Chaotic properties of dilute two- and three dimensional random Lorentz gases: Equilibrium systems

Abstract

We compute the Lyapunov spectrum and the Kolmogorov-Sinai entropy for a moving particle placed in a dilute, random array of hard-disk or hard-sphere scatterers, i.e., the dilute Lorentz gas model. This is carried out in two ways. First we use simple kinetic theory arguments to compute the Lyapunov spectrum for both two- and three-dimensional systems. In order to provide a method that can easily be generalized to nonuniform systems we then use a method based upon extensions of the Lorentz-Boltzmann (LB) equation to include variables that characterize the chaotic behavior of the system. The extended LB equations depend upon the number of dimensions and on whether one is computing positive or negative Lyapunov exponents. In the latter case the extended LB equation is closely related to an “anti-Lorentz-Boltzmann equation” where the collision operator has the opposite sign to the ordinary LB equation. Finally, we compare our results with computer simulations of Dellago and Posch [Phys. Rev. E 52, 2401 (1995); Phys. Rev. Lett. 78, 211 (1997)] and find very good agreement.