Derivation using projection techniques of a hierarchy of kinetic equations for relativistic gases

Abstract

The description of the non-equilibrium behavior of gases is one of the central problems of statistical mechanics. The description of such non-equilibrium behavior, which usually starts from the Liouville equation for the phase space distribution function f_N(x_i, ...,x_N, p_i, ..., p_N), has been, and continues to be, motivated by the following purposes:
1. The explicit calculation of transport parameters.
2. The recovery and generalization of Boltzmann type kinetic equations.
3. The description of phenomena which lie outside the domains of validity of the Boltzmann equation.
Many approaches have been introduced to obtain such information from the Liouville equation. Among these approaches one may cite, for instance:
1. The direct analysis of the BBGKY hierarchy of equations − a hierarchy of coupled equations for the reduced distribution function obtained from the Liouville equation by successive integration. Closure approximations are usually made to help analyze this hierarchy.
2. Cluster expansion techniques patterned after the cluster expansion techniques first used in equilibrium statistical mechanics.
3. Diagrammatic methods − the starting equations of which are obtained via Fourier analysis of the Liouville equation or BBGKY hierarchy.
4. Correlation function methods as exemplified by the linear response method of Kubo.
5. The use of explicit projection techniques resulting in a hierarchy of equations different from the BBGKY hierarchy. Projection techniques have also been used to study the evolution of systems with initial density discontinuities, derive the Reynolds equation, and obtain an integral formulation of hydrodynamics among others.
In this paper we use explicit projection techniques to derive a hierarchy of kinetic equations for a relativistic gas of identical particles interacting via position dependent two body forces.