Study of a Lorentz-gas based on correlated walks

The dynamics of a Lorentz-gas molecule is simulated in terms of correlated walks on cubic lattices. For a finite lattice with reflecting boundary, the ergodicity is established so that the probability distribution in position and direction approaches a stationary state which is homogeneous (site-independent) and isotropic (direction-independent). Exact expressions for probability distribution in direction are obtained for simple, body-centered, and face-centered cubic lattices. They approach equilibrium exponentially with a different number of relaxation times depending on the lattice. The directional probabilities for a single-site lattice with reflecting boundary approach equilibrium in an oscillatory manner. The Boltzmann H-function for this system, however, shows a monotonic behavior, and coincides with the H-function for the same system with periodic boundary. The parameters in the model, step length and unit time, can be eliminated in the kinetic theoretical limit. In this limit, the diffusion coefficient calculated for the lattice reproduces the result by means of the Boltzmann equation within the restriction imposed on the directions of the movement.