Euler hydrodynamics of one-dimensional attractive particle systems.(English)Zbl 1101.60075

Summary: We consider attractive irreducible conservative particle systems on $$\mathbb Z$$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of the authors and E. Saada [Stochastic Processes. Appl. 99, 1–30 (2002; Zbl 1058.60084)], from which we relax the assumption that the process has explicit invariant measures.

MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics

Zbl 1058.60084
Full Text:

References:

 [1] Andjel, E. D. (1982). Invariant measures for the zero range process. Ann. Probab. 10 525–547. · Zbl 0492.60096 [2] Andjel, E. D. and Vares, M. E. (1987). Hydrodynamic equations for attractive particle systems on $$\mathbbZ$$. J. Statist. Phys. 47 265–288. · Zbl 0685.58043 [3] Bahadoran, C. (2004). Blockage hydrodynamics of one-dimensional driven conservative systems. Ann. Probab. 32 805–854. · Zbl 1079.60076 [4] Bramson, M. and Mountford, T. (2002). Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 1082–1130. · Zbl 1042.60062 [5] Bahadoran, C., Guiol, H., Ravishankar, K. and Saada, E. (2002). A constructive approach to Euler hydrodynamics for attractive particle systems. Application to $$k$$-step exclusion. Stochastic Process. Appl. 99 1–30. · Zbl 1058.60084 [6] Bressan, A. (2000). Hyperbolic Systems of Conservation Laws : The One-Dimensional Cauchy Problem . Oxford Univ. Press. · Zbl 0997.35002 [7] Cocozza-Thivent, C. (1983). Processus attractifs sur $$\mathbbN^\mathbbZ^d$$. Rapport du laboratoire de probabilités, Univ. Paris VI. [8] Cocozza-Thivent, C. (1985). Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 509–523. · Zbl 0554.60097 [9] Ekhaus, M. and Gray, L. (1994). Convergence to equilibrium and a strong law for the motion of restricted interfaces. Unpublished manuscript. [10] Glimm, J. (1965). Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 697–715. · Zbl 0141.28902 [11] Godlewski, E. and Raviart, P. A. (1991). Hyperbolic Systems of Conservation Laws . Ellipses, Paris. · Zbl 0768.35059 [12] Guiol, H. (1999). Some properties of $$k$$-step exclusion processes. J. Statist. Phys. 94 495–511. · Zbl 0953.60091 [13] Keisling, J. D. (1998). An ergodic theorem for the symmetric generalized exclusion process. Markov Process. Related Fields 4 351–379. · Zbl 0926.60061 [14] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems . Springer, Berlin. · Zbl 0927.60002 [15] Kružkov, S. N. (1970). First order quasilinear equations with several independent variables. Math. URSS Sb. 10 217–243. · Zbl 0215.16203 [16] Landim, C. (1993). Conservation of local equilibrium for attractive particle systems on $$\mathbbZ^d$$. Ann. Probab. 21 1782–1808. · Zbl 0798.60085 [17] Lax, P. D. (1957). Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 537–566. · Zbl 0081.08803 [18] Liggett, T. M. (1976). Coupling the simple exclusion process. Ann. Probab. 4 339–356. JSTOR: · Zbl 0339.60091 [19] Liggett, T. M. (1985). Interacting Particle Systems . Springer, New York. · Zbl 0559.60078 [20] Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on $$\mathbbZ^d$$. Comm. Math. Phys. 140 417–448. · Zbl 0738.60098 [21] Rezakhanlou, F. (2001). Continuum limit for some growth models. II. Ann. Probab. 29 1329–1372. · Zbl 1081.82016 [22] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis . Springer, Berlin. · Zbl 0888.49001 [23] Seppäläinen, T. (1999). Existence of hydrodynamics for the totally asymmetric simple $$K$$-exclusion process. Ann. Probab. 27 361–415. · Zbl 0947.60088 [24] Seppäläinen, T. (2006). Translation Invariant Exclusion Processes . Available at http://www.math.wisc.edu/ seppalai/excl-book/etusivu.html. [25] Serre, D. (1999). Systems of Conservation Laws. 1 . Hyperbolicity , Entropies , Shock Waves . (Translated from the 1996 French original by I. N. Sneddon.) Cambridge Univ. Press. · Zbl 0930.35001 [26] Vol’pert, A. I. (1967). The spaces $$\mathrmBV$$ and quasilinear equations. Math. Sb. ( N.S. ) 73 ( 115 ) 225–302. · Zbl 0168.07402
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.