## 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.

### MSC:

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

Zbl 0725.60113
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### References:

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