Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton

In the so-called "microscopic" models of vehicular traffic, attention is paid explicitly to each individual vehicle each of which is represented by a "particle"; the nature of the "interactions" among these particles is determined by the way the vehicles influence each others' movement. Therefore, vehicular traffic, modeled as a system of interacting "particles" driven far from equilibrium, offers the possibility to study various fundamental aspects of truly nonequilibrium systems which are of current interest in statistical physics. Analytical as well as numerical techniques of statistical physics are being used to study these models to understand rich variety of physical phenomena exhibited by vehicular traffic. Some of these phenomena, observed in vehicular traffic under different circumstances, include transitions from one dynamical phase to another, criticality and self-organized criticality, metastability and hysteresis, phase-segregation, etc. In this critical review, written from the perspective of statistical physics, we explain the guiding principles behind all the main theoretical approaches. But we present detailed discussions on the results obtained mainly from the so-called "particle-hopping" models, particularly emphasizing those which have been formulated in recent years using the language of cellular automata.

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“…By applying the UDM to Burgers' equation v t = 2vv x + v xx one obtains the (L + 1)-state, deterministic CA [237,238].…”

Section: Ca From Ultra-discretizationmentioning

confidence: 99%

Statistical physics of vehicular traffic and some related systems

2000

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“…We note our result is related to ultradiscretization procedure in the field of integrable systems that connects discrete max-plus equations, CA, and continuous differential or difference equations [9][10][11][12]. For example, the Burgers equation,…”

Section: Introductionmentioning

confidence: 95%

On solutions to evolution equations defined by lattice operators

Ikegami

Takahashi

Matsukidaira

2013

Japan J. Indust. Appl. Math.

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“…We mention that a related discrete Burgers equation appeared in a different context in ref. [19], and that an "ultradiscrete" version of it found an application to traffic flow modeling in [20]. A semidiscrete Burgers equation (with continuous time t) was also introduced earlier and its integrability was analyzed [21].…”

Section: The Discrete Burgers Equation As a Compatibility Conditionmentioning

confidence: 99%

Symmetries of the discrete Burgers equation

Heredero¹,

Levi²,

Winternitz³

1999

J. Phys. A: Math. Gen.

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A discrete Cole-Hopf transformation is used to derive a discrete Burgers equation that is linearizable to a discrete heat equation. A five-dimensional symmetry algebra is obtained that reduces to the Lie point symmetry algebra of the usual Burgers equation, in the continuous limit. This Lie algebra is used to obtain explicit invariant solutions. RésuméOn utilise une transformation de Cole-Hopf pour obtenir uneéquation de Burgers discrète qui est linéarisable et peutêtre ainsi amenéeà uneéquation de la chaleur discrète. Une algèbre de symétrie de dimension cinq est obtenue. Cette algèbre se réduit aux symétries locales de l'équation de Burgers usuelle dans la limite continue. Cette algèbre de symétrie est utilisée pour obtenir des solutions invariantes explicites.

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Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton

Cited by 138 publications

References 10 publications

Statistical physics of vehicular traffic and some related systems

Statistical physics of vehicular traffic and some related systems

On solutions to evolution equations defined by lattice operators

Symmetries of the discrete Burgers equation

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