From Soliton Equations to Integrable Cellular Automata through a Limiting Procedure

Tokihiro, Tetsuji; Takahashi, Daisuke; Matsukidaira, Junta; Satsuma, Junkichi

doi:10.1103/physrevlett.76.3247

Cited by 404 publications

(400 citation statements)

References 12 publications

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“…Several CA which can be interpreted as traffic models have been derived using the so-called ultra-discretization method (UDM) [236]. This approach allows to establish a direct connection between certain differential equations and CA.…”

Section: Ca From Ultra-discretizationmentioning

confidence: 99%

Statistical physics of vehicular traffic and some related systems

2000

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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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Section: Ca From Ultra-discretizationmentioning

confidence: 99%

Statistical physics of vehicular traffic and some related systems

2000

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“…In this section, we consider an ultradiscrete version of the third-order integrable equations [13,14]. Since ultradiscretization requires a positivity of parameters and dependent variables of the equations, (Y1),(Y2),(Y4),(Y6),(Y7) and (Y9) are ultradiscretizable.…”

Section: Ultradiscretizationmentioning

confidence: 99%

Third-order integrable difference equations generated by a pair of second-order equations

Matsukidaira¹,

Takahashi²

2006

J. Phys. A: Math. Gen.

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“…Recently, by the development of computer sciences, discrete and ultra-discrete versions of integral system have been attracting a great deal of attention. The ultra-discretization of soliton equations and other various important equations are intensively studied in [1,2,3,4,5,6,7,8]. Moreover, it is known that the various ultra-discrete soliton equations are obtained from solvable vertex model in statistical mechanics in a certain limiting procedure [9,10].…”

Section: Introductionmentioning

confidence: 99%