Two new deterministic lattice versions of the wind-tree model, with fixed or moving mirrors placed randomly on a square lattice are considered. In both cases, the diffusion coefficient apppears to be given by the Boltzmann expression for all densities ofthe mirrors.Two lattice versions of the continuous wind-tree whereas for some models in refs. [3,4] collision model [1,2] are considered, that possess determinevents differ according to whether they occur at even istic dynamics of particles between randomly placed or at odd times. scatterers on a square lattice. In these models, aThe calculations were performed for a square lat-(wind) particle moves on the lattice with initially tice of 360x 360 lattice sites. Instead of considering one of four possible velocities, of magnitude 1 in the the motion ofa single particle, the motions of a small ± x-or ± y-direction. Steps of unit length are taken group of 50 x 50 particles placed initially in the cenper unit time, carrying the particle one lattice site ter of the lattice with either parallel or random yeforward in the direction of its velocity. When the locities were followed. particle moves to a position on the lattice, where a For model A this implies the simultaneous conscatterer (tree) resides, it continues to move in a disideration of a number of independently moving rection perpendicular to its original direction, as if particles and a diffusion process under equilibrium it had hit a mirror positioned at a E/4-or 3~t/4-angle conditions, where the diffusion constant will only to its direction of motion. In both models, the scatdepend on the concentration of the scatterers. Howterers are randomly placed on the lattice with a fracever, for model B, the spatially inhomogeneous mition f 1 at an angle x/4 and a fraction f at an angle tial condition is not an equilibrium condition and 3ir/4, so that the particle describes a deterministic, the diffusion coefficient will depend, in principle, on but random-like motion on the lattice. In model A, the concentration of the scatterers as well as the parthe mirrors are fixed for all time, while in model B, tides, since both determine the amount of flipping the mirrors flip at collision over an angle it/2, changof the mirrors. ing from a it/4-angle to a 3it/4-angle mirror and viceIn the case of parallel initial velocities, a random versa. We studied the diffusion process for both velocity distribution was observed after a few collimodels and we present some results of our compusions. Subsequently, a much slower approach to a tations for the case f1 =1;, carried out by letting an diffusion regime followed, where the mean square Atari ST 1040 run for about 48 hours. There is an displacement of the particles 4(t) approached a beimportant difference with the deterministic models havior like 4DT with a diffusion constant D, which studied so far in the literature [3,4] in that no timeis independent of the time t. The characteristic time dependent dynamics is introduced in our models, to reach a level of about ninety percent of the ...
