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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2011, Issue 1(22), Pages 158–164 (Mi vsgtu887)  

Procedings of the 2nd International Conference "Mathematical Physics and its Applications"
Mathematical Physics

Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics

A. S. Trushechkinab

a Dept. of System Analysis, National Research Nuclear University “MEPhI”, Moscow
b Dept. of Mathematical Physics, Steklov Mathematical Institute, Russian Academy of Sciences, Moscow

Abstract: We propose a procedure for obtaining the Boltzmann equation from the Liouville equation in a non-thermodynamic limit. It is based on the BBGKY hierarchy, the functional formulation of classical mechanics, and the distinguishing between two scales of space-time, i.e., macro- and microscale. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which have errors. Hence, one can speak about accuracy of the initial probability density function in the Liouville equation. Let's assume that our measuring instruments can observe the variations of physical values only on the macroscale, which is much greater than the characteristic interaction radius (microscale). Then the corresponfing initial density function cannot be used as initial data for the Liouville equation, because the last one is a description of the microscopic dynamics, and the particle interaction potential (with the characteristic interaction radius) is contained in it explicitly. Nevertheless, for a macroscopic initial density function we can obtain the Boltzmann equation using the BBGKY hierarchy, if we assume that the initial data for the microscopic density functions are assigned by the macroscopic one. The $H$-theorem (entropy growth) is valid for the obtained equation.

Keywords: statistical mechanics, physical kinetics, Boltzmann equation, Liouville equation, BBGKY hierarchy

DOI: https://doi.org/10.14498/vsgtu887

Full text: PDF file (568 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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Bibliographic databases:

Document Type: Article
UDC: 517.958
MSC: 82C05, 82C40
Original article submitted 21/XII/2010
revision submitted – 21/II/2011

Citation: A. S. Trushechkin, “Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1(22) (2011), 158–164

Citation in format AMSBIB
\Bibitem{Tru11}
\by A.~S.~Trushechkin
\paper Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2011
\vol 1(22)
\pages 158--164
\mathnet{http://mi.mathnet.ru/vsgtu887}
\crossref{https://doi.org/10.14498/vsgtu887}


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  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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