Ch. Dellago

Kolmogorov-Sinai entropy and Lyapunov spectra of a hard-sphere gas

The mixing behavior of a hard-sphere gas has its origin in the exponential growth of small perturbations in phase space. This instability is characterized by the so-called Lyapunov exponents. In this work, we compute full spectra of Lyapunov exponents for the hard-sphere gas for a wide range of densities ϱ and particle numbers by using a recently developed algorithm. In the dilute-gas regime, the maximum Lyapunov exponent is found to obey the Krylov relation λ ∝ ϱ ln ϱ, a formula exactly derived for the low-density Lorentz gas by Dorfman and van Beijeren. We study the system-size dependence and the effect of the fluid-solid-phase transition on the spectra. In the second part of this work we describe and test a direct simulation Monte Carlo method (DSMC) for the computation of Lyapunov spectra and present results for dilute hard-sphere gases. Excellent agreement is obtained with the results of the deterministic simulations. This suggests that the Lyapunov instability of a hard sphere gas may be analyzed within the framework of kinetic theory.

Lyapunov instability, local curvature, and the fluid-solid phase transition in two-dimensional particle systems

We study the relation between the Lyapunov spectrum and the multidimensional geometry of the potential energy surface in terms of the distribution of stable and unstable modes for different models. For this purpose we determined Lyapunov exponents for the so-called correlated cell model and its smooth generalization as a function of the density for various energies. In the smooth case averaged structural quantities, such as the fraction of unstable modes, the Gaussian curvature, and the Riemannian curvature were computed and compared to the mechanical instability of the system in the sense of Lyapunov. A similar analysis was also carried out for 36-disk systems representing various fluid and solid states. In all studied cases the most-positive Lyapunov exponent exhibits a maximum at the phase transition in agreement with results for the fluid-solid transition in many-particle systems.

Lyapunov instability in the extended XY model: Equilibrium and nonequilibrium molecular dynamics simulations

The extended XY-model, a classical system of planar rotators with nearest neighbor interaction, is studied with molecular dynamics simulations. This model exhibits a first-order phase transition between a low-temperature phase with algebraic decay of spatial correlations and a high-temperature phase, where the correlations decay exponentially. The nature of this phase transition can be changed from continuous to discontinuous by changing a control parameter for the interaction potential. For both phases full spectra of Lyapunov exponents are computed, and their dependence on the energy of the system is investigated. The maximum Lyapunov exponent is determined over a wide range of temperatures, and is shown to exhibit a maximum at the transition temperature if the transition is discontinuous. Furthermore, the behavior of the system in nonequilibrium steady states is considered, and the coefficient for energy transport is calculated by means of equilibrium (Green-Kubo theory) and nonequilibrium molecular dynamics.

Field dependent collision frequency of the two-dimensional driven random Lorentz gas

In the field-driven, thermostated Lorentz gas the collision frequency increases with the magnitude of the applied field due to long-time correlations. We study this effect with computer simulations and confirm the presence of nonanalytic terms in the field dependence of the collision rate as predicted by kinetic theory.

Field and density dependence of the Lyapunov spectrum for the driven random Lorentz gas

Abstract We report on extensive numerical computations of full Lyapunov spectra and related Kaplan-Yorke dimensions for the two-and three-dimensional random Lorentz gases subjected to an external field. This model is one of the simplest nonequilibrium steady-state systems displaying macroscopic irreversibility in spite of its time-reversible microscopic dynamics, in accordance with the Second Law. Nonequilibrium steady states are maintained by Gaussian constraints, keeping the kinetic energy constant. We study the variation of the Ly apunov exponents with the applied field in a wide range of scatterer densities. For dilute systems and small fields our numerical results are in perfect agreement with recent analytical expressions based on Boltzmann equation methods.