Abstract
The probability of first return to the initial intervalx and the diffusion tensorD xβ are calculated exactly for a ballistic Lorentz gas on a Bethe lattice or Cayley tree. It consists of a moving particle and a fixed array of scatterers, located at the nodes, and the lengths of the intervals between scatterers are determined by a geometric distribution. The same values forx andD xβ apply also to a regular space lattice with a fraction ρ of sites occupied by a scatterer in the limit of a small concentration of scatterers. If backscattering occurs, the results are very different from the Boltzmann approximation. The theory is applied to different types of lattices and different types of scatterers having rotational or mirror symmetries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. A. Lorentz,Proc. R. Acad. Amsterdam 7:438, 585, 684 (1905).
P. Ehrenfest and T. Ehrenfest,The Conceptual Foundations to the Statistical Approach in Mechanics, (Cornell University Press, Ithaca, New York, 1959).
B. J. Alder and W. E. Alley,J. Stat. Phys. 19:341 (1978).
J. C. Lewis and J. A. Tjon,Phys. Lett. A 66:349 (1978); C. Bruin,Physica 72:261 (1974).
D. Frenkel,Phys. Lett. 121A:385 (1987).
J. J. Brey, J. Gomez Ordoñez, and A. Santos,Phys. Lett. 127A:5 (1988).
P. M. Binder and M. H. Ernst,Physica A 164:91 (1990).
M. H. Ernst, G. A. van Velzen, and P. M. Binder,Phys. Rev. A 39:4327 (1989).
M. H. Ernst and G. A. van Velzen,J. Phys. A: Math. Gen. 22:4611 (1989).
Th. W. Ruijgrok and E. G. D. Cohen,Phys. Lett. 133A:415 (1988).
X. P. Kong and E. G. D. Cohen,Phys. Rev. B 540:4838 (1989).
G. A. van Velzen,J. Phys. A: Math. Gen. 23:4953 (1990).
G. A. van Velzen, Lattice Lorentz gases, Ph.D. dissertation, University of Utrecht (1990);J. Phys. A: Math. Gen. 24:787, 807 (1991).
M. H. Ernst and G. A. van Velzen,J. Stat. Phys. 57:455 (1989).
P. M. Binder and D. Frenkel,Phys. Rev. A 42:2463 (1990).
M. H. Ernst, inOrdering Phenomena in Condensed Matter Physics, Z. M. Galasiewicz and A. Pekalski, eds. (World Scientific, Singapore, 1991), p. 291.
X. P. Kong and E. G. D. Cohen,Physica D 47:9 (1991).
E. G. D. Cohen, inMicroscopic Simulations of Complex Hydrodynamic Phenomena (Proceedings Nato Summer School, Alghero, July 1991), M. Maréschal, ed. (World Scientific, Singapore, 1992).
R. M. Ziff, X. P. Kong, and E. G. D. Cohen,Phys. Rev. A 44:2410 (1991).
L. A. Bunimovich and S. E. Troubetzkoy, Preprint, University of Bielefeld (1991).
J. M. F. Gunn and M. Ortuño,J. Phys. A: Math. Gen. 18:L1035 (1985).
G. A. van Velzen,J. Phys. A: Math. Gen. 24:789, 807 (1991).
D. Frenkel, F. van Luyn, and P.M. Binder,Europhys. Lett. (1992).
A. J. H. Ossendrijver, Diffusion in lattice Lorentz gases, M.A. thesis, University of Utrecht (1991); A. J. H. Ossendrijver, A. Santos, and M. H. Ernst, University of Utrecht (1992).
D. W. Jepsen,J. Math. Phys. 6:405 (1965).
H. van Beijeren,Rev. Mod. Phys. 54:195 (1982).
P. Grassberger,Physica A 103:558 (1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Beijeren, H., Ernst, M.H. Diffusion in Lorentz lattice gas automata with backscattering. J Stat Phys 70, 793–809 (1993). https://doi.org/10.1007/BF01053595
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01053595