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 Probability Surveys > Vol. 15 (2018) open journal systems 


TASEP hydrodynamics using microscopic characteristics

Pablo A. Ferrari, Universidad de Buenos Aires


Abstract
The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. Along the way we show laws of large numbers for tagged particles, fluxes and second class particles, and simplify existing proofs in the shock cases. The presentation is self contained.

AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35.

Keywords: Totally asymmetric simple exclusion process.

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Ferrari, Pablo A., TASEP hydrodynamics using microscopic characteristics, Probability Surveys, 15, (2018), 1-27 (electronic). DOI: 10.1214/17-PS284.

References

[1]    G. Amir, O. Angel, and B. Valkó. The TASEP speed process. Ann. Probab., 39(4):1205–1242, 2011. MR2857238

[2]    E. Andjel, P. A. Ferrari, and A. Siqueira. Law of large numbers for the simple exclusion process. Stochastic Process. Appl., 113(2):217–233, 2004. MR2087959

[3]    E. D. Andjel, M. D. Bramson, and T. M. Liggett. Shocks in the asymmetric exclusion process. Probab. Theory Related Fields, 78(2):231–247, 1988. MR0945111

[4]    E. D. Andjel and M. E. Vares. Hydrodynamic equations for attractive particle systems on Z. J. Statist. Phys., 47(1–2):265–288, 1987.

[5]    O. Angel. The stationary measure of a 2-type totally asymmetric exclusion process. J. Combin. Theory Ser. A, 113(4):625–635, 2006.

[6]    C. Bahadoran, H. Guiol, K. Ravishankar, and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab., 34(4):1339–1369, 2006. MR2257649

[7]    C. Bahadoran, H. Guiol, K. Ravishankar, and E. Saada. Strong hydrodynamic limit for attractive particle systems on . Electron. J. Probab., 15:no. 1, 1–43, 2010.

[8]    M. Balázs, E. Cator, and T. Seppäläinen. Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab., 11:no. 42, 1094–1132 (electronic), 2006.

[9]    G. Ben Arous and I. Corwin. Current fluctuations for TASEP: a proof of the Prähofer-Spohn conjecture. Ann. Probab., 39(1):104–138, 2011.

[10]    A. Benassi and J.-P. Fouque. Hydrodynamical limit for the asymmetric simple exclusion process. Ann. Probab., 15(2):546–560, 1987.

[11]    A. Benassi, J.-P. Fouque, E. Saada, and M. E. Vares. Asymmetric attractive particle systems on Z: hydrodynamic limit for monotone initial profiles. J. Statist. Phys., 63(3–4):719–735, 1991.

[12]    P. J. Burke. The output of a queuing system. Operations Res., 4:699–704 (1957), 1956.

[13]    A. De Masi, N. Ianiro, A. Pellegrinotti, and E. Presutti. A survey of the hydrodynamical behavior of many-particle systems. In Nonequilibrium phenomena, II, Stud. Statist. Mech., XI, pages 123–294. North-Holland, Amsterdam, 1984. MR0757003

[14]    A. De Masi, C. Kipnis, E. Presutti, and E. Saada. Microscopic structure at the shock in the asymmetric simple exclusion. Stochastics Stochastics Rep., 27(3):151–165, 1989.

[15]    A. De Masi and E. Presutti. Mathematical methods for hydrodynamic limits, volume 1501 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991.

[16]    B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R. Speer. Exact solution of the totally asymmetric simple exclusion process: shock profiles. J. Statist. Phys., 73(5–6):813–842, 1993. MR1251221

[17]    L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.

[18]    P. A. Ferrari. The simple exclusion process as seen from a tagged particle. Ann. Probab., 14(4):1277–1290, 1986.

[19]    P. A. Ferrari. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields, 91(1):81–101, 1992. MR1142763

[20]    P. A. Ferrari. Shocks in the Burgers equation and the asymmetric simple exclusion process. In Statistical physics, automata networks and dynamical systems (Santiago, 1990), volume 75 of Math. Appl., pages 25–64. Kluwer Acad. Publ., Dordrecht, 1992. MR1263704

[21]    P. A. Ferrari and L. R. G. Fontes. Shocks in asymmetric one-dimensional exclusion processes. Resenhas, 1(1):57–68, 1993.

[22]    P. A. Ferrari and L. R. G. Fontes. Current fluctuations for the asymmetric simple exclusion process. Ann. Probab., 22(2):820–832, 1994.

[23]    P. A. Ferrari and L. R. G. Fontes. Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Related Fields, 99(2):305–319, 1994.

[24]    P. A. Ferrari and L. R. G. Fontes. Poissonian approximation for the tagged particle in asymmetric simple exclusion. J. Appl. Probab., 33(2):411–419, 1996.

