Abstract
We study a one-dimensional stochastic Lorentz gas where a light particle moves in a fixed array of nonidentical random scatterers arranged in a lattice. Each scatterer is characterized by a random transmission/reflection coefficient. We consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables. A symbolic program is presented which generates the exact velocity autocorrelation function (VACF) in terms of the moments of the transmission coefficients. The VACF is found for different types of disorder for times up to 20 collision times. We then consider a specific type of disorder: a two-state Lorentz gas in which two types of scatterers are arranged randomly in a lattice. Then a lattice point is occupied by a scatterer whose transmission coefficient is η with probability p or η+ε with probability 1−p. A perturbation expansion with respect to ε is derived. The ε2 term in this expansion shows that the VACF oscillates with time, the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor. The coarse-grained VACF decays for long times like t −3/2, which is similar to the decay of the VACF of the random Lorentz gas with a single type of scatterer. The perturbation results and the exact ones (found up to 20 collision times) show good agreement.
Similar content being viewed by others
REFERENCES
M. H. Ernst and A. Weyland, Phys. Lett. 34A:39 (1971).
L. A. Bunimovich and Ya. G. Sinai, Commun. Math. Phys. 78:479 (1981).
H. van Beijeren, Rev. Mod. Phys. 54:195 (1982).
J. Machta and R. Zwanzig, Phys. Rev. Let. 50:1959 (1983).
J. P. Bouchaud and P. Le Doussal, J. of Stat. Phys. 41:225 (1985).
A. Zacharel, T. Geisel, J. Nierwetberg, and G. Radons, Phys. Let. A. 114:315 (1986).
P. Gaspard and G. Nicolis, Phys. Rev. Let. 65:1693 (1990).
P. M. Bleher, J. of Stat. Phys. 66:315 (1992).
H. van Beijeren and J. R. Dorfman, Phys. Rev. Let. 74:4412 (1995).
Matsuoka and R. F. Martin, J. of Stat. Phys. 88:81 (1997).
P. Levitz, Europhys. Lett. 39:593 (1997).
E. Barkai, V. Fleurov, and J. Klafter (1998), submitted.
P. Grassberger, Physica A 103:558 (1980).
H. van Beijeren and H. Spohn, J. Stat. Phys. 31:231 (1983).
C. Olesky, J. Phys. A. Math. Gen. 23:1275 (1990).
P. M. Binder and D. Frenkel, Phys. Rev. A. 42:2463 (1990).
C. B. Briozzo, C. E. Budde, and M. O. Caceres, Physica A. 160:225 (1989).
J. M. F. Gunn and M. Ortuño, J. Phys. A. Math. 18:1095 (1985).
M. H. Ernst and G. A. van Velzen, J. Stat. Phys. 57:455 (1989).
H. van Beijeren and M. H. Ernst, J. Stat. Phys. 70:793 (1993).
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).
M. H. Ernst, J. R. Dorfman, R. Nix, and D. Jacobs, Phys. Rev. Let. 74:4416 (1995).
C. Apert, C. Bokel, J. R. Dorfman, and M. H. Ernst, Physica D 103:357 (1997).
S. Wolfram, Mathematica A System for Doing Mathematics by Computer (Addison–Wesley Publishing Company, Inc., New York, Amsterdam, Tokyo 1988).
J. W. Haus and K. W. Kehr, Physics Report 150:263 (1987).
G. Doetch, Guide to the Applications of the Laplace and Z transforms (Van Nostrand Reinhold Company, London, 1971).
R. Zwanzig, J. Stat. Phys. 28:127 (1982).
P. J. H. Denteneer and M. H. Ernst, Phys. Rev. B 29:1755 (1984).
H. Scher and M. Lax, Phys. Rev. B 7:4491 (1973).
H. Scher and M. Lax, Phys. Rev. B 7:4502 (1973).
H. Scher and E. W. Montroll, Phys. Rev. B 12:2455 (1975).
G. H. Weiss, Aspects and Applications of the Random Walk (North Holland, Amsterdam, 1994).
http://www.tau.ac.il:81/cc/
J. W. Haus and K. W. Kehr, J. Phys. Chem. Solids 40:1019 (1979).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barkai, E., Fleurov, V. Stochastic One-Dimensional Lorentz Gas on a Lattice. Journal of Statistical Physics 96, 325–359 (1999). https://doi.org/10.1023/A:1004532702233
Issue Date:
DOI: https://doi.org/10.1023/A:1004532702233