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This article is cited in 11 scientific papers (total in 11 papers)
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.
Received: 10.02.2016
Citation:
V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290
Linking options:
https://www.mathnet.ru/eng/rm9707https://doi.org/10.1070/RM9707 https://www.mathnet.ru/eng/rm/v71/i2/p81
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Abstract page: | 994 | Russian version PDF: | 254 | English version PDF: | 69 | References: | 126 | First page: | 72 |
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