Invariant measures for a two-species asymmetric process. (English) Zbl 0841.60085

Summary: 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 denity \(\rho\) and \(\lambda\) for the first- and first- plus second-class particles, respectively, was shown by the first author, C. Kipnis, and E. Saada [Ann. Probab. 19, No. 1, 226–244 (1991; Zbl 0725.60113)]. Recently B. Derrida, S. Janowsky, J. L. Lebowitz and E. Speer [J. Stat. Phys. 73, No. 5/6, 813–842 (1993)] 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 \(\rho\) and \(\lambda\) is equivalent to the product measure with densities \(\rho\) to the left of the origin and \(\lambda\) to the right of the origin.


82C22 Interacting particle systems in time-dependent statistical mechanics
60K40 Other physical applications of random processes


Zbl 0725.60113
Full Text: DOI


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