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On the Blockage Problem and the Non-analyticity of the Current for Parallel TASEP on a Ring

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Abstract

The Totally Asymmetric Simple Exclusion Process (TASEP) is an important example of a particle system driven by an irreversible Markov chain. In this paper we give a simple yet rigorous derivation of the chain stationary measure in the case of parallel updating rule. In this parallel framework we then consider the blockage problem (aka slow bond problem). We find the exact expression of the current for an arbitrary blockage intensity \(\varepsilon \) in the case of the so-called rule-184 cellular automaton, i.e. a parallel tasep where at each step all particles free-to-move are actually moved. Finally, we investigate through numerical experiments the conjecture that for parallel updates other than rule-184 the current may be non-analytic in the blockage intensity around the value \(\varepsilon = 0\).

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Notes

  1. Although it is possible to define the TASEP on the whole \(\mathbb {Z}\) by considering continuous time, in this paper we will consider only finite space (and discrete time).

  2. Please bear in mind that the right-neighbour is actually site \((i+1)\!\! \mod 2L\) in view of the periodic boundary conditions.

  3. Except if the train has length bigger than 1 and one of the particles composing it occupies site 1. In this case engine and caboose are the elements with smallest and highest index, respectively.

  4. Actually, this is the stationary probability due to Remark 4.2.

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Acknowledgments

We wish to thank Elisabetta Scoppola for useful discussions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Rome, Italy

    Benedetto Scoppola, Carlo Lancia & Riccardo Mariani

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  1. Benedetto Scoppola
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  2. Carlo Lancia
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  3. Riccardo Mariani
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Correspondence to Benedetto Scoppola.

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Scoppola, B., Lancia, C. & Mariani, R. On the Blockage Problem and the Non-analyticity of the Current for Parallel TASEP on a Ring. J Stat Phys 161, 843–858 (2015). https://doi.org/10.1007/s10955-015-1352-4

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  • DOI: https://doi.org/10.1007/s10955-015-1352-4

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