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 ${D}^{{-}1}={lim}_{s{→}0}{[{ ̃ \~{}{}}{{ψ}}(s)]}^{{-}1}={(2{π}{β}m)}1/2n$, 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 ${Σ}{{B}ₗ{n}ˡ}{{s}^{l{-}1}}$, leading to divergence of the density coefficients for $l{≥}2$ 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 $O({{ρ}}ˡ{t}^{l{-}1})$. 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.