Self-organized states in cellular automata: Exact solution

Medvedev, Mikhail V.; Diamond, P. H.

doi:10.1103/physreve.58.6824

Cited by 3 publications

(4 citation statements)

References 25 publications

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“…where ∆B 1 = B 1 − B 0 and ∆B 2 = B 2 − B 0 and we used that cos θ = (1 − sin 2 θ) 1/2 = (1 − B 0 /B) 1/2 = (∆B/B) 1/2 . To proceed further we introduce a mathematical model of a one-dimensional Markov chain [10,11]. It is shown in Fig.…”

Section: Figmentioning

confidence: 99%

“…To proceed further we introduce a mathematical model of a one-dimensional Markov chain [10,11]. It is shown in Fig.…”

Section: Asymmetric Diffusionmentioning

confidence: 99%

“…Hereafter we assume the steady state, for simplicity, so that all time derivatives vanish, d t n j = 0. We solve the system of N coupled equations ( 9)-( 12) by calculating the generating function [10,11] defined as follows…”

Section: Asymmetric Diffusionmentioning

confidence: 99%

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Asymmetric diffusion of cosmic rays

Medvedev

2015

Physics of Plasmas

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Cosmic ray propagation is diffusive because of pitch angle scattering by waves. We demonstrate that if the high-amplitude magnetohydrodynamic turbulence withB=hBi $ 1 is present on top of the mean field gradient, the diffusion becomes asymmetric. As an example, we consider the vertical transport of cosmic rays in our Galaxy propagating away from a point-like source. We solve this diffusion problem analytically using a one-dimensional Markov chain analysis. We obtained that the cosmic ray density markedly differs from the standard diffusion prediction and has a sizable effect on their distribution throughout the galaxy. The equation for the continuous limit is also derived, which shows limitations of the convection-diffusion equation. V C 2015 AIP Publishing LLC.

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Section: Figmentioning

confidence: 99%

“…To proceed further we introduce a mathematical model of a one-dimensional Markov chain [10,11]. It is shown in Fig.…”

Section: Asymmetric Diffusionmentioning

confidence: 99%

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Asymmetric diffusion of cosmic rays

Medvedev

2015

Physics of Plasmas

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“…2; an explicit estimation of the average slope in the running sandpile is given, for instance, in Ref. 24; an alternative estimation is provided in Ref. 25).…”

Section: The Running Sandpile Modelmentioning

confidence: 99%

Modeling transport across the running-sandpile cellular automaton by means of fractional transport equations

Sánchez

2018

Phys. Rev. E

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Fractional transport equations are used to build an effective model for transport across the running sandpile cellular automaton [Hwa et al., Phys. Rev. A 45, 7002 (1992)PLRAAN1050-294710.1103/PhysRevA.45.7002]. It is shown that both temporal and spatial fractional derivatives must be considered to properly reproduce the sandpile transport features, which are governed by self-organized criticality, at least over sufficiently long or large scales. In contrast to previous applications of fractional transport equations to other systems, the specifics of sand motion require in this case that the spatial fractional derivatives used for the running sandpile must be of the completely asymmetrical Riesz-Feller type. Appropriate values for the fractional exponents that define these derivatives in the case of the running sandpile are obtained numerically.

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The physics of complex systems: applications to earthquake

Mulargia

Geller

2003

Earthquake Science and Seismic Risk Reduction

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Self-organized states in cellular automata: Exact solution

Cited by 3 publications

References 25 publications

Asymmetric diffusion of cosmic rays

Asymmetric diffusion of cosmic rays

Modeling transport across the running-sandpile cellular automaton by means of fractional transport equations

The physics of complex systems: applications to earthquake

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