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The two species totally asymmetric simple exclusion process. (English) Zbl 0872.60084

Fannes, Mark (ed.) et al., On three levels. Micro-, meso-, and macro-approaches in physics. Proceedings of a NATO Advanced Research Workshop, Leuven, Belgium, July 19–23, 1993. New York, NY: Plenum Press. NATO ASI Ser., Ser. B, Phys. 324, 91-102 (1994).
The two species totally asymmetric simple exclusion process [see B. Derrida, S. A. Janowsky, J. L. Lebowitz and the author, J. Stat. Phys. 73 (1993) and Europhys. Lett. 22, 651-656 (1993)], or two species TASEP, is an interacting particle system [see T. M. Liggett, “Interacting particle systems” (1985; Zbl 0559.60078) and references therein] in which two types of particles, first class and second class, live on the sites of a one-dimensional lattice, hopping at random times to the adjacent site to their right; the model is called totally asymmetric because particles can jump only to the right, while more general ASEP models permit jumps in both directions, with a preference for one or the other. Here we study steady states – time invariant measures on the space of all configurations – for this model, and in particular discuss a family of explicitly computable translation invariant steady states on the infinite lattice; we show that these are precisely the extremal members of the set of all translation invariant steady states for the system. The family is parametrized by the densities \(\rho_1\) and \(\rho_2\) of the two species, and includes states with all densities lying within the triangle \(\rho_1\), \(\rho_2 \geq 0\), \(\rho_1+ \rho_2\leq 1\). On the boundary of this triangle the model reduces to the standard one species TASEP [see Liggett, loc. cit.] and the states constructed here reduce to the product states invariant for this simpler model.
For the entire collection see [Zbl 0845.00058].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 0559.60078
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