CHAOTIC PROPERTIES OF DILUTE 2-DIMENSIONAL AND 3-DIMENSIONAL RANDOM LORENTZ GASES - EQUILIBRIUM SYSTEMS

We compute the Lyapunov spectrum and the Kolmogorov-Sinai entropy for a moving particle placed in a dilute, random array of hard-disk or har d-sphere scatterers, i.e., the dilute Lorentz gas model. This is carri ed out in two ways. First we use simple kinetic theory arguments to co mpute the Lyapunov spectrum for both two-and three-dimensional systems . In order to provide a method that can easily be generalized to nonun iform systems we then use a method based upon extensions of the Lorent z-Boltzmann (LB) equation to include variables that characterize the c haotic behavior of the system. The extended LB equations depend upon t he number of dimensions and on whether one is computing positive or ne gative Lyapunov exponents. In the latter case the extended LB equation is closely related to an ''anti-Lorentz-Boltzmann equation'' where th e 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 (19 97)] and find very good agreement.