Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods

Abstract

We continue our investigation of the time evolution of a one-dimensional system of hard rods. At $t=0$ there is one particle with a specified position ${r}^{\ensuremath{'}}$ and velocity ${v}^{\ensuremath{'}}$, and the remainder are in "equilibrium." Since in this system collisions merely interchange velocities, the "equilibrium" velocity distribution ${h}_{0}(v)$ need not be Maxwellian. Exact solutions are obtained for the time-dependent one-particle position-velocity distribution function $f(r\ensuremath{-}{r}^{\ensuremath{'}}, v, \frac{t}{{v}^{\ensuremath{'}}})$. We investigate in particular the averaged positional part of $f$, viz., $G(r\ensuremath{-}{r}^{\ensuremath{'}}, t)$, which is the time-dependent pair correlation function whose space-time Fourier transform $S(k,\ensuremath{\omega})$ describes coherent neutron scattering in realistic systems. It is shown that $S(k,\ensuremath{\omega})$ does not generally contain modes corresponding to sound propagation. The exceptions are systems with discrete velocity distributions. In the latter case the space Fourier transform $\ensuremath{\chi}(k,t)$ of $G(r,t)$ is rigorously a sum of simple damped oscillations. An exact kinetic equation for the time evolution of $f$ is derived and investigated. Also found is an approximate kinetic equation which, however, gives exact values of $S(k,\ensuremath{\omega})$.