Collision Rules Research Articles

Statistical mechanics of fluidized granular media: short-range velocity correlations.

A statistical mechanical study of fluidized granular media is presented. Using a special energy injection mechanism, homogeneous fluidized stationary states are obtained. Molecular dynamics simulations and theoretical analysis of the inelastic hard-disk model show that there is a large asymmetry in the two-particle distribution function between pairs that approach and separate. Large velocity correlations appear in the postcollisional states due to the dissipative character of the collision rule. These correlations can be well-characterized by a state dependent pair correlation function at contact. It is also found that velocity correlations are present for pairs that are about to collide. Particles arrive at collisions with a higher probability that their velocities are parallel rather than antiparallel. These dynamical correlations lead to a decrease of the pressure and of the collision frequency as compared to their Enskog values. A phenomenological modified equation of state is presented.

Computational rock physics at the pore scale: Transport properties and diagenesis in realistic pore geometries

Transport properties, such as permeability, are important in many geophysical and petroleum applications. However, complex pore geometry often makes modeling and simulation of transport properties in porous media very difficult. Conventional methods are usually based on partial differential equations, but the implementation of these techniques becomes very complicated when the geometry is extremely complex. As a result, simplified geometry is often used; thus the implementation depends heavily on the model. Solving the same problem in a different geometry often means that many parts of the implementation have to be changed. Therefore, a more robust and simple tool, which can handle complex pore space without oversimplification or modification of the model, is needed. The Lattice-Boltzmann method (LBM), based on statistical description of microscopic phenomena, is one alternative. LBM describes fluid motion as collisions of imaginary particles, which are much bigger than the real fluid molecules. These particles have nearly nothing in common with real fluids, but they show almost the same behavior at a macroscopic scale. This simple collision rule is exactly equivalent to the Navier-Stokes equation within certain appropriate limits. Figure 1 shows a synthetic pore geometry (a random dense pack of identical spheres) and the corresponding digital 3-D structure. Uniform spheres in a random dense pack are a reasonable first approximation to real sandstones. Another advantage for this model is that changing porosity is relatively easy, and laboratory measurements can be mimicked using sintered glass beads. Figure 2 shows electric flux (current) calculated by the finite-element method (FEM) and hydraulic flux calculated by LBM. Both are normalized by each maximum value. The hydraulic flux shows much more sensitivity to the grain boundary than the electric current; the hydraulic flux decreases rapidly near the boundary. Also, the high flux regions are different in Figure 2, even though pore geometry …

Thermostating by deterministic scattering: construction of nonequilibrium steady states

We present a novel approach for constructing nonequilibrium steady states. It is based on a deterministic and time-reversible mechanism for dissipating energy from a subsystem into a thermal reservoir. The key idea is to thermalize a moving particle by appropriately modeling its microscopic collision rules with a boundary mimicking a thermal reservoir with arbitrarily many degrees of freedom. We demonstrate our method for the periodic Lorentz gas with an external electric field. By applying our thermostat we do not find an ergodic breakdown with increasing field strength.

Thermostating by Deterministic Scattering: The Periodic Lorentz Gas

We present a novel mechanism for thermalizing a system of particles in equilibrium and nonequilibrium situations, based on specifically modeling energy transfer at the boundaries via a microscopic collision process. We apply our method to the periodic Lorentz gas, where a point particle moves diffusively through an ensemble of hard disks arranged on a triangular lattice. First, collision rules are defined for this system in thermal equilibrium. They determine the velocity of the moving particle such that the system is deterministic, time-reversible, and microcanonical. These collision rules can systematically be adapted to the case where one associates arbitrarily many degrees of freedom to the disk, which here acts as a boundary. Subsequently, the system is investigated in nonequilibrium situations by applying an external field. We show that in the limit where the disk is endowed by infinitely many degrees of freedom it acts as a thermal reservoir yielding a well-defined nonequilibrium steady state. The characteristic properties of this state, as obtained from computer simulations, are finally compared to those of the so-called Gaussian thermostated driven Lorentz gas.

