The Domany-Kinzel cellular automaton revisited

Kohring, G.A.; Schreckenberg, Michael

doi:10.1051/jp1:1992264

Cited by 20 publications

(28 citation statements)

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“…1. The resulting value of rriri^) in the thermodynamic limit is in very good agreement with the extrapolated value for the up to now largest systems investigated by Kohring and Schreckenberg [20]. Again note that these results do not contain any uncontrolled dependencies on the system size A^ as all previous results.…”

Section: Note That the Internal Field H^it-1) Was Already Calculated supporting

confidence: 89%

“…Especially for this quantity finite-size effects were found to be very strong [17][18][19][20] and it is still unclear what is the correct finite-size scaling function to be used. So the extrapolated value mr(oo) obtained from finite-size scaling changed over the years by going to larger maximum system sizes.…”

Section: Note That the Internal Field H^it-1) Was Already Calculated mentioning

confidence: 99%

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New method for studying the dynamics of disordered spin systems without finite-size effects

1992

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A new method is presented which allows the determination of dynamical properties of disordered spin systems avoiding finite-size effects. The method is based on exact dynamical mean-field equations for the infinite large system. The resulting single-spin dynamics is solved by Monte Carlo simulations. We outline the formalism for the parallel dynamics of a fully connected model with random couplings. The decay of remanent magnetization of the model is studied. We find a power-law decay: m(t) -m(oo) oc/ ~"^ with a =0.474 and m{oo) =0.184 for the infinite system, PACS numbers: 87.10.+e Networks composed of spinlike elements which interact via long-range random interactions are of growing interest. Such models served as approximations for disordered materials as, for example, the Sherrington-Kirkpatrick (SK) model for spin glasses [l]. More recently similar systems have become popular as models for neural networks [2].The equilibrium behavior of such random spin systems is now well understood by applying the replica approach to determine thermal averages for disordered systems [3].Because of the lack of ergodicity these models show many interesting features that are essentially of nonequilibrium nature. Usually the dynamics at low temperatures will reach a state which will strongly depend on its initial conditions. This property made networks of spins useful for the models of associative memories [2].To study the dynamics of fully connected spin models, there are mainly two different approaches. The first one is to calculate the time evolution of dynamic quantities analytically using the so-called dynamical functional method [4]. This method has been successfully applied to the behavior of disordered spin systems at long time scales being able to recover and understand the equilibrium results mentioned above from a purely dynamical viewpoint [5][6][7].Unfortunately the situation is less satisfactory for the nonequilibrium, transient behavior. Apart from approximate treatments [8,9] exact calculations are only possible for very few time steps, e.g., up to four time steps for the SK model and two time steps for the Hopfield model [10].So most of the studies on dynamics of disordered spin systems rest on numerical simulations, which is the other possible approach. However, there are strong finite-size effects, which for example do not even allow a decision if there is a finite remanent magnetization for the infinite [Z(/)]y = spin system. In this Letter we propose a new method to avoid these finite-size problems. Our method combines the dynamical functional method, which allows us to perform the limit TV-• <» exactly, and a Monte Carlo simulation of the resulting stochastic one-particle equations. The method is demonstrated for a disordered spin system with synchronous dynamics.This model consists of TV Ising spins 5*, = ± 1, where every spin St is connected to all other spins Sj by couplings Jij, which are Gaussian random variables with distribution PUij) =VTV/2;rexp(-jNJ^j).Additionally the symmetry of the couplings ...

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Section: Note That the Internal Field H^it-1) Was Already Calculated supporting

confidence: 89%

Section: Note That the Internal Field H^it-1) Was Already Calculated mentioning

confidence: 99%

New method for studying the dynamics of disordered spin systems without finite-size effects

1992

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“…Tomé [4] studied this joint evolution and showed that the CA's can evolve following two different prescriptions: prescription A, used by Martins et al [2], which corresponds to updating both automata always using the same random number, and prescription B, introduced by Kohring and Schereckenberg [5], which implies that one must sometimes use two different random numbers (z 1 and z 2 ) to update the original and the replica. The later occurs when we have (σ i−1 + σ i+1 ) = 1 and (̺ i−1 + ̺ i+1 ) = 2, or vice-versa.…”

Section: Active-chaotic Transitionmentioning

confidence: 99%

Growth exponent in the Domany-Kinzel cellular automaton

Atman¹,

Moreira²

2000

Eur. Phys. J. B

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In a roughening process, the growth exponent β describes how the roughness w grows with the time t: w ∼ t β . We determine the exponent β of a growth process generated by the spatiotemporal patterns of the one dimensional Domany-Kinzel cellular automaton. The values obtained for β shows a cusp at the frozen/active transition which permits determination of the transition line. The β value at the transition depends on the scheme used: symmetric (β ∼ 0.83) or non-symmetric (β ∼ 0.61). Using damage spreading ideas, we also determine the active/chaotic transition line; this line depends on how the replicas are updated.

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“…In a very broad sense (and perhaps a rather vague one) a mean field approximation is one in which fluctuations are neglected. Different systematic mean field approaches has been proposed for the DK cellular automaton 6,8,7 . They all coincide (at least qualitatively) in the prediction for the shape of the frozen-active transition line, but they differ in the prediction of the damage spreading transition line.…”

Section: The Model: Long-range Cellular Automatonmentioning

confidence: 99%

“…The phase diagram of this model was calculated using different techniques, such as mean field [6][7][8] and Monte Carlo 2,5,9 . In this work we introduce a generalization of the above model, in which the state of any site depends on the state of all the rest of the sites at the previous time through a distance dependent transition probability. In this case the interaction between a pair of sites separated by a distance r decays with a power law of the type 1/r α (α ≥ 0).…”

Section: Introductionmentioning

confidence: 99%

Phase diagram of a stochastic cellular automaton with long-range interactions

Cannas

1998

Physica A: Statistical Mechanics and its Applications

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A stochastic one-dimensional cellular automaton with long range spatial interactions is introduced. In this model the probability state at time t of a given site depends on the state of all the other sites at time t − 1 through a power law of the type 1/r α , r being the distance between sites. For α → ∞ this model reduces to the Domany-Kinzel cellular automaton. The dynamical phase diagram is analyzed using Monte Carlo simulations for 0 ≤ α ≤ ∞. We found the existence of two different regimes: one for 0 ≤ α ≤ 1 and the other for α > 1. It is shown that in the first regime the phase diagram becomes independent of α. Regarding the frozen-active phase transition in this regime, a strong evidence is found that the mean field prediction for this model becomes exact, a result already encountered in magnetic systems. It is also shown that, for replicas evolving under the same noise, the long-range interactions fully suppress the spreading of damage for 0 ≤ α ≤ 1.

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The Domany-Kinzel cellular automaton revisited

Cited by 20 publications

References 0 publications

New method for studying the dynamics of disordered spin systems without finite-size effects

New method for studying the dynamics of disordered spin systems without finite-size effects

Growth exponent in the Domany-Kinzel cellular automaton

Phase diagram of a stochastic cellular automaton with long-range interactions

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