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

We continue our investigation of the time evolution of a one-dimensional system of hard rods. At $t=0\mathrm{}$ there is one particle with a specified position $r^{\prime }$ and velocity $v^{\prime }$, and the remainder are in "equilibrium." Since in this system collisions merely interchange velocities, the "equilibrium" velocity distribution $h_0\mathrm{}\left(v\right)\mathrm{}$ need not be Maxwellian. Exact solutions are obtained for the time-dependent one-particle position-velocity distribution function $f(r-r^{\prime }, v, \frac{t}{v^{\prime }})$. We investigate in particular the averaged positional part of $f$, viz., $G(r-r^{\prime }, t)$, which is the time-dependent pair correlation function whose space-time Fourier transform $S(k,\omega )$ describes coherent neutron scattering in realistic systems. It is shown that $S(k,\omega )$ 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 $\chi \mathrm{}(k,t)\mathrm{}$ of $G(r,t)\mathrm{}$ 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 $S(k,\omega )$.