Kinetic equations and density expansions: Exactly solvable one-dimensional system

We have made a detailed study of the time evolution of the distribution function f(q,v,t) of a labeled (test) particle in a one-dimensional system of hard rods of diameter a. The system has a density ρ and is in equilibrium at t=0. (Some properties of this system were studied earlier by Jepsen.) When the distribution function f at t=0 corresponds to a delta function in position and velocity, then f(q,v,t) is essentially the time-displaced self-distribution function fs. This function fs (which can be found in an explicit closed form) and all of the system properties which can be derived from it depend on ρ and a only through the combination n=ρ(1-ρa). In particular, the diffusion constant D is given by D-1=lims→0[ψ(s)]-1=(2πβm)12n, where ψ(s) is the Laplace transform of the velocity autocorrelation function ψ(t)=v(t)v. An expansion of [ψ(s)]-1 in powers of n, on the other hand, has the form Blnlsl-1, leading to divergence of the density coefficients for l2 when s→0. This is similar to the divergences found in higher dimensional systems. Similar results are found as well in the expansion of the collision operator describing the time evolution of f(q,v,t). The lowest-order term in the expansion is the ordinary (linear) Boltzmann equation, while higher terms are O(ρltl-1). Thus any attempt to write a Bogoliubov, Choh-Uhlenbeck-type Markoffian kinetic equation as a power series in the density leads to divergence in the terms beyond the Boltzmann equation. A Markoffian collision operator can, however, be constructed, without using a density expansion, which, e.g., describes the stationary distribution of a charged test particle in the system in the presence of a constant electric field. The distribution of the test particle in the presence of an oscillating external field is also found. Finally, the short- and long-time behavior of the self-distribution is examined.