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Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System.

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Title: Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System.
Authors: Großkinsky, Stefan1 stefan@statslab.cam.ac.uk, Schütz, Gunter2, Willmann, Richard2
Source: Journal of Statistical Physics. Aug2007, Vol. 128 Issue 3, p587-606. 20p. 1 Diagram, 6 Graphs.
Subjects: Cellular automata, Sequential machine theory, Martingales (Mathematics), Stochastic processes, Symmetry breaking, Machine theory
Abstract: We study spontaneous symmetry breaking in a one-dimensional driven two-species stochastic cellular automaton with parallel sublattice update and open boundaries. The dynamics are symmetric with respect to interchange of particles. Starting from an empty initial lattice, the system enters a symmetry broken state after some time T 1 through an amplification loop of initial fluctuations. It remains in the symmetry broken state for a time T 2 through a traffic jam effect. Applying a simple martingale argument, we obtain rigorous asymptotic estimates for the expected times 〈 T 1〉 ∝ Lln L and ln 〈 T 2〉 ∝ L, where L is the system size. The actual value of T 1 depends strongly on the initial fluctuation in the amplification loop. Numerical simulations suggest that T 2 is exponentially distributed with a mean that grows exponentially in system size. For the phase transition line we argue and confirm by simulations that the flipping time between sign changes of the difference of particle numbers approaches an algebraic distribution as the system size tends to infinity. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Statistical+Physics%22">Journal of Statistical Physics</searchLink>. Aug2007, Vol. 128 Issue 3, p587-606. 20p. 1 Diagram, 6 Graphs.
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  Data: <searchLink fieldCode="DE" term="%22Cellular+automata%22">Cellular automata</searchLink><br /><searchLink fieldCode="DE" term="%22Sequential+machine+theory%22">Sequential machine theory</searchLink><br /><searchLink fieldCode="DE" term="%22Martingales+%28Mathematics%29%22">Martingales (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+processes%22">Stochastic processes</searchLink><br /><searchLink fieldCode="DE" term="%22Symmetry+breaking%22">Symmetry breaking</searchLink><br /><searchLink fieldCode="DE" term="%22Machine+theory%22">Machine theory</searchLink>
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  Data: We study spontaneous symmetry breaking in a one-dimensional driven two-species stochastic cellular automaton with parallel sublattice update and open boundaries. The dynamics are symmetric with respect to interchange of particles. Starting from an empty initial lattice, the system enters a symmetry broken state after some time T 1 through an amplification loop of initial fluctuations. It remains in the symmetry broken state for a time T 2 through a traffic jam effect. Applying a simple martingale argument, we obtain rigorous asymptotic estimates for the expected times 〈 T 1〉 ∝ Lln L and ln 〈 T 2〉 ∝ L, where L is the system size. The actual value of T 1 depends strongly on the initial fluctuation in the amplification loop. Numerical simulations suggest that T 2 is exponentially distributed with a mean that grows exponentially in system size. For the phase transition line we argue and confirm by simulations that the flipping time between sign changes of the difference of particle numbers approaches an algebraic distribution as the system size tends to infinity. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Martingales (Mathematics)
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              Text: Aug2007
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