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 there is one particle with a specified position r and velocity v, and the remainder are in "equilibrium." Since in this system collisions merely interchange velocities, the "equilibrium" velocity distribution h0(v) need not be Maxwellian. Exact solutions are obtained for the time-dependent one-particle position-velocity distribution function f(r-r, v, tv). We investigate in particular the averaged positional part of f, viz., G(r-r, t), which is the time-dependent pair correlation function whose space-time Fourier transform S(k,) describes coherent neutron scattering in realistic systems. It is shown that S(k,) 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 (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,).