Memory functions for transport fluxes: Explicit formulae for high order memory kernels

Memory kernels of the fourth order and fifth order in Mori's continued fraction representation of time correlation functions are derived here explicitly in terms of the transport fluxes and their time derivatives. Collective modes and mode-mode coupling effects are included in the basic dynamic variable set. It is observed that the damping matrix terms in the continued fraction, are equilibrium ensemble averages of the transport fluxes and are thus amenable to computer simulation calculations. Since equilibrium ensemble averages can be obtained with less numerical uncertainties than dynamic time correlations, the expressions presented for K4(0) and K5(0) are useful in numerical works.