Systematic Density Expansion of the Lyapunov Exponents for a Two-Dimensional Random Lorentz Gas
- 39 Downloads
We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form A 0ñln ñ+B 0ñ, where A 0 and B 0 are known constants and ñ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order (ñ2), the positive Lyapunov exponent is of the form A 0ñln ñ+B 0ñ+A 1ñ2ln ñ +B 1ñ2. Explicit numerical values of the new constants A 1 and B 1 are obtained by means of a systematic analysis. This takes into account, up to O(ñ2), the effects of all possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.
KeywordsLyapunov exponent Lorentz gas density expansion
Unable to display preview. Download preview PDF.
- 2.D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London (1990).Google Scholar
- 3.J. R. Dorfman, An Introduction to Chaos in Non-Equilibrium Statistical Mechanics, Cambridge Univ. Press (1999).Google Scholar
- 4.W. G. Hoover, Computational Statistical Mechanics Elsevier Publ. Co., Amsterdam (1991).Google Scholar
- 6.N. S. Krylov, Works on the Foundations of Statistical Mechanics, Princeton University Press, Princeton (1979).Google Scholar
- 10.Since we use the full collision frequency in our equations this should perhaps rather be called a Lorentz-Enskog equation. Substituting νc = 2nav one indeed recovers the Lorentz Boltzmann equation of .Google Scholar
- 11.H. van Beijeren, A. Latz, and J. R. Dorfman, Phys. Rev. E 63: 016312 (2001).Google Scholar
- 12.H. V. Kruis, Masters thesis, University of Utrecht, The Netherlands (1997).Google Scholar
- 13.D. Panja, Ph.D. thesis, University of Maryland, College Park, USA (2000).Google Scholar
- 14.D. Panja (unpublished).Google Scholar
- 15.J. M. J. van Leeuwen and A. Weyland, Physica 36:457 (1967); J. M. J. van Leeuwen and A. Weyland, Physica 39:35 (1968).Google Scholar