Lattice Wind‐Tree Models. I. Absence of Diffusion

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.