12 - Divergences, Resummations, and Logarithms

Summary

Cluster expansion methods provide a power series expansion in the density for the collision operator in the equation for the time dependence of the one particle distribution function. Successive terms depend on the dynamics of successively larger numbers of particles. All but the first few terms grow with time, for long times, due to contributions from correlated sequences of binary collisions. Useful expressions are obtained by summing the fastest growing terms in each order of the density. This “ring resummation” predicts that the density expansion for transport coefficients contains terms proportional to logarithms of the gas density. The leading logarithmic terms in the expansion have been calculated for several systems and are in good agreement with the results of computer simulations. The ring sum also provides a microscopic foundation for mode coupling theory needed for a description of the long time behavior of Green-Kubo correlation functions and other quantities.