Abstract
This paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics, in particular to a study of nonequilibrium phenomena in Lorentz lattice gases with stochastic collision rules. Using simple arguments, based upon discussions in the mathematical literature, we show that such lattice gases belong to the category of dynamical systems with positive Lyapunov exponents. This is accomplished by showing how such systems can be expressed in terms of continuous phase space variables. Expressions for the Lyapunov exponent of a one-dimensional Lorentz lattice gas with periodic boundaries are derived. Other quantities of interest for the theory of irreversible processes are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. A. Posch and W. G. Hoover,Phys. Lett. A 123:227 (1987);Phys. Rev. A 39:2175 (1989).
D. J. Evans, E. G. D. Cohen and G. P. Morriss,Phys. Rev. A 42:5990 (1990); see also D. J. Evans and G. P. Morriss,Statistical Mechanics of Nonequilibrium Liquids (Academic Press, London, 1990).
N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai,Phys. Rev. Lett. 70: 2209 (1993);Commun Math. Phys. 154:569 (1993).
P. Gaspard and G. Nicolis,Phys. Rev. Lett. 65:1693 (1990).
P. Gaspard,J. Stat. Phys. 68:673 (1992); P. Gaspard and F. Baras, inMicroscopic Simulation of Complex Hydrodynamic Phenomena, M. Maréchal and B. L. Holian, eds. (Plenum Press, New York, (1992), p. 301; P. Gaspard and S. A. Rice,J. Chem. Phys. 90:2225 (1989);91:3279 (1989).
J. R. Dorfman and P. Gaspard,Phys. Rev. E 51:28 (1995).
G. A. van Velzen, Lorentz lattice gases, Ph.D. dissertation, University of Utrecht (1990); M. H. Ernst, InOrdering Phenomena in Condensed Matter Physics, Z. M. Galasiewicz and A. Pekalski, eds. (World Scientific, Singapore, p. 291); M. H. Ernst and G. A. van Velzen,J. Phys. A 22:4337 (1989);J. Stat. Phys. 71:1015 (1993); H. van Beijeren and M. H. Ernst,J. Stat. Phys. 70:793 (1993).
M. H. Ernst, J. R. Dorfman, R. Nix, and D. Jacobs,Phys. Rev. Lett. 74:4416 (1995).
P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).
P. Walters,An Introduction to Ergodic Theory (Springer-Verlag, New York, 1982).
K. Peterson,Ergodic Theory (Cambridge University Press, Cambridge, 1983).
P. Gaspard and X. J. Wang,Phys. Rep. 235:291 (1993); P. Gaspard and J. R. Dorfman,Phys. Rev. E (to appear).
F. Bagnoli, R. Rechtman, and S. Ruffo,Phys. Lett. 172A:34 (1992); F. Bagnoli,Int. J. Mod. Phys. 3c:307 (1992); M. A. Shereshevsky,J. Nonlinear Sci. 2:1 (1992).
J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985); E. Ott,Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
S. Goldstein, O. E. Lanford III, and J. L. Lebowitz,J. Math Phys. 14:1228 (1973).
H. van Beijeren and J. R. Dorfman,Phys. Rev. Lett. 74:4412 (1995).
C. Beck and F. Schlögl,Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dorfman, J.R., Ernst, M.H. & Jacobs, D. Dynamical chaos in the Lorentz lattice gas. J Stat Phys 81, 497–513 (1995). https://doi.org/10.1007/BF02179990
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179990