Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

We compute the Lyapunov exponent, the generalized Lyapunov exponents, and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, are used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. The Lyapunov exponent for the corresponding map agrees with the conjectured limit _map = -2 log(R) + C + O(R) and we derive an approximate value for the constantC in good agreement with numerical simulations. We also find a diverging diffusion constantD(t)logt and a phase transition for the generalized Lyapunov exponents.     

Dynamical chaos in the Lorentz lattice gas

This paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics, in particular to a study of nonequilibrium phenomena in Lorentz lattice gases with stochastic collision rules. Using simple arguments, based upon discussions in the mathematical literature, we show that such lattice gases belong to the category of dynamical systems with positive Lyapunov exponents. This is accomplished by showing how such systems can be expressed in terms of continuous phase space variables. Expressions for the Lyapunov exponent of a one-dimensional Lorentz lattice gas with periodic boundaries are derived. Other quantities of interest for the theory of irreversible processes are discussed.     

An approximate formula for the diffusion coefficient for the periodic Lorentz gas

An approximate stochastic model for the topological dynamics of the periodic triangular Lorentz gas is constructed. The model, together with an extremum principle, is used to find a closed form approximation to the diffusion coefficient as a function of the lattice spacing. This approximation is superior to the popular Machta and Zwanzig result and agrees well with a range of numerical estimates.     

Example of diffusion in a disordered Lorentz gas

We prove a diffusion law for a disordered Lorentz gas obtained by modification of a model of Gates, Gerst, Kac in Ref. 1, even though the motion is not a Markovian one in the technical sense of the word.     

New results for diffusion in Lorentz lattice gas cellular automata

New calculations to over ten million time steps have revealed a more complex diffusive behavior than previously reported of a point particle on a square and triangular lattice randomly occupied by mirror or rotator scatterers. For the square lattice fully occupied by mirrors where extended closed particle orbits occur, anomalous diffusion was still found. However, for a not fully occupied lattice the superdiffusion, first noticed by Owczarek and Prellberg for a particular concentration, obtains for all concentrations. For the square lattice occupied by rotators and the triangular lattice occupied by mirrors or rotators, an absence of diffusion (trapping) was found for all concentrations, except on critical lines, where anomalous diffusion (extended closed orbits) occurs and hyperscaling holds for all closed orbits withuniversal exponentsd _f =7/4 and =15/7. Only one point on these critical lines can be related to a corresponding percolation problem. The questions arise therefore whether the other critical points can be mapped onto a new percolation-like problem and of the dynamical significance of hyperscaling.     

Some Estimates for 2-Dimensional Infinite and Bounded Dilute Random Lorentz Gases

We present a (mostly) rigorous approach to unbounded and bounded (open) dilute random Lorentz gases. Relying on previous rigorous results on the dilute (Boltzmann–Grad) limit we compute the asymptotics of the Lyapunov exponent in the unbounded case. For the bounded open case in a circular region we give here an incomplete rigorous analysis which gives the asymptotics for large radius of the escape rate and of the rescaled quasi-invariant (q.i., or quasi-stationary) measure. We finally give a complete proof on existence and asymptotic properties of the q.i. measure in a one-dimensional caricature.     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

Lorentz lattice gases,abnormal diffusion,and polymer statistics

Diffusive behavior in various Lorentz lattice gases, especially wind-tree-like models, is discussed. Comparisons between lattice and continuum models as well as deterministic and probabilistic models are made. In one deterministic model, where the scatterers behave like double-sided mirrors, a new kind of abnormal diffusion is found, viz., the mean square displacement is proportional to the time, but the probability density distribution function is non-Gaussian. The connections of this mirror model with the percolation problem and the statistics of polymer chains on a lattice are also discussed.     

The runaway effect in a Lorentz gas

The Lorentz gas of charged particles in a constant and uniform electric field is studied. The gas flows through the medium of immobile, randomly distributed scatterers. Particles with velocity v suffer collisions with frequency proportional to ¦v¦ ⁿ . Forn < 0 runaway of the gas is forced by the field: the mean velocity of the flow increases without bounds. By a simple physical argument an integral relation is established between the probability of collisionless motion and the velocity distribution. It is then shown that whenn < –1 a fraction of particles moves as if the scattering centers were absent. The detailed discussion of this uncollided runaway is presented. Some qualitative features of the velocity distribution are illustrated on rigorous solutions in one dimension.     

Systematic Density Expansion of the Lyapunov Exponents for a Two-Dimensional Random Lorentz Gas

We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form A ₀ñln ñ+B ₀ñ, where A ₀ and B ₀ are known constants and ñ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order (ñ²), the positive Lyapunov exponent is of the form A ₀ñln ñ+B ₀ñ+A ₁ñ²ln ñ +B ₁ñ². Explicit numerical values of the new constants A ₁ and B ₁ are obtained by means of a systematic analysis. This takes into account, up to O(ñ²), the effects of all possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.     

