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.     

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.     

Diffusion in concentrated lattice gases

Journal of Statistical Physics - The diffusion of many particles on a lattice is an example of a correlated random-walk process. Recently the waiting-time distributions for two consecutive jumps of...     

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.     

Diffusion approximation for billiards with totally accommodating scatterers

We study the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accommodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long-time and large-scale regime. The proof relies on a multiscale asymptotics and gives the order of approximation.     

Diffusion in Lorentz lattice gas automata with backscattering

The probability of first return to the initial intervalx and the diffusion tensorD _x are calculated exactly for a ballistic Lorentz gas on a Bethe lattice or Cayley tree. It consists of a moving particle and a fixed array of scatterers, located at the nodes, and the lengths of the intervals between scatterers are determined by a geometric distribution. The same values forx andD _x apply also to a regular space lattice with a fraction of sites occupied by a scatterer in the limit of a small concentration of scatterers. If backscattering occurs, the results are very different from the Boltzmann approximation. The theory is applied to different types of lattices and different types of scatterers having rotational or mirror symmetries.     

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.     

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.     

Stress-stress correlation functions in lattice gases beyond Boltzmann's approximation

The complete time dependence of the stress-stress correlation functions in lattice gas cellular automata is calculated from the ring kinetic theory using numerical and analytical methods. This provides corrections, typically of 10–20%, to the usual molecular chaos calculations, where correlation functions decay exponentially. The resulting correlation function crosses over from an initial exponential decay to the long-time behavior calculated from mode coupling theory. The present theory, applied to the viscosity, accounts for a substantial part of the observed difference between the Boltzmann theory and simulations.     

Rayleigh approximation for axisymmetric scatterers

The Rayleigh approximation is known to be designed only for small ellipsoidal scatterers. We suggest an approach that allows one to find a simple, often analytical, long-wavelength approximation for nonellipsoidal particles. We apply the approach to axisymmetric scatterers and utilize Chebyshev particles to study the main properties of the obtained approximation. To a certain degree, it can be considered as an extension of the Rayleigh approximation to nonspheroidal scatterers.     

Diffusion in lattices with anisotropic scatterers

We study diffusion in lattices with periodic and random arrangements of anisotropic scatterers. We show, using both analytical techniques based upon our previous work on asymptotic properties of multistate random walks and computer calculation, that the diffusion constant for the random arrangement of scatterers is bounded above and below at an arbitrary density by the diffusion constant for an appropriately chosen periodic arrangement of scatterers at the same density. We also investigate the accuracy of the low-density expansion for the diffusion constant up to second order in the density for a lattice with randomly distributed anisotropic scatterers. Comparison of the analytical results with numerical calculations shows that the accuracy of the density expansion depends crucially on the degree of anisotropy of the scatterers. Finally, we discuss a monotonicity law for the diffusion constant with respect to variation of the transition rates, in analogy with the Rayleigh monotonicity law for the effective resistance of electric networks. As an immediate corollary we obtain that the diffusion constant, averaged over all realizations of the random arrangement of anisotropic scatterers at density, is a monotone function of the density.     

Diffusion of lattice gases without double occupancy on three-dimensional percolation lattices

We examined the diffusion of lattice gases, where double occupancy of sites is excluded, on three-dimensional percolation lattices at the percolation thresholdp _c . The critical exponent for the root-mean-square displacement was determined to bek=0.183±0.010, which is similiar to the result of Roman for the problem of the ant in the labyrinth. Furthermore, we found a plateau value fork at intermediate times for systems with higher concentrations of lattice gas particles.     

Diffusion in the Lorentz Gas

The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann-Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.     

Lorentz transformation on the lattice

We study the possibility of representing the kinematical variables of a free particle in terms of scale factors and integers. The action of a set of transformations from the Lorentz group parametrized by integers on this system of variables are investigated, and it is shown that one can effectively characterize these symmetries on a lattice in this way. By taking the scales sufficiently small, one can arbitrarily closely approach the continuous case.Work supported in part by the Binational Science Foundation (BSF) Jerusalem. Israel.     

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.     

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,     

Diffusion in Lorentz lattice gas cellular automata: The honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found, ranging from propagation, superdiffusion, normal, quasi-normal, and anomalous, to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far is summarized.     

Low-viscosity lattice gases

A class of lattice gas models are studied which are variants of the FCHC model. The aim is to achieve the highest possible Reynolds coefficient (inverse dimensionless viscosity) for efficient simulations of the three-dimensional incompressible Navier-Stokes equations. The models include an arbitrary number of rest particles and violation of semi-detailed balance. Within the framework of the Boltzmann approximation exact expressions are obtained for the Reynolds coefficients. The minimization of the viscosity is done by solving a Hitchcock-type optimization problem for the fine tuning of the collision rules. When the number of rest particles exceeds one, there is a range of densities at which the viscosity takes negative values. Various optimal models with up to 26 bits per node have been implemented on a CRAY-2 and their true transport coefficients have been measured with good accuracy. Fairly large discrepancies with Boltzmann values are observed when semi-detailed balance is violated; in particular, no negative viscosity is obtained. Still, the best model has a Reynolds coefficient of 13.5, twice that of the best previously implemented model, and thus is about 16 times more efficient computationally. Suggestions are made for further improvements. It is proposed to use models with very high Reynolds coefficients for sub-grid-scale modeling of turbulent flows.     

Negative-viscosity lattice gases

A new irreversible collision rule is introduced for lattice-gas automata. The rule maximizes the flux of momentum in the direction of the local momentum gradient, yielding a negative shear viscosity. Numerical results in 2D show that the negative viscosity leads to the spontaneous ordering of the velocity field, with vorticity resolvable down to one lattice-link length. The new rule may be used in conjunction with previously proposed collision rules to yield a positive shear viscosity lower than the previous rules provide. In particular, Poiseuille flow tests demonstrate a decrease in viscosity by more than a factor of 2.