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
Many authors consider ASEP on the circle or the lattice [1,L] with open boundary conditions.
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.
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).
References
Alcaraz, F.C., Bariev, R.Z.: Exact solution of asymmetric diffusion with N classes of particles of arbitrary size and hierarchical order. Braz. J. Phys. 30, 655–666 (2000)
Alcaraz, F.C., Droz, M., Henkel, M., Rittenberg, V.: Reaction-diffusion processes, critical dynamics and quantum chains. Ann. Phys. 230, 250–302 (1994)
Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math. 64, 0466–0537 (2011)
Arita, C., Kuniba, A., Sakai, K., Sawabe, T.: Spectrum of a multi-species asymmetric simple exclusion process on a ring. J. Phys. A, Math. Theor. 42, 345002 (2009) (41 pp.)
Arita, C., Ayyer, A., Mallick, K., Prolhac, S.: Recursive structures in multispecies TASEP. J. Phys. A, Math. Theor. 44, 335004 (2011) (27 pp.)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)
Cantini, L.: Algebraic Bethe ansatz for the two species ASEP with different hopping rates. J. Phys. A, Math. Theor. 41, 095001 (2008) (16 pp.)
Derrida, B., Evans, M.R.: Bethe ansatz solution for a defect particle in the asymmetric exclusion process. J. Phys. A, Math. Gen. 32, 4833 (1999) (18 pp.)
Ferrari, P.A., Fontes, L.R.G., Kohayakawa, Y.: Invariant measures for a two-species asymmetric process. J. Stat. Phys. 76, 1153–1177 (1994)
Golinelli, O., Mallick, K.: The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. J. Phys. A, Math. Gen. 39, 12679–12705 (2006)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976)
Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)
Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (2005). [Reprint of the 1985 original]
Perk, J.H.H., Schultz, C.L.: Families of commuting transfer matrices in q-state vertex models. In: Jimbo, M., Miwa, T. (eds.) Non-linear Integrable Systems: Classical Theory and Quantum Theory, pp. 137–152. World Scientific, Singapore (1983)
Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834, 523–542 (2010)
Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008)
Tracy, C.A., Widom, H.: Erratum to “Integral formulas for the asymmetric simple exclusion process”. Commun. Math. Phys. 304, 875–878 (2011)
Wehefritz-Kaufmann, B.: Dynamical critical exponent for two-species totally asymmetric diffusion on a ring. SIGMA 6, 039 (2010) (15 pp.)
Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interactions. Phys. Rev. Lett. 19, 1312–1315 (1967)
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This work was supported by the National Science Foundation through grants DMS-0906387 (first author) and DMS-0854934 (second author).
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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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DOI: https://doi.org/10.1007/s10955-012-0531-9