Long time tail for spacially inhomogeneous random walks

Abstract

The Lorentz gas consists of a single classical particle moving through in R 2 randomly distributed, infinitely heavy hard disk scatte-rers. The Lorentz particle starts at the origin with a uniform distribution of velocities of magnitude one. The hard disks are in equilibrium conditioned not to overlap the origin. The time integral over the correlation function of the stationary velocity process v(t) gives the diffusion constant of the Lorentz gas. One deep problem is to show that this diffusion constant is finite, which would follow from a sufficiently fast decay of. Theoretical and numerical results indicate a power law decay as-t-(d/2 + I) in d dimensions. To understand qualitatively the origin of the long time tail I use an old recipe in non-equilibrium statistical mechanics which advices to replace the deterministic motion by a stochastic one. Then at each site of the lattice Z d there is with probability p a scatterer, with probability 1-p no scatterer. The particle performs a random walk: At a site with a scatterer the particle continues with probability I/2d in one of the 2d directions. At a site with no scatterer the particle continues in the direction it came from. One wants to prove that the velocity au-tocorrelation function has a long time tail. This seems to be difficult. To simplify: Suppose that at each site except the origin there is a scatterer. Then in one dimension, for the particle starting at the origin, since the only contribution comes from paths where the particle is at the origin at time t, =-r(t) + (r~r) (t)-..., where r(t) is the probability of a first return to the origin at time t for the simple random walk. Since r(t) = t-3/2 for long times, this implies the expected long time tail for this model. A symmetry argument due to M. Aizenman proves that ~-t-(d/2+I) in d dimensions. The long time tail originates from a spacially inhomo-geneous distribution of scatterers. A periodic distribution gives an exponential decay. Together with H. van Beijeren I studied the one dimensional random scatterer model. Time is continuous and the distances ~. between scatterers are independently and identically distributed. For 1 ~ Z t + /z )2>/ small z we prove o /dte v(t)'v(O)> /8 + O(zI/2+~), 5>0. The negative time tail cannot decrease faster than°t-3/2.