Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas

Abstract

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 $A_0\tilde{n}\ln \tilde{n}+B_0\tilde{n}$, where $A_0$ and $B_0$ are known constants and $\tilde{n}$ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order $({\tilde{n}}^2)$, the positive Lyapunov exponent is of the form $A_0\tilde{n}\ln \tilde{n}+B_0\tilde{n}+A_1{\tilde{n}}^2\ln \tilde{n}+B_1{\tilde{n}}^2$. 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 $O({\tilde{n}}^2)$, the effects of {\it all\/} possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.

12 pages, 9 figures, minor changes in this version, to appear in J. Stat. Phys