Two-dimensional Lattice Gas Automaton Research Articles

Thermodynamic entropy and chaos in a discrete hydrodynamical system

We show that the thermodynamic entropy density is proportional to the largest Lyapunov exponent (LLE) of a discrete hydrodynamical system, a deterministic two-dimensional lattice gas automaton. The definition of the LLE for cellular automata is based on the concept of Boolean derivatives and is formally equivalent to that of continuous dynamical systems. This relation is justified using a Markovian model. In an irreversible process with an initial density difference between both halves of the system, we find that Boltzmann's H function is linearly related to the expansion factor of the LLE although the latter is more sensitive to the presence of traveling waves.

Simple stochastic lattice gas automaton model for formation of river networks

A stochastic lattice gas automata model for formation of river networks is proposed. The model is based on two-dimensional lattice gas automata with three fundamental principles at each node. The water source is regarded as a fixed point where a drop of water drips every time step. This system can be treated as a memory network: the probability of water moving along a direction relies on the history of the channel segment along which water drops have moved. Last, we find that the width of the river channel and the number of channels with this width meet a scaling law when the system reaches a critical status.

Anomalous spreading of a density front from an infinite continuous source in a concentration-dependent lattice gas automaton diffusion model

A two-dimensional lattice gas automaton (LGA) is used for simulating concentration-dependent diffusion in a microscopically random heterogeneous structure. The heterogeneous medium is initialized at a low density ρ0 and then submitted to a steep concentration gradient by continuous injection of particles at a concentration ρ1>ρ0 from a one-dimensional source to model spreading of a density front. Whereas the nonlinear diffusion equation generally used to describe concentration-dependent diffusion processes predicts a scaling law of the type ϕ = xt−1/2 in one dimension, the spreading process is shown to deviate from the expected t1/2 scaling. The time exponent is found to be larger than ½, i.e. diffusion of the density front is enhanced with respect to standard Fickian diffusion. It is also established that the anomalous time exponent decreases as time elapses: anomalous spreading is thus not a timescaling process.We demonstrate that occurrence of anomalous spreading results from the diffusivity gradient (dD(ρ)/dρ) existing in the concentration-dependent LGA diffusion model. Standard Fickian diffusion appears as a special case which only occurs when (dD(ρ)/dρ)≈0. Decrease of the anomalous exponent with time may indicate that anomalous diffusion is only transient. In any case, the LGA system possesses a very long transitory regime and spreading remains an anomalous superdiffusive process over large period of time. A simple qualitative model, based on the supply and demand principle, is proposed to account for anomalous spreading. A correspondence is finally established between LGA simulations and experimental measurements of one-dimensional water absorption in non-saturated porous materials in which evidence of anomalous spreading was recently reported.

Scale invariance in coarsening of binary and ternary fluids.

Phase separation in binary and ternary fluids is studied using a two-dimensional lattice gas automata. The lengths given by the the first zero crossing point of the correlation function and the total interface length is shown to exhibit power law dependence on time. In binary mixtures, our data clearly indicate the existence of a regime having more than one length scale, where the coarsening process proceeds through the rupture and reassociation of domains. In ternary fluids; in the case of symmetric mixtures there exists a regime with a single length scale having dynamic exponent 1/2, while in asymmetric mixtures our data establish the break down of scale invariance.

A lattice gas automaton simulation of the nonlinear diffusion equation: A model for moisture flow in unsaturated porous media

We investigate a two-dimensional lattice gas automaton (LGA) for simulating the nonlinear diffusion equation in a random heterogeneous structure. The utilility of the LGA for computation of nonlinear diffusion arises from the fact that, the diffusion coefficient in the LGA depends on the local density ρ of ‘fluid’ particles which statistically determines the collision rate and thus, the mean free path λ of the particles at the microscopic scale. The LGA may therefore be used as a physical analogue to simulate moisture flow in unsaturated porous media. The capability of the LGA to account for unsaturated flow is tested through a set of numerical experiments simulating one-dimensional infiltration in a simplified semi-infinite homogenous isotropic porous material. Different mechanisms of interactions are used between the fluid and the solid phase to simulate various fluid–solid interfaces. The heterogeneous medium, initially at low density is submitted to a steep density gradient by continuously injecting fluid particles at high concentration and zero velocity along one face of the model. The propagation of the infiltration front is visualized at different time steps through concentration profiles parallel to the applied concentration gradient and the infiltration rate is measured continuously until steady-state flow is reached. The numerical results show close agreement with the classical theory of flow in unsaturated porous media. The cumulative absorption exhibits the expected t 1/2 dependence. The evolution of the effective diffusion coefficient with the particle concentration is estimated from the measured density profiles for the various porous materials. Depending on the applied fluid–solid interactions, the macroscopic effective diffusivity may vary by more than two orders of magnitude with density.

A Lattice Gas Automaton Capable of Modeling Three-Dimensional Electromagnetic Fields

A lattice gas automaton (LGA) capable of modeling Maxwell's equations in three dimensions is described. The automaton is a three-dimensional interconnection of two-dimensional LGA cells, with appropriate operations at the junctions between cells to include the properties of polarization. A homogeneous mathematical description of the heterogeneous three-dimensional automaton is provided in terms of the underlying binary variables. The dynamics of the automaton conserve the scalar components of the electric and magnetic fields. The implementation of the automaton on the CAM-8 cellular automata machine is described. The LGA has been validated through calculation of resonant frequencies within various cavities. The numerical results indicate the success of the automaton in analyzing three-dimensional EM field problems. We have not proven analytically that this model reproduces Maxwell's equations in the macroscopic limit, as this is a topic of future study.

Long-time tails in lattice gases violating detailed balance.

Using analytic and simulation techniques, we study the long- and intermediate-time behavior of the velocity autocorrelation function in two-dimensional lattice gas automata with stationary states that violate detailed balance. Such models are prototypes of dissipative systems, have stationary states that are very different from Gibbs states, and exhibit long range spatial correlations. Such static correlations are absent in models with detailed balance symmetry. In some lattice gases with strong violation of detailed balance, the simulations show negative velocity correlations at intermediate times (cage effect), which, to our knowledge, has never been observed in lattice gases before. A mode coupling calculation is used to analyze the long-time tail, whose amplitude is very different from the mean field prediction. When the above static correlations are taken into account, our theoretical predictions agree very well with the results of computer simulations.

An investigation of fractal dimensions in two-dimensional lattice gas turbulence

The authors investigate the dissipation mechanisms acting in a two-dimensional lattice gas automaton by inspecting the structure functions of the turbulent velocity field associated with the Boolean configuration of the automaton. In particular, they investigate whether the Boolean noise produced by the automaton can promote fractal structures within the flow. They show that this is not the case and the presence of the noise only results in a non-analyticity of the flow field which can be progressively eliminated upon averaging the Boolean field on coarser and coarser grids. As a result, the authors find that the non-fractal nature of homogeneous two-dimensional turbulence is not affected by the presence of the microscopic noise.