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

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## 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.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)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 3.Schadschneider, A.: Traffic flow: a statistical physics point of view. Phys. A
**313**(1), 153–187 (2002). (Fundamental Problems in Statistical Physics)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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)ADSCrossRefzbMATHGoogle Scholar - 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.Uchiyama, M.: Two-species asymmetric simple exclusion process with open boundaries. Chaos Solitons Fractals
**35**(2), 398–407 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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.Crampe, N., Mallick, K., Ragoucy, E., Vanicat, M.: Open two-species exclusion processes with integrable boundaries. J. Phys. A
**48**(17), 175002 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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.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)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 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.Ferrari, P.A., Martin, J.B.: Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab.
**35**(3), 807–832, 05 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Ayyer, A., Linusson, S.: Correlations in the multispecies TASEP and a conjecture by lam. Trans. Am. Math. Soc.
**369**(2), 1097–1125 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Crampe, N., Finn, C., Ragoucy, E., Vanicat, M.: Integrable boundary conditions for multi-species ASEP. J. Phys. A
**49**(37), 375201 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar