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 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.     

Statistical properties of the periodic Lorentz gas. Multidimensional case

In 1981 Bunimovich and Sinai established the statistical properties of the planar periodic Lorentz gas with finite horizon. Our aim is to extend their theory to the multidimensional Lorentz gas. In that case the Markov partitions of the Bunimovich-Sinai type, the main tool of their theory, are not available. We use a crude approximation to such partitions, which we call Markov sieves. Their construction in many dimensions is essentially different from that in two dimensions; it requires more routine calculations and intricate arguments. We try to avoid technical details and outline the construction of the Markov sieves in mostly qualitative, heuristic terms, hoping to carry out our plan in full detail elsewhere. Modulo that construction, our proofs are conclusive. In the end, we obtain a stretched-exponential bound for the decay of correlations, the central limit theorem, and Donsker's Invariance Principle for multidimensional periodic Lorentz gases with finite horizon.     

Molecular dynamical simulation of the canonical ensemble

We apply a technique to simulate the canonical ensemble, mixing molecular dynamics and Monte Carlo techniques, in which particles suffer virtual hard shocks. In the limit of infinite time the system approaches a Boltzmann distribution. A good approximation to the Boltzmann distribution is achieved in computationally accessible time for some model systems including the one-dimensional jellium.     

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.     

Topological dynamics of flipping Lorentz lattice gas models

We study the topological dynamics of the flipping mirror model of Ruijgrok and Cohen with one or an infinite number of particles. In particular we prove the topological transitivity and topological mixing up to a natural first integral for the one-particle model.     

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.     

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.     

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.     

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.     

On localization of vorticity in Lorentz lattice gases

We study the generalized deterministic Lorentz lattice gases, in a fixed as well as in varying environments, in lattices with dimensionsd3. We show that bounded orbits (vortices) in these models are often contained in some lower dimensional subsets (vortex sheets) of these lattices.     

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.     

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. Based on computer simulations, and by applying straightforward analytical arguments, we systematically improve the Machta–Zwanzig random walk approximation [Phys. Rev. Lett. 50:1959 (1983)] by including microscopic correlations. We furthermore, show that, on a fine scale, the diffusion coefficient is a non-trivial function of the density. 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.     

Deviations from Fick's law in Lorentz gases

We have calculated the self-dynamic structure factorF(k,t) for tagged particle motion in hopping Lorentz gases. We find evidence that, even at long times, the probability distribution function for the displacement of the particles is highly non-Gaussian. At very small values of the wave vector this manifests itself as the divergence of the Burnett coefficient (the fourth moment of the distribution never approaching a value characteristic of a Gaussian). At somewhat larger wave vectors we find thatF(k,t) decays algebraically, rather than exponentially as one would expect for a Gaussian. The precise form of this power-law decay depends on the nature of the scatterers making up the Lorentz gas. We find different power-law exponents for scatterers which exclude certain sites and scatterers which do not.     

Correlation dimension of the sheared hard-disk Lorentz gas

The correlation dimension for the isokinetic Lorentz gas is calculated for hard disks using nonequilibrium molecular dynamics simulation. The trajectories are confined to a strange attractor embedded in a four-dimensional phase space—the additional degree of freedom having not been included properly until this work. This degree of freedom accounts for the explicit time dependence of the system (as quantified by the moving periodic cells under shear) and is significant because the collisions tend to synchronize with the periodic change of symmetry of the lattice at high shear rates.     

Exact solutions to the time-dependent Lorentz gas Boltzmann Equation: The approach to hydrodynamics

New exact solutions to the time-dependent Lorentz gas Boltzmann equation are presented for two classes of nonequilibrium initial value problems: thedecay of localized disturbances and theresponse to applied electric fields. These exact results are used to gain some insight into the crossover of the nonequilibrium state from the early-timekinetic regime to the late-timehydrodynamic regime.     

Atomistic simulation of dynamical and defect properties of multiferroic hexagonal YMnO_3

Atomistic simulation has been performed to investigate the dynamical and defect properties of multiferroic hexagonal YMnO3 with newly developed interaction potentials. Dynamical calculation reveals that phonon vibrations of hexagonal YMnO3 are quite different from those of orthorhombic YMnO3. Defect calculation finds that O Frenkel is the most probable intrinsic disorder, and Mn antisite defect is favorable to exist, especially for Mn ions entering the Y2 sites. It is also found that holes prefer to localiz...     

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.     

Molecular dynamics simulation of gas adsorption on defected graphene

Gas sensing is one of the most promising applications for graphene. Using molecular dynamics simulation method, adsorption isotherm of xenon (Xe) gas on defected and perfect graphene is studied in order to investigate sensing properties of graphene for Xe gas. In this method, first generation of Brenner many-body potential is used to simulate the interaction of carbon–carbon (C) atoms in graphene, and Lennard–Jones two-body potential is used to simulate interaction of Xe–Xe and Xe–C atoms. In the simulated systems, adsorption coverage, radial distribution function, heat of adsorption, binding energy and specific heat capacity at constant volume are calculated for several temperatures between 90 K and 130 K, and various pressures. It was found that both of the defected and perfect graphene could be introduced as very good candidates for adsorption of Xe gas.