Identifying rare chaotic and regular trajectories in dynamical systems with Lyapunov weighted path sampling

Depending on initial conditions, individual finite time trajectories of dynamical systems can have very different chaotic properties. Here we present a numerical method to identify trajectories with atypical chaoticity, pathways that are either more regular or more chaotic than average. The method is based on the definition of an ensemble of trajectories weighted according to their chaoticity, the Lyapunov weighted path ensemble. This ensemble of trajectories is sampled using algorithms borrowed from transition path sampling, a method originally developed to study rare transitions between long-lived states. We demonstrate our approach by applying it to several systems with numbers of degrees of freedom ranging from one to several hundred and in all cases the algorithm found rare pathways with atypical chaoticity. For a double-well dimer embedded in a solvent, which can be viewed as simple model for an isomerizing molecule, rare reactive pathways were found for parameters strongly favoring chaotic dynamics.Comment: 8 pages, 5 figure

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

In the field-driven, thermostatted 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 non-analytic terms in the field dependence of the collision frequency as predicted by kinetic theory.Comment: 6 pages, 2 figures. Submitted to Phys. Rev.

Density-dependent diffusion in the periodic Lorentz gas

We study the deterministic diffusion coefficient of the two-dimensional periodic Lorentz gas as a function of the density of scatterers. Results obtained from computer simulations are compared to the analytical approximation of Machta and Zwanzig [Phys.Rev.Lett. 50, 1959 (1983)] showing that their argument is only correct in the limit of high densities. We discuss how the Machta-Zwanzig argument, which is based on treating diffusion as a Markovian hopping process on a lattice, can be corrected systematically by including microscopic correlations. We furthermore show that, on a fine scale, the diffusion coefficient is a non-trivial function of the density. We finally argue that, on a coarse scale and for lower densities, the diffusion coefficient exhibits a Boltzmann-like behavior, whereas for very high densities it crosses over to a regime which can be understood qualitatively by the Machta-Zwanzig approximation.Comment: 9 pages (revtex) with 9 figures (postscript

The Kolmogorov-Sinai Entropy for Dilute Gases in Equilibrium

We use the kinetic theory of gases to compute the Kolmogorov-Sinai entropy per particle for a dilute gas in equilibrium. For an equilibrium system, the KS entropy, h_KS is the sum of all of the positive Lyapunov exponents characterizing the chaotic behavior of the gas. We compute h_KS/N, where N is the number of particles in the gas. This quantity has a density expansion of the form h_KS/N = a\nu[-\ln{\tilde{n}} + b + O(\tilde{n})], where \nu is the single-particle collision frequency and \tilde{n} is the reduced number density of the gas. The theoretical values for the coefficients a and b are compared with the results of computer simulations, with excellent agreement for a, and less than satisfactory agreement for b. Possible reasons for this difference in b are discussed.Comment: 15 pages, 2 figures, submitted to Phys. Rev.