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 $f(q,v,t)\mathrm{}$ of a labeled (test) particle in a one-dimensional system of hard rods of diameter $a$. The system has a density $\rho$ and is in equilibrium at $t=0\mathrm{}$. (Some properties of this system were studied earlier by Jepsen.) When the distribution function $f$ at $t=0\mathrm{}$ corresponds to a delta function in position and velocity, then $f(q,v,t)\mathrm{}$ 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 $\rho$ and $a$ only through the combination $n=\frac{\rho }{(1-\rho a)}$. In particular, the diffusion constant $D$ is given by $D^{-1\mathrm{}}={\lim }_{s\to 0}{\left[\tilde{\psi }\mathrm{}\right(s\left)\right]}^{-1\mathrm{}}={\left(2\mathrm{}\pi \beta m\right)}^{\frac{1}{2}}n$, where $\tilde{\psi }\mathrm{}\left(s\right)\mathrm{}$ is the Laplace transform of the velocity autocorrelation function $\psi \mathrm{}\left(t\right)=〈v\left(t\right)v〉$. An expansion of ${\left[\tilde{\psi }\mathrm{}\right(s\left)\right]}^{-1\mathrm{}}$ in powers of $n$, on the other hand, has the form $\Sigma \frac{B_l{n}^l}{s^{l-1\mathrm{}}}$, leading to divergence of the density coefficients for $l\ge 2$ when $s\to 0$. 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 $f(q,v,t)\mathrm{}$. The lowest-order term in the expansion is the ordinary (linear) Boltzmann equation, while higher terms are $O\left({\rho }^l{t}^{l-1\mathrm{}}\right)$. 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.