Electronic Journal of Probability

Diffusion and Scattering of Shocks in the Partially Asymmetric Simple Exclusion Process

Vladimir Belitsky and Gunter Schütz

Full-text: Open access

Abstract

We study the behavior of shocks in the asymmetric simple exclusion process on $Z$ whose initial distribution is a product measure with a finite number of shocks. We prove that if the particle hopping rates of this process are in a particular relation with the densities of the initial measure then the distribution of this process at any time is a linear combination of shock measures of the structure similar to that of the initial distribution. The structure of this linear combination allows us to interpret this result by saying that the shocks of the initial distribution perform continuous time random walks on $Z$ interacting by the exclusion rule. We give explicit expressions for the hopping rates of these random walks. The result is derived with a help of quantum algebra technique. We made the presentation self-contained for the benefit of readers not acquainted with this approach, but interested in applying it in the study of interacting particle systems.

Article information

Source
Electron. J. Probab., Volume 7 (2002), paper no. 11, 21 pp.

Dates
Accepted: 21 February 2002
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1463434884

Digital Object Identifier
doi:10.1214/EJP.v7-110

Mathematical Reviews number (MathSciNet)
MR1902844

Zentralblatt MATH identifier
1017.60097

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82C23: Exactly solvable dynamic models [See also 37K60]

Keywords
Asymmetric simpleexclusion process evolution of shock measures quantum algebra

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Belitsky, Vladimir; Schütz, Gunter. Diffusion and Scattering of Shocks in the Partially Asymmetric Simple Exclusion Process. Electron. J. Probab. 7 (2002), paper no. 11, 21 pp. doi:10.1214/EJP.v7-110. https://projecteuclid.org/euclid.ejp/1463434884


Export citation

References

  • A. Boldrighini, C. Cosimi, A. Frigio, M. Grasso-Nunes, Computer simulations of shock waves in the completely asymmetric simple exclusion process, J. Stat. Phys.55, (1989), 611-623.
  • B. Derrida, S. A. Janowsky, J. L. Lebowitz, E. R. Speer, Exact solution of the totally asymmetric simple exclusion process: Shock profiles, J. Stat. Phys.73 (1993), 813-842.
  • B. Derrida, J. L. Lebowitz, E. R. Speer, Shock profiles in the asymmetric simple exclusion process in one dimension, J. Stat. Phys.89 (1997), 135-167.
  • W. Feller, Introduction to the probability theory and its applications. Vol. I, Wiley P, 1968 (3-d edition).
  • P. A. Ferrari, Shocks in one-dimensional processes with drift. In: Probability and Phase Transition. (Ed. G. Grimmett), Cambridge, 1993.
  • P. A. Ferrari, L. R. G. Fontes, Shock fluctuations in the asymmetric simple exclusion process, Probab. Theor. Rel. Fields 99, (1994), 305-319.
  • P. A. Ferrari, L. R. G. Fontes and M. E. Vares, The asymmetric simple exclusion model with multiple shocks, Ann. Inst. Henri Poincaré, Probabilités et Statistiques 36, 2 (2000) 109-126.
  • P. A. Ferrari, C. Kipnis, S. Saada, Microscopic structure of traveling waves in the asymmetric simple exclusion process, Ann. Prob. 19, No.1 (1991), 226-244.
  • A. N. Kirillov, N. Yu. Reshetikhin. In: Proceedings of the 1988 Luminy Conference on Infinite-Dimensional Lie Algebras and Groups, V. G. Kac (ed.), World Scientific, Singapore (1988).
  • A. B. Kolomeisky, G. M. Schütz, E. B. Kolomeisky and J. P. Straley, Phase diagram of one-dimensional driven lattice gases with open boundaries, J. Phys. A 31 (1998), 6911-6919.
  • K. Krebs, Ph.D. thesis, University of Bonn (2001).
  • T. M. Liggett, Interacting Particle Systems Springer, Berlin (1985).
  • T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999).
  • V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B 330 (1990), 523–556.
  • G. M. Schütz, Duality relations for asymmetric exclusion process, J. Stat. Phys. 86, Nos. 5/6 (1997), 1265-1287.
  • G. M. Schütz, Exactly solvable models for many-body systems far from equilibrium,in Phase Transitions and Critical Phenomena. Vol. 19, Eds. C. Domb and J. Lebowitz, Academic Press, London, (2000).
  • G. M. Schütz, Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427-445.