Numerical Simulations of Atomization of Droplets by 9-Velocity Lattice Gas Automata.

The lattice gas automata method was used to simulate fluid motion. In 7-velocity models, it is known as a ILG model was proposed as one of the models which can be manage two fluid components. Although the ILG model of 7-volocity model meets Laplace's formula, the pressure is independent of the density in 9-volocity model. In the present study, the ILG model was applied to 9-velocity model and collision rule was improved to handle the interface energy transport. As a result, it was clarified the improved 9-velocity ILG model meets not only Lapalce's formula, but also can deal with difference of density and heat transfer at the interface

Absorbing boundary and free-surface conditions in the phononic lattice solid by interpolation

We have recently developed a new lattice-Boltzmann-based approach for modelling compressional wave propagation in heterogeneous media, which we call the phononic lattice solid by interpolation (PLSI). In this paper, we propose an absorbing boundary condition for the PLSI method in which the microscopic reflection coefficients at the boundaries of a model are set to zero and viscous layers are added to the boundaries. Numerical simulation examples using the PLSI method and comparisons with exact solutions demonstrate that artificial boundary reflections can be almost completely eliminated when the incidence angle is less than approximately 70°, Beyond this angle, remanent artificial boundary reflections become visible. We propose four methods for modelling free-surface reflections in PLSI simu-lations. In the first three methods, special collision rules at a free surface are specified to take into account the effect of a free surface on quasi-particle movements (i.e. wave propagation) They are termed the specular bouncing, backward bouncing I, and combined bouncing methods. They involve quasi-particle reflections with a coefficient of − 1 and require the free surface to be located exactly along lattice nodes. For the fourth method, we modify the backward bouncing I model for the case when a free surface is located at any position along lattice links and thus term it the backward bouncing II model. It uses the reflection coefficient at the free surface to calculate the reflected number densities during PLSI simulations. Hence, the free surface is handled in the same way as an interface within a model. Numerical examples and comparisons with exact solutions show that these four methods used at the microscopic scale are all appropriate for modelling macroscopic waves reflected from free surfaces.

A multi-species lattice-gas automaton model to study passive and reactive tracer migration in 2D fractures

We developed a numerical model based on a multi-species lattice gas cellular automaton to study passive and reactive tracer migration in saturated geological media. The model was made of multiple lattice gases interacting via a two-species collision rule. For a binary mixture, the model displayed a negative deviation from Raoult's law and therefore behaved as a real solution. By biasing the initial two-species collision rule, our model was made to obey the tracer assumption which requires that the tracer species does not affect the velocity of the vehicle fluid. In a 2D fracture, we checked the Taylor-Aris relation. An irreversible adsorption between the tracer and the solid phase was numerically added to perform filtration of the colloids. A good agreement was found with the solution of the filtration equation. An attachment efficiency was defined and was found to bear a linear relationship to the filtration coefficient. We added a third species to study the potential role of colloids in the transport of contaminants. Contaminant migration was enhanced when contaminants were bound to colloids and was slightly reduced when colloids were allowed to adsorb on the solid phase.

Lattice Boltzmann solutions of the three-dimensional planetary geostrophic equations

We use the lattice Boltzmann method as the basis for a three-dimensional, numerical ocean circulation model in a rectangular basin. The fundamental dynamical variables are the populations of mass- and buoyancy-particles with prescribed discrete velocities. The particles obey collision rules that correspond, on the macroscopic scale, to planetary geostrophic dynamics. The advantages of the model are simplicity, stability, and massively parallel construction. By the special nature of its construction, the lattice Boltzmann model resolves upwelling boundary layers and unsteady convection. Solutions of the model show many of the features predicted by ocean circulation theories.

