The nonequilibrium Lorentz gas

We study the conductivity of a Lorentz gas system, composed of a regular array of fixed scatterers and a point-like moving particle, as a function of the strength of an applied external field. In order to obtain a nonequilibrium stationary state, the speed of the point particle is fixed by the action of a Gaussian thermostat. For small fields the system is ergodic and the diffusion coefficient is well defined. We show that in this range the Periodic Orbit Expansion can be successfully applied to compute the values of the thermodynamic variables. At larger values of the field we observe a variety of possible dynamics, including the breakdown of ergodic behavior, and later the existence of a single stable trajectory for the largest fields. We also study the behavior of the system as a function of the orientation of the array of scatterers with respect to the external field. Finally, we present a detailed dynamical study of the transitions in the bifurcation sequence in both the elementary cell and the fundamental domain. The consequences of this behavior for the ergodicity of the system are explored. (c) 1995 American Institute of Physics.     

Diffusion and propagation in triangular Lorentz lattice gas cellular automata

The diffusion process of point particles moving on regular triangular and random lattices, randomly occupied with stationary scatterers (a Lorentz lattice gas cellular automaton), is studied, for strictly deterministic scattering rules, as a function of the concentration of the scatterers. In addition to the normal and various kinds of retarded diffusion found before on the regular square lattice, straight-line propagation through the scatterers is observed.     

Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of these projections are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted.     

The Nosé–Hoover Thermostated Lorentz Gas

We apply the Nosé–Hoover thermostat and three variations of it, which control different combinations of velocity moments, to the periodic Lorentz gas. Switching on an external electric field leads to nonequilibrium steady states for the four models. By performing computer simulations we study the probability density, the conductivity and the attractor in nonequilibrium. The results are compared to the Gaussian thermostated Lorentz gas and to the Lorentz gas as thermostated by deterministic scattering. We find that slight modifications of the Nosé–Hoover thermostat lead to different dynamical properties of our models. However, in all cases the attractor appears to be multifractal.     

Steady-state electrical conduction in the periodic Lorentz gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a thermostat constructed according to Gauss principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.Dedicated to Elliott Lieb     

Dynamical Conductivity of the Dilute Lorentz Gas with Spherically Symmetric Scatterers

The dynamical conductivity of the Lorentz gas with spherically symmetric potentials is studied to lowest order in the density of scatterers. The frequency-dependent friction coefficient is calculated from the Fourier transform of the force–force time-correlation function determined by the dynamics of a single scattering process. The corresponding dynamical conductivity varies with frequency on the scale of the inverse collision time. As an example, the conductivity is calculated for a scattering potential of the Maxwell type.     

Anomalous transport in heterogeneous media

The diffusion dynamics of particles in heterogeneous media is studied using particle-based simulation techniques. A special focus is placed on systems where the transport of particles at long times exhibits anomalies such as subdiffusive or superdiffusive behavior. First, a two-dimensional model system is considered containing gas particles (tracers) that diffuse through a random arrangement of pinned, disk-shaped particles. This system is similar to a classical Lorentz gas. However, different from the original Lorentz model, soft instead of hard interactions are considered and we also discuss the case where the tracer particles interact with each other. We show that the modification from hard to soft interactions strongly affects anomalous-diffusive transport at high obstacle densities. Second, non-linear active micro-rheology in a glass-forming binary Yukawa mixture is investigated, pulling single particles through a deeply supercooled state by applying a constant force. Here, we observe superdiffusion in force direction and analyze its origin. Finally, we consider the Brownian dynamics of a particle which is pulled through a two-dimensional random force field. We discuss the similarities of this model with the Lorentz gas as well as active micro-rheology in glass-forming systems.     

The diffusion coefficient of a model with multiple collisions

A model Lorentz gas, in which each scatterer may be struck more than once, is analyzed, and the diffusion coefficient obtained explicitly as a function of the density of the scatterers.