Lyapunov instability in a system of hard disks in equilibrium and nonequilibrium steady states

We present a generalization of Benettin’s classical algorithm for the calculation of full Lyapunov spectra to the case of dynamical systems where the smooth streaming is interrupted by a differentiable map at discrete times. With this formalism we derive the transformation rules for the offset vectors in tangent space for a system of hard particles in equilibrium and nonequilibrium steady states. In particular, we study the color conductivity of a system of hard disks carrying color charges subjected to an external color field. Full Lyapunov spectra are obtained numerically for equilibrium systems of 64 and 144 hard disks. Furthermore, the maximum Lyapunov exponent and the Kolmogorov-Sinai entropy are studied over a wide range of densities. Both mimic the collision rate very well. In the low density regime the maximum Lyapunov exponent is found to follow the relation ${\lambda }_1$∝-ρ lnρ, as conjectured by Krylov. Full Lyapunov spectra are also reported for nonequilibrium steady-state systems of 64 hard disks, which carry color charges and are externally perturbed by an applied color field. The simulations cover a wide range of densities and fields. From a careful study of small three- and four-particle systems the validity of the conjugate pairing rule is established numerically with an error less than 0.1%. Also the number of vanishing Lyapunov exponents due to the conserved quantities—center of mass, linear momentum, and kinetic energy—is discussed in some detail. © 1996 The American Physical Society.