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