Derivation of the Vlasov Equation

The solution of the Liouville equation is expressed in the interaction representation. We show that the early-time behavior of the single-particle distribution function $F_1$ for a classical gas consisting of particles interacting through weak two-body central forces is governed by the Vlassov equation. The two-body potentials are assumed to be "good" functions. The derivation fails if initial correlations are present. The argument is carried out in configuration space. The analysis is extended to the derivation of differential equations governing the time evolution of the multiparticle distribution functions $F_k$. In this "Vlasov approximation" to the solution of the Liouville equation, we find that the Vlassov equation for $F_1$, together with the differential equations for the $F_k$ ($k<\mathrm{}1\mathrm{}$), amounts to the statement that the multiparticle distribution functions are factorable into products of single-particle distribution functions.