The exact solution of an elimination problem in kinetic theory

A class of initial value problems is considered where a specified particle has a given distribution ƒ(1)(z1;0) and the rest are in equilibrium at t=0. A formally exact expansion is obtained for a certain n-particle reduced distribution-function ƒ(1)(z1, z2, …, zn;t) in terms of the one-particle reduced distribution-function ƒ(1)(z1;t) for the specified particle at time t. We start with separate expansions for these functions in terms of ƒ(1)(z1; 0) and use direct inversion and combinatorial procedures to obtain a convenient representation of the general term in the resulting expansion.