VARIOUS VELOCITY CORRELATION-FUNCTIONS IN A LORENTZ GAS - SIMULATION AND MODE-COUPLING THEORY

We present computer simulation results for several types of velocity c orrelation function in the two dimensional, overlapping Lorentz gas. O nly the normal velocity autocorrelation function, whose integral gives the diffusion constant, shows obvious anomalous behaviour at the perc olation transition. The other functions are fairly well approximated b y the Lorentz-Boltzmann equation, even for densities at which the trav elling particle is trapped. We do, however, at a sub-percolation densi ty, examine the long time behaviour of the autocorrelation function co rresponding to the second rank, irreducible tenser of the velocity, an d find an algebraic decay with an exponent of 3.0 +/- 0.1, consistent with the theoretically expected value of 3. With these observations in mind we re-examine the mode coupling theory of Gotze, Leutheusser and Yip (Phys. Rev. A 23 (1981) 2634,) replacing their one (frequency dep endent) relaxation time approximation to a kinetic operator by a two ( frequency dependent) relaxation time model. We find that this leads to a significantly better estimate of the diffusion constant at low dens ity. Furthermore the theory correctly predicts no striking anomalous b ehaviour in the types of velocity correlation function that are unrela ted to diffusion as the percolation threshold is crossed.