Abstract
We consider a process of two classes of particles jumping on a one-dimensional lattice. The marginal system of the first class of particles is the one-dimensional totally asymmetric simple exclusion process. When classes are disregarded the process is also the totally asymmetric simple exclusion process. The existence of a unique invariant measure with product marginals with density ρ and λ for the first- and first- plus second-class particles, respectively, was shown by Ferrari, Kipnis, and Saada. Recently Derrida, Janowsky, Lebowitz, and Speer have computed this invariant measure for finite boxes and performed the infinite-volume limit. Based on this computation we give a complete description of the measure and derive some of its properties. In particular we show that the invariant measure for the simple exclusion process as seen from a second-class particle with asymptotic densities ρ and λ is equivalent to the product measure with densities ρ to the left of the origin and λ to the right of the origin.
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Ferrari, P.A., Fontes, L.R.G. & Kohayakawa, Y. Invariant measures for a two-species asymmetric process. J Stat Phys 76, 1153–1177 (1994). https://doi.org/10.1007/BF02187059
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DOI: https://doi.org/10.1007/BF02187059