Une équation cinétique pour les gaz quantiques

Abstract

This article is the second of a series where we propose a theory of the dynamics of quantum gases with a precise treatment of the short range two body correlations. For this purpose, we use a variant of the Wigner transform, the , which has a useful property in this context: if, before collision, two particles are uncorrelated, the incoming part of the free Wigner transform remains always exactly factorized. Therefore, instead of asssuming a factorization of the two particle distribution, it is less restrictive to make a similar assumption on the free transform. Time symmetry is broken when the correlations of the outgoing part of the free transform are ignored; this approximation, which is well in the spirit of Boltzmann, amounts to considering successive collisions as independent processes (the correlations present in the outgoing part could play a role if the same particles rapidly collided again, an unlikely event in dilute gases). Therefore, the succession of collisions is not treated exactly, but each collision is, even during interaction. One then obtains a closed set of equations which leads directly to a kinetic equation for the free distribution $ f $. The equation includes retardation and quantum refraction effects, and provides a generalization of the Boltzmann equation. The reconstruction of the two particle distribution function $ f_{II} $ from $ f $ reintroduces the effects of short range correlations and, more generally, gives the values of all one particle and two particle physical observables. The price to pay for not using the usual one particle distribution is that the relation between the local physical quantities (density of particles, energy density, etc.) becomes more complicated than in the usual theory, and includes terms which are non linear in $ f $. We nevertheless show how this formalism automatically satisfies the local conservation of all hydrodynamic quantities: particle, momentum and energy densities. The pressure tensor includes quadratic terms which correspond to second virial corrections. The temperature of the gas is defined from the kinetic energy of the particles when they are far apart from each other. We finally briefly discuss the connections of this formalism with the classical Enskog theory, or mean field type theories such as the Landau theory (in particular, we note the presence of recoil effects in the molecular field)