TASEP hydrodynamics using microscopic characteristics. (English) Zbl 1434.60288

Ferrari, Pablo A.

Probab. Surv. 15, 1-27 (2018).

Summary: 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, H. Rost [Z. Wahrscheinlichkeitstheor. Verw. Geb. 58, 41–53 (1981; Zbl 0451.60097)] 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.

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

totally asymmetric simple exclusion process

Citations:

Zbl 0451.60097

References:

[1]	G. Amir, O. Angel, and B. Valkó. The TASEP speed process. Ann. Probab., 39(4):1205-1242, 2011. · Zbl 1225.82039
[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. · Zbl 1080.60089
[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. · Zbl 0632.60107
[4]	E. D. Andjel and M. E. Vares. Hydrodynamic equations for attractive particle systems on \(\mathbf{Z}\). J. Statist. Phys., 47(1-2):265-288, 1987. · Zbl 0685.58043
[5]	O. Angel. The stationary measure of a 2-type totally asymmetric exclusion process. J. Combin. Theory Ser. A, 113(4):625-635, 2006. · Zbl 1087.60067
[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. · Zbl 1101.60075
[7]	C. Bahadoran, H. Guiol, K. Ravishankar, and E. Saada. Strong hydrodynamic limit for attractive particle systems on \(\mathbb{Z}\). Electron. J. Probab., 15:no. 1, 1-43, 2010. · Zbl 1193.60113
[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. · Zbl 1139.60046
[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. · Zbl 1208.82036
[10]	A. Benassi and J.-P. Fouque. Hydrodynamical limit for the asymmetric simple exclusion process. Ann. Probab., 15(2):546-560, 1987. · Zbl 0623.60120
[11]	A. Benassi, J.-P. Fouque, E. Saada, and M. E. Vares. Asymmetric attractive particle systems on \(\mathbf{Z}\): 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. · Zbl 1414.90097
[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. · Zbl 0567.76006
[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. · Zbl 0679.60094
[15]	A. De Masi and E. Presutti. Mathematical methods for hydrodynamic limits, volume 1501 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991. · Zbl 0754.60122
[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. · Zbl 1102.60320
[17]	L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002
[18]	P. A. Ferrari. The simple exclusion process as seen from a tagged particle. Ann. Probab., 14(4):1277-1290, 1986. · Zbl 0628.60103
[19]	P. A. Ferrari. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields, 91(1):81-101, 1992. · Zbl 0744.60117
[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. · Zbl 0764.60105
[21]	P. A. Ferrari and L. R. G. Fontes. Shocks in asymmetric one-dimensional exclusion processes. Resenhas, 1(1):57-68, 1993. · Zbl 0849.60096
[22]	P. A. Ferrari and L. R. G. Fontes. Current fluctuations for the asymmetric simple exclusion process. Ann. Probab., 22(2):820-832, 1994. · Zbl 0806.60099
[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. · Zbl 0801.60094
[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. · Zbl 0855.60097
[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. · Zbl 0841.60085
[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. · Zbl 1196.60162
[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. · Zbl 0813.60095
[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. · Zbl 0725.60113
[29]	P. A. Ferrari and J. B. Martin. Stationary distributions of multi-type totally asymmetric exclusion processes. Ann. Probab., 35(3):807-832, 2007. · Zbl 1117.60089
[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. · Zbl 1185.60109
[31]	P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab., 33(4):1235-1254, 2005. · Zbl 1078.60083
[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. · Zbl 1118.82032
[33]	T. E. Harris. Additive set-valued Markov processes and graphical methods. Ann. Probability, 6(3):355-378, 1978. · Zbl 0378.60106
[34]	K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys., 209(2):437-476, 2000. · Zbl 0969.15008
[35]	C. Kipnis. Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab., 14(2):397-408, 1986. · Zbl 0601.60098
[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. · Zbl 0927.60002
[37]	C. Landim. Hydrodynamical equation for attractive particle systems on \(\mathbf{Z}^{d}\). Ann. Probab., 19(4):1537-1558, 1991. · Zbl 0798.60084
[38]	C. Landim. Hydrodynamical limit for asymmetric attractive particle systems on \(\mathbf{Z}^{d}\). Ann. Inst. H. Poincaré Probab. Statist., 27(4):559-581, 1991. · Zbl 0751.60097
[39]	C. Landim. Conservation of local equilibrium for attractive particle systems on \(\mathbf{Z}^{d}\). Ann. Probab., 21(4):1782-1808, 1993. · Zbl 0798.60085
[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. · Zbl 0268.35062
[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). · Zbl 1086.60531
[42]	T. M. Liggett. Ergodic theorems for the asymmetric simple exclusion process. Trans. Amer. Math. Soc., 213:237-261, 1975. · Zbl 0322.60086
[43]	T. M. Liggett. Coupling the simple exclusion process. Ann. Probability, 4(3):339-356, 1976. · Zbl 0339.60091
[44]	T. M. Liggett. Ergodic theorems for the asymmetric simple exclusion process. II. Ann. Probability, 5(5):795-801, 1977. · Zbl 0378.60104
[45]	T. M. Liggett. Interacting particle systems. Classics in Mathematics. Springer-Verlag, Berlin, 2005. Reprint of the 1985 original. · Zbl 1103.82016
[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. · Zbl 1069.60091
[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. · Zbl 1015.60093
[48]	F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on \(\mathbf{Z}^{d}\). Comm. Math. Phys., 140(3):417-448, 1991. · Zbl 0738.60098
[49]	F. Rezakhanlou. Evolution of tagged particles in non-reversible particle systems. Comm. Math. Phys., 165(1):1-32, 1994. · Zbl 0811.60094
[50]	F. Rezakhanlou. Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire, 12(2):119-153, 1995. · Zbl 0836.76046
[51]	H. Rost. Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete, 58(1):41-53, 1981. · Zbl 0451.60097
[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. · Zbl 0617.60096
[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. · Zbl 0947.60088
[57]	F. Spitzer. Interaction of Markov processes. Advances in Math., 5:246-290 (1970), 1970. · Zbl 0312.60060
[58]	W. D. Wick. A dynamical phase transition in an infinite particle system. J. Statist. Phys., 38(5-6):1015-1025, 1985. · Zbl 0625.76080