SCALING OF PARTICLE TRAJECTORIES ON A LATTICE

The scaling behavior of the closed trajectories of a moving particle g enerated by randomly placed rotators or mirrors on a square or triangu lar lattice is studied numerically. On both lattices, for most concent rations of the scatterers the trajectories close exponentially fast. F or special critical concentrations infinitely extended trajectories ca n occur which exhibit a scaling behavior similar to that of the perime ters of percolation clusters. At criticality, in addition to the two c ritical exponents tau = 15/7 and d(f) = 7/4 found before, the critical exponent sigma = 3/7 appears. This exponent determines structural sca ling properties of closed trajectories of finite size when they approa ch Infinity. New scaling behavior was found for the square lattice par tially occupied by rotators, indicating a different universality class than that of percolation clusters. Nea, criticality, in the critical region, two scaling functions were determined numerically: f(x), relat ed to the trajectory length (S) distribution n(S), and h(x), related t o the trajectory size R-S (gyration radius) distribution, respectively . The scaling function f(x:) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent sigma = 0.4 3 approximate to 3/7 as at criticality, leading to a stretched exponen tial dependence of n(S) on S, n(S) similar to exp(-S-6/7). However, fo r the rotator model on the partially occupied square lattice an altern ative scaling function is found, leading to a new exponent sigma' = 1. 6 +/- 0.3 and a superexponential dependence of n(S) on S. h(x) is esse ntially a constant, which depends on the type of lattice and the conce ntration of the scatterers. The appearance of the same exponent sigma = 3/7 at and near a critical point is discussed.