Shock fluctuations in asymmetric simple exclusion. (English) Zbl 0744.60117

The one-dimensional nearest neighbors asymmetric simple exclusion process is used as a microscopic approximation to the Burgers equation. We study the process with rates of jumps \(p>q\) to the right and left, respectively, and with initial product measure with densities \(\rho<\lambda\) to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density \((\rho=0)\) we prove, in the scale \(t^{1/2}\), that the position of the shock at time \(t\) depends only on the initial configuration in a region depending on \(t\). The proofs are based on laws of large numbers for the second class particle.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI


[1] [ab] Andjel, E.D., Bramson, M., Liggett, T.M.: Shoeks in the asymmetric simple exclusion process. Probab. Theory Relat. Fields78, 231-247 (1988) · Zbl 0632.60107
[2] [ak] Andjel, E.D., Kipnis, C.: Derivation of the hydrodynamical equations for the zero-range interaction process: a nonlinear Euler equation. Ann. Probab.12, 325-334 (1984) · Zbl 0536.60097
[3] [av] Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on ? J. Stat. Phys.47, 265-288 (1987) · Zbl 0685.58043
[4] [befg] Boldrighini, C., Cosimi, C., Frigio, A., Grasso-Nunes, M.: Computer simulations of shock waves in completely asymmetric simple exclusion process. J. Stat. Phys.55, 611-623 (1989)
[5] [b] Bramson, M.: Front propagation in certain one dimensional exelusion models. J. Stat. Phys.51, 863-869 (1988) · Zbl 1086.60529
[6] [df] De Masi, A., Ferrari, P.A.: Self difusion in one dimensional lattice gases in the presence of an external field. J. Stat. Phys.38, 603-613 (1985) · Zbl 0624.60117
[7] [dkps] De Masi, A., Kipnis, C., Presutti, E., Saada, E.: Microscopic structure at the shock in the asymmetric simple exclusion. Stochastics27, 151-165 (1988) · Zbl 0679.60094
[8] [f] Ferrari, P.A.: The simple exclusion process as seen from a tagged particle. Ann. Probab.14, 1277-1290 (1986) · Zbl 0628.60103
[9] [lks] Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves for asymmetric simple exclusion process. Ann. Probab.19, 226-244 (1991). · Zbl 0725.60113
[10] [gp] Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. Henri Poincare B53, 1-14 (1990) · Zbl 0705.76054
[11] [k] Kipnis, C.: Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab.14, 397-408 (1986) · Zbl 0601.60098
[12] [lps] Lebowitz, J.L., Presutti, E., Spohn, H.: Microscopic models of hydrodynamical behavior. J. Stat. Phys.51, 841-862 (1988) · Zbl 1086.60531
[13] [l] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab.4, 339-356 (1976) · Zbl 0339.60091
[14] [L] Liggett, T.M.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985 · Zbl 0559.60078
[15] [s] Saada, E.: A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab.15, 375-381 (1987) · Zbl 0617.60096
[16] [S] Spohn, H.: Large scale dynamics of interacting particles. Part B: Stochastic lattice gases. (Preprint, 1989)
[17] [vb] Van Beijeren, H.: Fluctuations in the motions of mass and of patterns in one-dimensional driven diffusive systems. J. Stat. Phys.63, 47-58 (1991)
[18] [w] Wick, D.: A dynamical phase transition in an infinite particle system. J. Stat. Phys.38, 1015-1025 (1985) · Zbl 0625.76080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.