Some fundamental techniques of kinetic theory for N-particle systems are introduced. In particular, binary collision operators and the binary collision expansion are defined for both smooth and hard sphere potentials. The Liouville equation for the phase space distribution 7 function is presented for smooth interaction potentials, and the pseudo-Liouville equation is given for hard sphere interactions. Integration of the Liouville or pseudo-Liouville equation over a number of particle variables leads to the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy equations. It is shown that the binary collision expansion is a correct representation of the dynamics of a system of N hard sphere particles. The Green-Kubo formulas for transport coefficients in terms of time correlation functions are derived.

The contents of the book are summarized, briefly. Important applications of kinetic theory not covered in the book are mentioned. For classical systems, these applications include kinetic equations for plasmas, for nucleation and aggregation, and for active matter. For quantum systems the phenomenon of localization electrons in amorphous solids, closely related to the kinetic theory of the classical Lorentz gas, is not discussed. Also not discussed are: (1) the theory for electron transport in metals involving electron-phonon interactions, (2) the theory describing long time tail effects in metallic systems that lead to non analytic terms in magnetic susceptibilities, and (3) the kinetic theory of the quark-gluon plasma formed in heavy ion collisions. The book concludes with the observation that one can find parallel developments in gravitational physics, particularly the appearance of hydrodynamic behavior and long time tails in the holographic AdS/CFT theory of black holes.

Time correlation functions appearing in the Green-Kubo expressions for transport coefficients are studied by using kinetic theory. Of particular interest is the theory for the algebraic, t??d=2 long time time decays, or long time tails, first seen in computer simulations of the velocity autocorrelation function for tagged particle diffusion. Kinetic equations for distribution functions that determine time correlation functions are obtained using cluster expansions, and divergences appear due to the effects of correlated sequences of binary collisions. The ring resummation introduced in Chapter 12 leads to mode-coupling expressions containing products of two hydrodynamic mode eigenfunctions. The sum of these contributions leads directly to the long time tails, quantitatively in agreement with computer simulations. Mode coupling theory also leads to an explanation of the observed, intermediate time, molasses tails, and to the existence of fractional powers in sound dispersion relations, for which there is strong experimental evidence.

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.