Derivation of the Generalized Vlasov Equation for Classical Fluids

Abstract

The generalized Vlasov equation for classical fluids is derived from the Liouville equation. The derivation is valid in the limiting case where the one-particle phase-space distribution function is Maxwellian with respect to the momentum and is slowly varying in the space and time. alized Vlasov equation has been used, with or without a collision term. 4 ),5) The equation .was obtained by Zwanzig,6) when he discussed collective coordinate for classical fluids. It is the purpose of the present paper to give a derivation of the generalized Vlasov equation, starting from the Liouville equation, and to show the range of its validity. It is hoped that the result will become a basis on which one con­ siders the ways of introducing the corrections to the equation. The problem of deriving the Boltzmann equation or its generalization from the Liouville equation has been taken up by various authors for classical gases. 7 ) The general terms for non-uniform classical fluids was achieved by Severne S ) with the aid of diagram technique developed by Prigogine and his coworkers.g) The result was generalized by Fujita to the case where quantum mechanics must be used. IO