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On the Asymmetric Simple Exclusion Process with Multiple Species

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Abstract

In previous work the authors, using the Bethe Ansatz, found for the N-particle asymmetric simple exclusion process on the integers a formula—a sum of multiple integrals—for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations hold.

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Notes

  1. Many authors consider ASEP on the circle or the lattice [1,L] with open boundary conditions.

  2. This is sometimes called the M+1 species model, empty sites behaving as particles of another species. With our convention, a particle of species M is first-class, having priority over all others.

  3. For example, suppose σ=(3 2 5 4 1), for which B={4,1}. Then the steps might be

    $$(1\ 2\ 3\ 4\ 5)\to(1\ 3\ 2\ 4\ 5)\to(3\ 1\ 2\ 4\ 5)\to(3\ 2\ 1\ 4\ 5) \to(3\ 2\ 1\ 5\ 4)\to(3\ 2\ 5\ 1\ 4)\to(3\ 2\ 5\ 4\ 1). $$

    The only S-factors involving ξ 5 come from steps four and five, and are S(ξ 4,ξ 5) and S(ξ 1,ξ 5).

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Acknowledgements

This work was supported by the National Science Foundation through grants DMS-0906387 (first author) and DMS-0854934 (second author).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Craig A. Tracy

  2. Department of Mathematics, University of California, Santa Cruz, CA, 95064, USA

    Harold Widom

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  1. Craig A. Tracy
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Correspondence to Harold Widom.

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Tracy, C.A., Widom, H. On the Asymmetric Simple Exclusion Process with Multiple Species. J Stat Phys 150, 457–470 (2013). https://doi.org/10.1007/s10955-012-0531-9

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