A Lorentz invariant thermal lattice gas model

In current lattice gas models particles are constrained to travel along the principal lattice directions between sites, and must advance from one site to another at each time-step. The collision rules are therefore restricted to redistributing momentum along these principal lattice directions. We introduce a computational technique that does not suffer from these limitations. We associate with each particle a continuous momentum which is used to control the motion of the particle on the lattice. This motion is probabilistic, a quasi-random walk, so after many time-steps the particle's average motion is a straight line. The collision rules are no longer constrained by the lattice and can be more realistically implemented; here we use Lorentz invariant binary elastic collisions. We show that the equilibrium distribution of our model is the relativistic Boltzmann distribution which permits us to find the temperature of the lattice gas.

A thermodynamic lattice gas model of Poiseuille flow

Plane Poiseuille flow is simulated using a thermodynamic automaton model. This well-understood problem is used to demonstrate theconsistency of the physical predictions obtained from this new lattice gas model with established physical theory. The model presented is a nonrelativistic version of a model constructed by other researchers in which the collision and propagation rules could be used to derive the relativistic Boltzmann equation. The present model has great potential for describing hydrodynamic processes since it does not suffer from many of the limitations of models that rely on differential equations. For example, it can be used to model highly nonlinear processes or flow past complicated boundaries as easily as it can be used for the Poiseuille flow problems presented here. In the present paper, the physical foundations for using this model to study hydrodynamic processes are demonstratedby showing the Boltzmann nature of the gas, the construction of thermodynamic boundary conditions and consistency with physical theory when the temperature dependence of viscosity is modeled. The principal objective of this paper is to establish the physical validity of this model by comparison with well-established theoretical results from equilibrium thermodynamics. PACS No.: 05.10

Thermodynamic Automaton Simulations of Fluid Flow and Diffusion in Porous Media

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.

Lyapunov instability of two-dimensional fluids: Hard dumbbells.

We generalize Benettin's classical algorithm for the computation of the full Lyapunov spectrum to the case of a two-dimensional fluid composed of linear molecules modeled as hard dumbbells. Each dumbbell, two hard disks of diameter sigma with centers separated by a fixed distance d, may translate and rotate in the plane. We study the mixing between these qualitatively different degrees of freedom and its influence on the full set of Lyapunov exponents. The phase flow consists of smooth streaming interrupted by hard elastic collisions. We apply the exact collision rules for the differential offset vectors in tangent space to the computation of the Lyapunov exponents, and of time-averaged offset-vector projections into various subspaces of the phase space. For the case of a homogeneous mass distribution within a dumbbell we find that for small enough d/sigma, depending on the density, the translational part of the Lyapunov spectrum is decoupled from the rotational part and converges to the spectrum of hard disks. (c) 1998 American Institute of Physics.

Automaton Simulations of Dispersion in Porous Media

A thermodynamic lattice gas (automaton) model is used to simulate dispersion in porous media. Simulations are constructed at two distinctly different scales, the pore scale at which capillary models are constructed and large scale or Darcy scale at which probabilistic collision rules are introduced. Both models allow for macroscopic (pore scale) phase separation. The pore scale models clearly show the effect of pore structure on dispersion. The large scale (mega scale) simulations indicate that when the pressure difference between the displacing phase and displaced phase is properly chosen (representing the average pressure gradient between the phases). The simulation results are consistent with both theoretical predictions and experimental observations.

Probability of first return on Cayley trees

A Lorentz lattice gas with a fraction of the scatterers being pure backscatterers, the remaining being stochastic (right and left) rotators, is considered. The problem at hand is the evaluation of the probability x that the moving particle returns to its original site, in the limit where the density of sites occupied by scatterers is small and the lattice becomes a Cayley tree. In the special case of deterministic collision rules (Gunn - Ortuno model on a Cayley tree), an exact cubic equation for the Laplace transform of the distribution function of first-return times is derived and its asymptotic form near the threshold value (beyond which x = 1) is obtained. In the general case of stochastic models, a mean-field approximation is proposed for x and the mean return time . The approximation reduces to the exact result in the case of the deterministic model, as well as in the stochastic model without backscatterers. Comparison with Monte Carlo simulations shows a reasonable agreement in the dependence of x and on . The relevance of the results to the development of approximate analytic expressions for the diffusion coefficient is discussed.

