Reactive lattice gas automata

Ławniczak, Anna T.; Dab, David; Kapral, Raymond; Boon, Jean‐Pierre

doi:10.1016/0167-2789(91)90286-i

Cited by 55 publications

(26 citation statements)

References 22 publications

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“…We consider a general lattice gas with a collision rule preserving particle number and momentum in each direction on a lattice in D dimensions. We assume that bit i at a site represents the presence or absence of a particle of unit mass 9 and momentum c i /∆t. We can then write the (ensemble-averaged) mass and momentum densities as…”

Section: Figure 1: Fhp-i Collision Rulesmentioning

confidence: 99%

Correlations and renormalization in lattice gases

Boghosian

Taylor

1995

Phys. Rev. E

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A complete formulation is given of an exact kinetic theory for lattice gases. This kinetic theory makes possible the calculation of corrections to the usual Boltzmann / Chapman-Enskog analysis of lattice gases due to the buildup of correlations. It is shown that renormalized transport coefficients can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the diagrammatic expansions of continuum kinetic theory and quantum field theory. A closed-form expression for the coefficients associated with the vertices of these diagrams is given. This method is applied to several standard lattice gases, and the results are shown to correctly predict experimentally observed deviations from the Boltzmann analysis.

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Section: Figure 1: Fhp-i Collision Rulesmentioning

confidence: 99%

Correlations and renormalization in lattice gases

Boghosian

Taylor

1995

Phys. Rev. E

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“…[l3, 14,15,24,26,271 In order to incorporate other aspects of a modeled medium, such as some types of flow or external forces acting on the system, we can consider two types of random propagations: cell dependent and node dependent random propagations. [23,241 …”

Section: Propagationmentioning

confidence: 99%

“…For other types of LGA diffusion dynamics the macroscopic equations can be derived by analogy. [23,24] Theorem 3. where N!j' =< NTi (3) (x, t ) >j=l,.,.,ml for j = 0,1,2. It is easy to check that evolution of N,,(r, k ) is governed by (3.14), then on the time scale t = t2k the local concentration pr(x, t ) satisfies the LGA diffusion equation: …”

Section: Lga Diffusion Equationsmentioning

confidence: 99%

“…It follows along the lines of the proof of Theorem 3.2 For details see[23].Downloaded by [University of Otago] at 21:43 07 October 2015 Corollary 3.2 Under the assumptions of Theorem 3.3, when m = 4 and v,i satisfy (3.46), the equation (3.49) becomes (3.52) If v,; = v, for every i t then the diffusion is isotropic with the diffusion coeficient D, : (3.53) D, = PTO -PTz +' 2 ~( P T z -P T~+~)~~' 3.3.2 LGA reaction-diffusion equations Theorem 3.4 Assume: i) the automaton space and time scale as in (3.22), ii) the mixing operator R, is a state, space and time independent with uniform probabilities, such as in Theorem 3.2, iv) the reaction probabilities P ( a P ) , for a # P , for each lattice L, , scale as microscopic macroscopic P ( a P ) -c Z P ( a P ) , (3.54) then the local concentration p,(x, t ) satisfies the following equations. I.…”

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From reactive lattice gas automaton rules to its partial differential equations

Ławniczak

2000

Transport Theory and Statistical Physics

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We present a general statistical mechanical theory of multispecies lattice gas automata (LGA) for reaction-diffusion systems. We describe how reactive and elastic collisions and free streaming of molecules between collisions are modeled by LGA chemical transformation, mixing and propagation steps. We characterize these steps in terms of microdynamical variables and transition probabilities. From these steps we construct diffusion and reaction-diffusion automaton dynamics. We describe these dynamics in terms of LGA microdynamical equations and discrete lattice Boltzmann equations. From the LGA discrete Boltzmann equations we derive the corresponding LGA macroscopic diffusion and reactiondiffusion PDEs. By comparing these equations with those arising from phe-261 Copyright 0 2000 by Marcel Dekker, Inc www.dekker.com Downloaded by [University of Otago] at 21:43 07 October 2015 262 LAWNICZAKnomenological description we establish relationships between LGA parameters and those of the physical system. We provide a survey of various applications of reactive LGA and discuss the usefullness of the method to investigate the dynamics on mesoscopic scales.

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“…Other areas of lattice gas research include: thermohydrodynamics [38,28,27,61,63], immiscible fluids [9,37,72,73], reaction-diffusion systems [74,75,76], magnetohydrodynamics [77,78,79,80], flow through porous media [81,36], and renormalized kinetic theory [82,83,84,85,86,87,88]. Numerical measurements taken from classical lattice gas simulations are generally in excellent agreement with mean-field theory predictions and, in the rare instance when this is not the case, with more exact field theoretic calculations [82,85,88].…”

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confidence: 99%

The Classical Lattice-Gas Method

Yepez¹

1999

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Reactive lattice gas automata

Cited by 55 publications

References 22 publications

Correlations and renormalization in lattice gases

Correlations and renormalization in lattice gases

From reactive lattice gas automaton rules to its partial differential equations

The Classical Lattice-Gas Method

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