Advertisement

The Phase Diagram for a Multispecies Left-Permeable Asymmetric Exclusion Process

  • Arvind Ayyer
  • Caley Finn
  • Dipankar Roy
Article
  • 10 Downloads

Abstract

We study a multispecies generalization of a left-permeable asymmetric exclusion process (LPASEP) in one dimension with open boundaries. We determine all phases in the phase diagram using an exact projection to the LPASEP solved by us in a previous work. In most phases, we observe the phenomenon of dynamical expulsion of one or more species. We explain the density profiles in each phase using interacting shocks. This explanation is corroborated by simulations.

Keywords

Asymmetric exclusion process Left-permeable Multispecies Phase diagram Interacting shocks Dynamical expulsion 

Notes

Acknowledgements

We thank the referees for a number of useful suggestions. The first and third authors are supported by UGC Centre for Advanced Studies (Grant No. F. 510/25/CAS-II/2018(SAP-I)). The first author was also partly supported by Department of Science and Technology Grant EMR/2016/006624.

References

  1. 1.
    Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1d asymmetric exclusion model using a matrix formulation. J. Phys. A 26(7), 1493 (1993)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Blythe, R.A., Evans, M.R.: Nonequilibrium steady states of matrix-product form: a solver’s guide. J. Phys. A 40(46), R333 (2007)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Schadschneider, A.: Traffic flow: a statistical physics point of view. Phys. A 313(1), 153–187 (2002). (Fundamental Problems in Statistical Physics)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329(4), 199–329 (2000)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Penington, C.J., Hughes, B.D., Landman, K.A.: Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena. Phys. Rev. E 84, 041120 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Evans, M.R., Foster, D.P., Godrèche, C., Mukamel, D.: Asymmetric exclusion model with two species: spontaneous symmetry breaking. J. Stat. Phys. 80(1), 69–102 (1995)ADSCrossRefGoogle Scholar
  7. 7.
    Arita, C.: Phase transitions in the two-species totally asymmetric exclusion process with open boundaries. J. Stat. Mech. 2006(12), P12008 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Uchiyama, M.: Two-species asymmetric simple exclusion process with open boundaries. Chaos Solitons Fractals 35(2), 398–407 (2008)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ayyer, A., Lebowitz, J.L., Speer, E.R.: On the two species asymmetric exclusion process with semi-permeable boundaries. J. Stat. Phys. 135(5), 1009–1037 (2009)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ayyer, A., Lebowitz, J.L., Speer, Eugene R.: On some classes of open two-species exclusion processes. Markov Process. Relat. Fields 18(5), 157–176 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Crampe, N., Mallick, K., Ragoucy, E., Vanicat, M.: Open two-species exclusion processes with integrable boundaries. J. Phys. A 48(17), 175002 (2015)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Crampe, N., Evans, M.R., Mallick, K., Ragoucy, E., Vanicat, M.: Matrix product solution to a 2-species TASEP with open integrable boundaries. J. Phys. A 49(47), 475001 (2016)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Ayyer, A., Finn, C., Roy, D.: Matrix product solution of a left-permeable two-species asymmetric exclusion process. Phys. Rev. E 97, 012151 (2018)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Evans, M.R., Ferrari, P.A., Mallick, K.: Matrix representation of the stationary measure for the multispecies TASEP. J. Stat. Phys. 135(2), 217–239 (2009)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Prolhac, S., Evans, M.R., Mallick, K.: The matrix product solution of the multispecies partially asymmetric exclusion process. J. Phys. A 42(16), 165004 (2009)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Ferrari, P.A., Martin, J.B.: Multi-class processes, dual points and M/M/1 queues. Markov Process. Relat. Fields 12(2), 175–201 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ferrari, P.A., Martin, J.B.: Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab. 35(3), 807–832, 05 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ayyer, A., Linusson, S.: Correlations in the multispecies TASEP and a conjecture by lam. Trans. Am. Math. Soc. 369(2), 1097–1125 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cantini, L., Garbali, A., de Gier, J., Wheeler, M.: Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries. J. Phys. A 49(44), 444002 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ayyer, A., Roy, D.: The exact phase diagram for a class of open multispecies asymmetric exclusion processes. Sci. Rep. 7, 13555 (2017)ADSCrossRefGoogle Scholar
  21. 21.
    Crampe, N., Finn, C., Ragoucy, E., Vanicat, M.: Integrable boundary conditions for multi-species ASEP. J. Phys. A 49(37), 375201 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Uchiyama, M., Sasamoto, T., Wadati, M.: Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials. J. Phys. A 37(18), 4985 (2004)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia

Personalised recommendations