Nonlinear Hydrodynamics of Lattice-Gas Automata with Semi-Detailed Balance

Equations governing the evolution of the hydrodynamic variables in a lattice-gas automaton, arbitrarily far from equilibrium, are derived from the micro-dynamical description of the automaton, under the condition that the local collision rules satisfy semi-detailed balance. This condition guarantees that a factorized local equilibrium distribution (for each node) of the Fermi–Dirac form is invariant under the collision step but not under propagation. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice-Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium. Linearization of the hydrodynamic equations derived here leads to Green–Kubo formulae for the transport coefficients.

Droplet Dynamics Using the Lattice-Gas Method

We apply the lattice-gas method to study droplet behavior. The ascending velocity and deformation of a drop under buoyancy are numerically analyzed. We develop an algorithm to adjust surface tension by changing collision rules, and deformation of a droplet is calculated with different surface tensions. The deformation of a drop driven by the pressure gradient under gravity is also investigated.

Nonlinear lattice gas hydrodynamics

Starting from a microdynamical description, we derive the equations governing the evolution of the hydrodynamic variables in a lattice gas automaton. The essential features are: (i) the local collision rules satisfy semi-detailed balance; this condition guarantees that a factorized local equilibrium distribution of the Fermi-Dirac form is invariant under the collision step, but not under propagation; (ii) particles entering a collision are uncorrelated (Boltzmann hypothesis); and (iii) the system can be arbitrarily far from global equilibrium; we do not not assume linear response, as usually imposed, to obtain the dissipative contributions. Linearization of the resulting hydrodynamic equations leads to Green-Kubo formulas for the transport coefficients. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium.

A new thermal lattice gas

We present a new thermal lattice gas in which the particles are distributed among 3 possible energy levels. Collision rules obey conservation of particles and momentum in the usual way, but conservation of energy is enforced by “demons” in a manner familiar from the micro-canonical Monte-Carlo method [6]. For this reason, the model is much simpler than others in the literature. We outline the statistical mechanical properties of our model, and describe the calculation of its transport coefficients within linear response theory, as functions of density and temperature.

Lattice Boltzmann simulations of liquid-gas and binary fluid systems

We present the details of a lattice Boltzmann approach to phase separation in nonideal one- and two-component fluids. The collision rules are chosen such that the equilibrium state corresponds to an input free energy and the bulk flow is governed by the continuity, Navier-Stokes, and, for the binary fluid, a convection-diffusion equation. Numerical results are compared to simple analytic predictions to confirm that the equilibrium state is indeed thermodynamically consistent and that the kinetics of the approach to equilibrium lie within the expected universality classes. The approach is compared to other lattice Boltzmann simulations of nonideal systems. \textcopyright{} 1996 The American Physical Society.

Microscopic theory for long-range spatial correlations in lattice gas automata.

Lattice gas automata with collision rules that violate the conditions of semidetailed balance exhibit algebraic decay of equal-time spatial correlations between fluctuations of conserved densities. This is shown on the basis of a systematic microscopic theory. Analytical expressions for the dominant long-range behavior of correlation functions are derived using kinetic theory. We discuss a model of interacting random walkers with x-y anisotropy whose pair correlation function decays as 1/${\mathit{r}}^{2}$, and an isotropic fluid-type model with momentum correlations decaying as 1/${\mathit{r}}^{2}$. The pair correlation function for an interacting random walker model with interactions satisfying all symmetries of the square lattice is shown to have 1/${\mathit{r}}^{4}$ density correlations. Theoretical predictions for the amplitude of the algebraic tails are compared with the results of computer simulations. \textcopyright{} 1996 The American Physical Society.