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

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.