The exact solution of an elimination problem in kinetic theory

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f(1)(z1; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f(n)(z1;…;zn; t) in terms of the 1-particle reduced distribution function f(1)(z1; t) for the specified particle by starting with separate expressions for these functions in terms of f(1)(z1; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.