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, t{v^'}). 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, ).