Lorentz lattice-gas and kinetic-walk model

Abstract

The Ruijgrok-Cohen (RC) mirror model [Phys. Lett. A 133, 415 (1988)] of a Lorentz lattice gas, in which particles are reflected by left and right diagonally oriented mirrors randomly placed on the sites of a square lattice, is further investigated. Extensive computer simulations of individual trajectories up to ${2}^{24}$ steps in length, on a lattice of 65 536\ifmmode\times\else\texttimes\fi{}65 536 sites, are carried out. This model generates particle trajectories that are related to a variety of kinetic growth and ``smart'' (nontrapping) walks, and provides a kinetic interpretation of them. When all sites are covered with mirrors of both orientations with equal probability, the trajectories are equivalent to smart kinetic walks that effectively generate the hulls of bond percolation clusters at criticality. For this case, ${10}^{6}$ trajectories were generated, yielding with unprecedented accuracy an orbit size-distribution exponent of \ensuremath{\tau}=2.1423\ifmmode\pm\else\textpm\fi{}0.0003 and a fractal dimension of ${\mathit{d}}_{\mathit{f}}$=1.750 47\ifmmode\pm\else\textpm\fi{}0.000 24 (without correcting for finite-size effects), compared with theoretical predictions of 15/7=2.142 857. . . and 7/4, respectively. When the total concentration of mirrors C is less than unity, so that the trajectories can cross, the size distribution of the closed orbits does not follow a power law, but appears to be described by a logarithmic function.This function implies that all trajectories eventually close. The geometry of the trajectories does not show clear self-similar or fractal behavior in that the dependence of the mean-square displacement upon the time also appears to follow a logarithmic function. These trajectories are related to the growing self-avoiding trail (GSAT) introduced by Lyklema [J. Phys. A 18, L617 (1985)], and the present work supports the conjecture of Bradley [Phys. Rev. A 41, 914 (1990)] that the GSAT (the RC model with C=2/3) is not at a critical point. It is observed that when C1, the trajectories behave asymptotically like an unrestricted random walk, and so for comparison the RC model in the random walk or Boltzmann approximation (BA) is also studied. In the BA, the size distribution of returning trajectories and the geometric properties of open trajectories are investigated; the time dependence of the mean-square displacement is derived explicitly and is shown to exhibit an Ornstein-Uhlenbeck type of behavior.