[25]    P. A. Ferrari, L. R. G. Fontes, and Y. Kohayakawa. Invariant measures for a two-species asymmetric process. J. Statist. Phys., 76(5–6):1153–1177, 1994.

[26]    P. A. Ferrari, P. Gonçalves, and J. B. Martin. Collision probabilities in the rarefaction fan of asymmetric exclusion processes. Ann. Inst. Henri Poincaré Probab. Stat., 45(4):1048–1064, 2009. MR2572163

[27]    P. A. Ferrari and C. Kipnis. Second class particles in the rarefaction fan. Ann. Inst. H. Poincaré Probab. Statist., 31(1):143–154, 1995.

[28]    P. A. Ferrari, C. Kipnis, and E. Saada. Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab., 19(1):226–244, 1991.

[29]    P. A. Ferrari and J. B. Martin. Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab., 35(3):807–832, 2007.

[30]    P. A. Ferrari, J. B. Martin, and L. P. R. Pimentel. A phase transition for competition interfaces. Ann. Appl. Probab., 19(1):281–317, 2009.

[31]    P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab., 33(4):1235–1254, 2005.

[32]    P. L. Ferrari and H. Spohn. Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys., 265(1):1–44, 2006.

[33]    T. E. Harris. Additive set-valued Markov processes and graphical methods. Ann. Probability, 6(3):355–378, 1978. MR0488377

[34]    K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys., 209(2):437–476, 2000.

[35]    C. Kipnis. Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab., 14(2):397–408, 1986.

[36]    C. Kipnis and C. Landim. Scaling limits of interacting particle systems, volume 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.

[37]    C. Landim. Hydrodynamical equation for attractive particle systems on Zd. Ann. Probab., 19(4):1537–1558, 1991.

[38]    C. Landim. Hydrodynamical limit for asymmetric attractive particle systems on Zd. Ann. Inst. H. Poincaré Probab. Statist., 27(4):559–581, 1991.

[39]    C. Landim. Conservation of local equilibrium for attractive particle systems on Zd. Ann. Probab., 21(4):1782–1808, 1993.

[40]    P. D. Lax. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.

[41]    J. L. Lebowitz, E. Presutti, and H. Spohn. Microscopic models of hydrodynamic behavior. J. Statist. Phys., 51(5–6):841–862, 1988. New directions in statistical mechanics (Santa Barbara, CA, 1987).

[42]    T. M. Liggett. Ergodic theorems for the asymmetric simple exclusion process. Trans. Amer. Math. Soc., 213:237–261, 1975. MR0410986

[43]    T. M. Liggett. Coupling the simple exclusion process. Ann. Probability, 4(3):339–356, 1976.

[44]    T. M. Liggett. Ergodic theorems for the asymmetric simple exclusion process. II. Ann. Probability, 5(5):795–801, 1977. MR0445644

[45]    T. M. Liggett. Interacting particle systems. Classics in Mathematics. Springer-Verlag, Berlin, 2005. Reprint of the 1985 original.

[46]    T. Mountford and H. Guiol. The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab., 15(2):1227–1259, 2005.

[47]    M. Prähofer and H. Spohn. Current fluctuations for the totally asymmetric simple exclusion process. In In and out of equilibrium (Mambucaba, 2000), volume 51 of Progr. Probab., pages 185–204. Birkhäuser Boston, Boston, MA, 2002.

[48]    F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys., 140(3):417–448, 1991.

[49]    F. Rezakhanlou. Evolution of tagged particles in non-reversible particle systems. Comm. Math. Phys., 165(1):1–32, 1994.

[50]    F. Rezakhanlou. Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire, 12(2):119–153, 1995. MR1326665

[51]    H. Rost. Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete, 58(1):41–53, 1981.

[52]    E. Saada. A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab., 15(1):375–381, 1987.

[53]    T. Seppalainen. Translation invariant exclusion processes. Available at https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf (2015/11/24).

[54]    T. Seppäläinen. Coupling the totally asymmetric simple exclusion process with a moving interface. Markov Process. Related Fields, 4(4):593–628, 1998. I Brazilian School in Probability (Rio de Janeiro, 1997).

[55]    T. Seppäläinen. Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields, 4(1):1–26, 1998.

[56]    T. Seppäläinen. Existence of hydrodynamics for the totally asymmetric simple K-exclusion process. Ann. Probab., 27(1):361–415, 1999.

[57]    F. Spitzer. Interaction of Markov processes. Advances in Math., 5:246–290 (1970), 1970. MR0268959

[58]    W. D. Wick. A dynamical phase transition in an infinite particle system. J. Statist. Phys., 38(5–6):1015–1025, 1985.




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