Diffusion in a periodic Lorentz gas

We use a constant driving forceF _d together with a Gaussian thermostatting constraint forceF _d to simulate a nonequilibrium steady-state current (particle velocity) in a periodic, two-dimensional, classical Lorentz gas. The ratio of the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. A regular Galton-board lattice of fixed particles is arranged in a dense triangular-lattice structure. The moving scatterer particle travels through the lattice at constant kinetic energy, making elastic hard-disk collisions with the fixed particles. At low field strengths the nonequilibrium conductivity is statistically indistinguishable from the equilibrium Green-Kubo estimate of Machta and Zwanzig. The low-field conductivity varies smoothly, but in a complicated way, with field strength. For moderate fields the conductivity generally decreases nearly linearly with field, but is nearly discontinuous at certain values where interesting stable cycles of collisions occur. As the field is increased, the phase-space probability density drops in apparent fractal dimensionality from 3 to 1. We compare the nonlinear conductivity with similar zero-density results from the two-particle Boltzmann equation. We also tabulate the variation of the kinetic pressure as a function of the field strength,     

Long-Time-Tail Effects on Lyapunov Exponents of a Random, Two-Dimensional Field-Driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, nonoverlapping hard-disk scatterers in a thermostatted electric field, . The low-density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz–Boltzmann equation. In this paper we develop a method to extend theses results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of the Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to nonanalytic, field-dependent contributions to both the positive and negative Lyapunov exponents which are of the form ~ ²ln~, where ~ is a dimensionless parameter proportional to the strength of the applied field. We show that these nonanalytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value due to the presence of the thermostatted field, and that the collision frequency also contains such nonanalytic terms.     

Numerical study of aD-dimensional periodic Lorentz gas with universal properties

We give the results of a numerical study of the motion of a point particle in ad-dimensional array of spherical scatterers (Sinai's billiard without horizon). We find a simple universal law for the Lyapounov exponent (as a function ofd) and a stretched exponential decay for the velocity autocorrelation as a function of the number of collisions. The diffusion seems to be anomalous in this problem. Ergodicity is used to predict the shape of the probability distribution of long free paths. Physical interpretations or clues are proposed.     

Long-time behavior of the Lorentz electron gas in a constant,uniform electric field

The long-time behavior of the Lorentz electron gas is studied in the presence of a uniform external field. A discussion of the rigorous solution of the one-dimensional Boltzmann equation is followed by the derivation of the asymptotic form of the velocity distribution in an arbitrary number of dimensions. The system is shown to absorb energy from the field without bounds, which excludes the usually assumed steady state with finite thermal energy density.Supported by the Polish Ministry of Higher Education, Science and Technology, Project MR.I.7.The authors are very grateful to Dr. Y. Pomeau for many valuable comments.     

Decay of correlations in the regular Lorentz gas

The regular Lorentz gas on triangular lattice is studied numerically and analytically. The velocity correlation function is shown to decay exponentially in the number of collisions with a decay rate which vanishes as the scatterers approach close packing. The crossover to power law decay at close packing is described by a scaling function.     

The computational complexity of the Lorentz lattice gas

The Lorentz lattice gas is studied from the perspective of computational complexity theory. It is shown that using massive parallelism, particle trajectories can be simulated in a time that scales logarithmically in the length of the trajectory. This result characterizes the logical depth of the Lorentz lattice gas and allows us to compare it to other models in statistical physics.     

Long-time tails in lattice Lorentz gases

We consider a Lorentz gas on a square lattice with a fraction c of scattering sites. The collision laws are deterministic (fixed mirror model) or stochastic (with transmission, reflection, and deflection probabilities ,, and respectively). If all mirrors are parallel, the mirror model is exactly solvable. For the general case a self-consistent ring kinetic equation is used to calculate the longtime tails of the velocity correlation function (0) (t) and the tensor correlation Q(0)Q(t) withQ= _{x y} . Both functions showt ^–2 tails, as opposed to the continuous Lorentz gas, where the tails are respectivelyt ^–2 andt ^–3. Inclusion of the self-consistent ring collisions increases the low-density coefficient of the tail in (0)(t) by 30–100% as compared to the simple ring collisions, depending on the model parameters.     

Recurrence properties of Lorentz lattice gas cellular automata

Recurrence properties of a point particle moving on a regular lattice randomly occupied with scatterers are studied for strictly deterministic, nondeterministic, and purely random scattering rules.On leave from Institute of Oceanology, USSR Academy of Sciences, 117218 Moscow, USSR     

Long-time tails of the velocity autocorrelation functions for the triangular periodic Lorentz gas

We present numrical results on the velocity autocorrelation function (VACF)C(t)=<ν(t)·ν(0)> for the periodic Lorentz gas on a two-dimensional triangular lattice as a function of the radiusR of the hard disk scatterers on the lattice. Our results for the unbounded horizon case confirm 1/t decay of the VACF for long times (out to 100 times the mean free time between collisions) and provide strong support for the conjecture by Friedman and Martin that the 1/t decay is due to long free paths along which a moving particle does not scatter up to timet. Even after new sets of long free paths become available forR<1/4, we continue to find good agreement between numerical results and an analytically estimated 1/t decay. For the bounded horizon case , our numerical VACFs decay exponentially, although it is difficult to discriminate among pure exponential decay, exponential decay with prefactor, and stretched exponential decay.     

Lorentz lattice gases: Basic theory

We present several ballistic models of the Lorentz gas in two-dimensional lattices with deterministic and stochastic deflection rules, and their corresponding Liouville equations. Boltzmann-level-equation results are obtained for the diffusion coefficient and velocity autocorrelation function for models with stochastic deflection rules. The long-time behavior of the mean square displacement is briefly discussed and the possibility of abnormal diffusion indicated. Even if the diffusion coefficient exists, its low-density limit may not be given correctly by the Boltzmann equation.