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

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.

About the authors

Anton S Trushechkin

Steklov Mathematical Institute, Russian Academy of SciencesNational Research Nuclear University "MEPhI"

Email: trushechkin@mi.ras.ru
(к.ф.-м.н.), научный сотрудник, отд. математической физики; доцент, каф. системного анализа; Математический институт им. В. А. Стеклова РАННациональный исследовательский ядерный университет МИФИ; Steklov Mathematical Institute, Russian Academy of SciencesNational Research Nuclear University "MEPhI"

References

  1. Козлов В. В. Тепловое равновесие по Гиббсу и Пуанкаре. М., Ижевск: Институт компьютерных исследований, 2002. 320 с.
  2. Боголюбов Н. Н. Проблемы динамической теории в статистической физике. М., Л.: Гостехиздат, 1946. 119 с.
  3. Боголюбов Н. Н. Кинетические уравнения и функции Грина в статистической механике / В сб.: Собрание научных трудов в двенадцати томах. Т. V. М.: Наука, 2005. С. 616- 638.
  4. Волович И. В. Проблема необратимости и функциональная формулировка классической механики // Вестн. Сам. гос. ун-та. Естественнонаучн. сер., 2008. № 8/1(67). С. 35-55, arXiv: 0907.2445 [cond-mat.stat-mech].
  5. Волович И. В. Уравнения Боголюбова и функциональная механика // ТМФ, 2010. Т. 164, № 3. С. 354-362
  6. Trushechkin A. S., Volovich I. V. Functional classical mechanics and rational numbers // p- Adic Numbers, Ultrametric Analysis, and Applications, 2009. Т. 1, № 4. С. 361-367, arXiv: 0910.1502 [math-ph].
  7. Трушечкин А. С. Необратимость и роль измерительного прибора в функциональной формулировке классической механики // ТМФ, 2010. Т. 164, № 3. С. 435-440
  8. Лифшиц Е. М., Питаевский Л. П. Физическая кинетика. М.: Физматлит, 2002. 536 с.
  9. Wigner E. The Unreasonable Effectiveness of Mathematics in the Natural Sciences // Commun. Pure Appl. Math., 1960. Vol. 13, no. 1. Pp. 1-14

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