Kinetic Equations and Density Expansions: Exactly Solvable One-Dimensional System

Abstract

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 $\ensuremath{\rho}$ 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 ${f}_{s}$. This function ${f}_{s}$ (which can be found in an explicit closed form) and all of the system properties which can be derived from it depend on $\ensuremath{\rho}$ and $a$ only through the combination $n=\frac{\ensuremath{\rho}}{(1\ensuremath{-}\ensuremath{\rho}a)}$. In particular, the diffusion constant $D$ is given by ${D}^{\ensuremath{-}1}={\mathrm{lim}}_{s\ensuremath{\rightarrow}0}{[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)]}^{\ensuremath{-}1}={(2\ensuremath{\pi}\ensuremath{\beta}m)}^{\frac{1}{2}}n$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)$ is the Laplace transform of the velocity autocorrelation function $\ensuremath{\psi}(t)=〈v(t)v〉$. An expansion of ${[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)]}^{\ensuremath{-}1}$ in powers of $n$, on the other hand, has the form $\ensuremath{\Sigma}\frac{{B}_{l}{n}^{l}}{{s}^{l\ensuremath{-}1}}$, leading to divergence of the density coefficients for $l\ensuremath{\ge}2$ when $s\ensuremath{\rightarrow}0$. 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 $O({\ensuremath{\rho}}^{l}{t}^{l\ensuremath{-}1})$. 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.