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 τ=15/7 andd f=7/4 found before, the critical exponent σ=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) distributionn s, andh(x), related to the trajectory sizeR s (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 σ=0.43≈3/7 as at criticality, leading to a stretched exponential dependence ofn S onS, nS∼exp(−S 6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent σ′=1.6±0.3 and a superexponential dependence ofn S 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 σ=3/7 at and near a critical point is discussed.
Similar content being viewed by others
References
Th. W. Ruijgrok and E. G. D. Cohen,Phys. Lett. A 133:415 (1988).
X. P. Kong and E. G. D. Cohen,Phys. Rev. B 40:4838 (1989).
X. P. Kong and E. G. D. Cohen,J. Stat. Phys. 62:737 (1991).
R. M. Ziff, X. P. Kong, and E. G. D. Cohen,Phys. Rev. A 44:2410 (1991).
E. G. D. Cohen, New types of diffusion in lattice gas cellular automata, inMicroscopic Simulations of Complex Hydrodynamic Phenomena, M. Maréchal and B. L. Holian, eds. (Plenum Press, New York, 1992), p. 137.
H. F. Meng and E. G. D. Cohen,Phys. Rev. E 50:2482 (1994).
E. G. D. Cohen, and F. Wang,J. Stat. Phys. 81:445 (1995).
F. Wang and E. G. D. Cohen,J. Stat. Phys. 81:467 (1995).
E. G. D. Cohen and F. Wang,Physica A 219:56 (1995).
E. G. D. Cohen and F. Wang,Fields Inst. Commun. 6:43 (1996).
F. Wang and E. G. D. Cohen,J. Stat. Phys. 84:233 (1996).
D. Stauffer and A. Aharony,Introduction to Percolation Theory (Taylor and Francis, London, 1992).
R. M. Ziff,Phys. Rev. Lett. 56:545 (1986).
R. M. Ziff, P. T. Cumming, and G. Stell,J. Phys. A 17:3009 (1984).
M.-S. Cao and E. G. D. Cohen, cond-mat (a xxx.lanl.gov #9608159, #9608160 (1996).
P. Grassberger,J. Phys. A 19:2675 (1986).
R. M. Ziff, Private communication.
M. Ortuño, J. Ruiz, and M. F. Gunn,J. Stat. Phys. 65:453 (1991).
L. A. Bunimovich and S. E. Troubetzkoy,J. Stat. Phys. 67:289 (1992);74: 1 (1994).
R. M. Bradley,Phys. Rev. B 41:914, (1989).
A. L. Owczarek and T. Prellberg,J. Stat. Phys. 79:951 (1995).
J. D. Catalá, J. Ruiz, and M. Ortuño,J. Phys. B 90:369 (1993).
P. L. Leath,Phys. Rev. Lett. 36:921 (1976);Phys. Rev. B 14:5046 (1976).
A. M. Ferrenberg and R. H. Swendsen,Phys. Rev. Lett. 61:2635 (1988).
J. P. Hansen and I. R. McDonald,Theory of Simple Liquids (Academic Press, London, 1986), p. 106.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cao, MS., Cohen, E.G.D. Scaling of particle trajectories on a lattice. J Stat Phys 87, 147–178 (1997). https://doi.org/10.1007/BF02181484
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02181484