Properties of the Boltzmann equation in the classical approximation
Abstract
We examine the Boltzmann equation with elastic pointlike scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the nonrenormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since one has also access to the nonapproximated result for comparison.
 Authors:
 Institut de Physique Theorique, GifsurYvette Cedex (France)
 RIKEN, Wako (Japan); Brookhaven National Lab., Upton, NY (United States)
 Publication Date:
 Research Org.:
 Brookhaven National Laboratory (BNL), Upton, NY (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Nuclear Physics (NP) (SC26)
 OSTI Identifier:
 1188218
 Report Number(s):
 BNL1078042015JA
Journal ID: ISSN 15507998; PRVDAQ; KB0301020
 Grant/Contract Number:
 SC00112704
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Physical Review. D, Particles, Fields, Gravitation and Cosmology
 Additional Journal Information:
 Journal Volume: 90; Journal Issue: 12; Journal ID: ISSN 15507998
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 73 NUCLEAR PHYSICS AND RADIATION PHYSICS
Citation Formats
Epelbaum, Thomas, Gelis, François, Tanji, Naoto, and Wu, Bin. Properties of the Boltzmann equation in the classical approximation. United States: N. p., 2014.
Web. doi:10.1103/PhysRevD.90.125032.
Epelbaum, Thomas, Gelis, François, Tanji, Naoto, & Wu, Bin. Properties of the Boltzmann equation in the classical approximation. United States. doi:10.1103/PhysRevD.90.125032.
Epelbaum, Thomas, Gelis, François, Tanji, Naoto, and Wu, Bin. 2014.
"Properties of the Boltzmann equation in the classical approximation". United States.
doi:10.1103/PhysRevD.90.125032. https://www.osti.gov/servlets/purl/1188218.
@article{osti_1188218,
title = {Properties of the Boltzmann equation in the classical approximation},
author = {Epelbaum, Thomas and Gelis, François and Tanji, Naoto and Wu, Bin},
abstractNote = {We examine the Boltzmann equation with elastic pointlike scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the nonrenormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since one has also access to the nonapproximated result for comparison.},
doi = {10.1103/PhysRevD.90.125032},
journal = {Physical Review. D, Particles, Fields, Gravitation and Cosmology},
number = 12,
volume = 90,
place = {United States},
year = 2014,
month =
}
Web of Science

On the Classical Boltzmann Equation for Gases
With great simplicity the Boltzmann equation for dilute classical gases is formulated from the reduced Liouville equation by using the three assumptions: no threebody effects, collision time much less than the experimental time, and negligible change in local properties of the system over distances of order of the range of the intermolecular forces. The nature of these assumptions sheds some light on recent attempts to generalize the Boltzmann equation. It was easy formally to relax the first assumption to permit more than twobody effects. The resulting equation is an intuitive extension of the Boltzmann equation. 
Boltzmann equation in classical and quantum field theory
Improving upon the previous treatment by Mueller and Son, we derive the Boltzmann equation that results from a classical scalar field theory. This is obtained by starting from the corresponding quantum field theory and taking the classical limit with particular emphasis on the path integral and perturbation theory. A previously overlooked Van Vleck determinant is shown to control the tadpole type of selfenergy that can still appear in the classical perturbation theory. Further comments on the validity of the approximations and possible applications are also given.