Progress in Kinetic Theory: Generalization of Boltzmann's Equation

Abstract

We have recently celebrated the hundredth birthday of Boltzmann's kinetic equation, an equation that has been enormously fruitful in physics and engineering. However, Boltzmann's model of transport processes has its limitations. It is based upon a view of Nature that is 'coarse-grained' in time and space, the intervals of coarse-graining being the duration of a collision, and the range of intermolecular force. In the coarse-grained world of rarefied systems, Boltzmann' s Stosszahlansatz works well. As more data on transport processes in liquids and dense gases accumulate, the need for systematic generalization of Boltzmann's equation (and the Chapman-Enskog analysis) grows. Computer experiments, neutron scattering experiments, and the scattering of laser beams have been particularly stimulating here, for they probe extremely short intervals of space and time in the dynamics of the target system. Coarse-graining, and the Stosszahlansatz are no longer acceptable. It is not widely appreciated that for the past decade, scientists have had a compact and elegant formalism at their disposal, for the construction of generalized kinetic equations. The new equations describe the response of the N-body system in the full frequency and wave-number domain. They are characterized by kernels that exhibit both spatial and temporal 'memory' and are, necessarily, very complicated. We shall review recent progress in the analysis, commenting on 1) the generalization of Boltzmann's equation to higher densities . . . the cluster expansion and the Bogoljubov Ansatz, revisited. 2) the generalized kinetic equations of Zwanzig and Mori 3) progress in analysis and solution.