Lattice Wind-Tree Models. I. Absence of Diffusion

Abstract

Some lattice versions of the wind-tree model of Ehrenfest are introduced and studied. They include two versions in which overlapping of tree particles is forbidden, a third in which it is allowed, and a fourth in which it is allowed but results in a finite repulsive force. In every case it is found that the mean-square displacement Δ(t) of the wind particles at time t is bounded independently of t at sufficiently high density of the trees. This is in sharp contrast with the Einstein relation Δ(t) = O(t) as t → ∞, which might be expected to hold at low densities. Randomization of the initial velocity of a wind particle is also shown to occur in a certain sense, and upper bounds on a recurrence time are obtained at high density. On the other hand, it is shown that thermalization does not occur even at low densities.