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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

Citations:

Zbl 0725.60113
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References:

[1] E. D. Andjel, Invariant measures for the zero range process,Ann. Prob. 10:525–547 (1982). · Zbl 0492.60096
[2] L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).
[3] A. De Masi, C. Kipnis, E. Presutti, and E. Saada, Microscopic structure at the shock in the asymmetric simple exclusion,Stochastics 27:151–165 (1990). · Zbl 0679.60094
[4] B. Derrida, S. Janowsky, J. L. Lebowitz, and E. Speer, Exact solution of the totally asymmetric simple exclusion process: Shock profiles,J. Stat. Phys. 73(5/6):813–842. · Zbl 1102.60320
[5] W. Feller,An Introduction to Probability Theory and its Applications (Wiley, New York, 1968). · Zbl 0155.23101
[6] P. A. Ferrari, The simple exclusion process as seen from a tagged particle,Ann. Prob. 14:1277–1290 (1986). · Zbl 0628.60103
[7] P. A. Ferrari, Shock fluctuations in asymmetric simple exclusion,Proc. Theory Related Fields 91:81–101 (1992). · Zbl 0744.60117
[8] P. A. Ferrari and L. R. G. Fontes, Shock fluctuations in the asymmetric simple exclusion process,Prob. Theory Related Fields, to appear. · Zbl 0801.60094
[9] P.A. Ferrari, C. Kipnis, and E. Saada, Microscopic structure of travelling waves for the asymmetric simple exclusion process,Ann. Prob. 19:226–244 (1991). · Zbl 0725.60113
[10] R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics (Addison-Wesley, Reading, Massachusetts, 1989).
[11] W. Hoeffding, Probability inequalities for sums of bounded random variables,J. Am. Stat. Assoc. 58:13–30 (1963). · Zbl 0127.10602
[12] T. M. Liggett, Coupling the simple exclusion process,Ann. Prob. 4:339–356 (1976). · Zbl 0339.60091
[13] T. M. Liggett,Interacting Particle Systems (Springer-Verlag, Berlin, 1985). · Zbl 0559.60078
[14] T. Lindvall,Lectures on the Coupling Method (Wiley, 1992). · Zbl 0850.60019
[15] C. J. H. McDiarmid, On the method of bounded differences inSurveys in Combinatorics 1989, J. Siemons, ed. (Cambridge University Press, Cambridge, 1989), pp. 148–188.
[16] G. N. Raney, Functional composition patterns and power series reversion,Trans. AMS 94:441–451 (1960). · Zbl 0131.01402
[17] E. R. Speer, The two species asymmetric simple exclusion process, inMicro, Meso, and Macroscopic Approaches in Physics, M. Fannes, C. Maes, and A. Verbeure, eds. (Plenum, New York, 1994), pp. 91–102. · Zbl 0872.60084
[18] F. Spitzer,Principles of Random Walk, 2nd ed. (Springer-Verlag, Berlin, 1976). · Zbl 0359.60003
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