Efforts for a systematic derivation as well as a generalization of the Boltzmann equation have been made by N.N. Bogolyubov in 1946. 10 The starting point for the derivation of the Boltzmann equation is the time reversal invariant Liouville equation for the N-particle phase space density

where {,} is the Poisson bracket and H the Hamiltonian of the system. This follows from the classical equations of motion and expresses the conservation of the probability in phase space. To arrive at the Boltzmann equation which violates time reversal invari ance, because the direction of increasing entropy is singled out, some coarse graining is necessary which is done by successively integrating over the coordinates and momenta of N-1 particles until one arrives at the one particle distribution function f : = f(x; v; t) This way one arrives at the so{called B.B.G.K.Y. chain of equations which stands for the rst letters of the physicists Bogolyubov, Born and Green, Kirkwood, Yvon. Then one has to perform the limit of low density, the so-called Boltzmann-Grad-Limit and make the assumption of molecular chaos for the initial dis tribution which implies factorization of the reduced n-particle densities into products of one particle densities. Furthermore one has to assume that the system is large enough so that the in uence of the walls of its container is negligible. There exist a number of such derivations of the Boltzmann equation of which I would like to mention the one by O.E. Lanford and one by an Italian group. While Lanford's derivation holds only for a very short time interval, the Italian derivation implies so low densities that nearly no collisions take place. These restrictions are not surprising since at higher den sities correlated collision sequences appear such as the "ring collisions" which introduce correlations between the colliding particles violating the assump tion of molecular chaos.

In this way the relaxation to equilibrium is slowed down. In computer simulations of a gas of hard spheres which are often called "relaxation exper- iments" one can study the approach to the one particle equilibrium distribution. Alder and Wainwright did pioneering experiments of this kind in 1958. Quite recently such experiments have also been performed at Vienna University. Following the analytic methods of Bogolyubov a modified Boltzmann equation of the type

is obtained where J(ff) containes the two particle collisions, K(fff) the three parti cle collisions, L(ffff) the four paticle collisions etc.. The solution of this generalized Boltzmann equation with the Chapman-Enskog method then leads to a density or virial expansion for the viscosity

which is a power series expansion in the density n with temperature dependent co eficients. For higher densities, however, it turns out that especially due to the "ring collision" terms the Bogolyubov collision integrals K(fff) and L(ffff) become di vergent and a cut off for the mean free path has to be introduced. The revised density expansion for the viscosity now contains a logarithmic term in the density n:

Similar expressions result for the other transport coeficients.