Scaling of particle trajectories on a lattice |
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Authors: | Meng-She Cao E. G. D. Cohen |
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Affiliation: | (1) The Rockefeller University, 10021 New York, New York |
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Abstract: | The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 anddf=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionns, andh(x), related to the trajectory sizeRs (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.43 3/7 as at criticality, leading to a stretched exponential dependence ofnS onS, nS exp(–S6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent ![sgr](/content/f3t4565455023668/xxlarge963.gif) =1.6±0.3 and a superexponential dependence ofnS onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed. |
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Keywords: | Lattice particle trajectories percolation scaling function criticality |
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