 13:1 (2017), 129–146
7. M. Bialy, A. E. Mironov, “Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane”, Russian Math. Surveys, 74:2 (2019), 187–209
8. V. Schastnyy, D. Treschev, “On local integrability in billiard dynamics”, Exp. Math., 28:3 (2019), 362–368
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