Geodesic
Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2011), pp. 158-164.

Voir la notice de l'article provenant de la source Math-Net.Ru

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, BBGKY hierarchy.
Mots-clés : Liouville equation 
@article{VSGTU_2011_1_a20,
     author = {A. S. Trushechkin},
     title = {Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {158--164},
     publisher = {mathdoc},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2011_1_a20/}
}
TY  - JOUR
AU  - A. S. Trushechkin
TI  - Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2011
SP  - 158
EP  - 164
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2011_1_a20/
LA  - ru
ID  - VSGTU_2011_1_a20
ER  - 
%0 Journal Article
%A A. S. Trushechkin
%T Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2011
%P 158-164
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2011_1_a20/
%G ru
%F VSGTU_2011_1_a20
A. S. Trushechkin. Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2011), pp. 158-164. http://geodesic.mathdoc.fr/item/VSGTU_2011_1_a20/

[1] Kozlov V. V., Thermal equilibrium in the sense of Gibbs and Poincaré, Institut Komp'yuternykh Issledovanij, Moscow, Izhevsk, 2002, 320 pp. | MR | Zbl

[2] Bogolyubov N. N., Problems of Dynamical Theory in Statistical Physics, Gostekhizdat, Moscow, Leningrad, 1946, 119 pp. | MR

[3] Bogolyubov N. N., “Kinetic equations and Green functions in statistical mechanics”, Collection of scientific works in twelve volumes, v. V, Nauka, Moscow, 2005, 616–638

[4] Volovich I. V., “Time Irreversibility Problem and Functional Formulation of Classical Mechanics”, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser., 2008, no. 8/1(67), 35–55

[5] Volovich I. V., “Bogoliubov equations and functional mechanics”, Theoret. and Math. Phys., 164:3 (2010), 1128–1135 | DOI | DOI | Zbl

[6] Trushechkin A. S., Volovich I. V., “Functional classical mechanics and rational numbers”, $p$-Adic Numbers, Ultrametric Analysis, and Applications, 1:4 (2009), 361–367, arXiv: 0910.1502 [math-ph] | DOI | MR | Zbl

[7] Trushechkin A. S., “Irreversibility and the role of an instrument in the functional formulation of classical mechanics”, Theoret. and Math. Phys., 164:3 (2010), 1198–1201 | DOI | DOI | Zbl

[8] Lifshitz E. M., Pitaevskii L. M., Physical Kinetics, Fizmatlit, Moscow, 2002, 536 pp. | Zbl

[9] Wigner E., “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Commun. Pure Appl. Math., 13:1 (1960), 1–14 ; Vigner E., “Nepostizhimaya effektivnost matematiki v estestvennykh naukakh”, Etyudy o simmetrii, Mir, M., 1971, 182–198 | DOI | Zbl