
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. Trushechkin^{ab} ^{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 nonthermodynamic limit. It is based on the BBGKY hierarchy, the functional formulation of classical mechanics, and the distinguishing between two scales of spacetime, 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
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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 158164
\mathnet{http://mi.mathnet.ru/vsgtu887}
\crossref{https://doi.org/10.14498/vsgtu887}
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