Blockage hydrodynamics of one-dimensional driven conservative systems. (English) Zbl 1079.60076

Author’s abstract: We consider an arbitrary one-dimensional conservative particle system with finite-range interactions and finite site capacity, governed on the hydrodynamic scale by a scalar conservation law with Lipshitz-continuous flux \(h\). A finite-size perturbation restricts the local current to some maximum value \(\varphi\). We show that the perturbed hydrodynamic behaviour is entirely determined by \(\varphi\) if \(\inf (h;\varphi)\) is first non-decreasing and then non-increasing, which we believe is a necessary condition.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
35L65 Hyperbolic conservation laws
Full Text: DOI


[1] Andjel, E. D. (1981). The asymmetric exclusion process on \(\ZZ^d\). Z. Wahrsch. Verw. Gebiete 58 423–432. · Zbl 0458.60097
[2] Bahadoran, C. (1997). Hydrodynamique des processus de misanthropes spatialement hétérogènes. Thèse de doctorat, Ecole Polytechnique.
[3] Bahadoran, C. (1998). Hydrodynamic limit for spatially heterogeneous simple exclusion processes. Probab. Theory Related Fields 110 287–331. · Zbl 0929.60082
[4] Bahadoran, C., Guiol, H., Ravishankhar, K. and Saada, E. (2002). A constructive approach to Euler hydrodynamics for attractive particle systems. Application to \(k\)-step exclusion. Stochastic Process. Appl. 99 . · Zbl 1058.60084
[5] Ballou, D. P. (1970). Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans. Amer. Math. Soc. 152 441–460. · Zbl 0207.40401
[6] Bramson, M. and Mountford, T. (2002). Stationary blocking measures for the asymmetric exclusion process. Ann. Probab. 30 1082–1130. · Zbl 1042.60062
[7] Chowdhury, D., Santen, S. and Schadschneider, A. (2000). Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329 199–329.
[8] Cocozza, C. (1985). Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 509–523. · Zbl 0554.60097
[9] Covert, P. and Rezakhanlou, F. (1997). Hydrodynamic limit for particle systems with nonconstant speed parameter. J. Statist. Phys. 88 383–426. · Zbl 0939.82032
[10] Di Perna, R. (1984). Measure-valued solutions to conservation laws. Arch. Rat. Mech. Anal. 223–270. · Zbl 0616.35055
[11] Guiol, H. (1999). Some properties of \(k\)-step exclusion process. J. Statist. Phys. 94 495–511. · Zbl 0953.60091
[12] Janowski, S. A. and Lebowitz, J. L. (1994). Exact results for the asymmetric simple exclusion process with a blockage. J. Statist. Phys. 77 . · Zbl 0838.60088
[13] Jensen, L. (2000). Large deviations of the asymmetric simple exclusion process in one dimension. Ph.D. dissertation, New York Univ.
[14] Katz, S., Lebowitz, J. L. and Spohn, H. (1984). Stationary nonequilibrium states for stochastic lattice gas models of ionic superconductors. J. Statist. Phys. 34 497–537.
[15] Kingman, J. F. C. (1968). The ergodic theory of subadditive processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 30 499–510. · Zbl 0182.22802
[16] Kipnis, C. and Landim, C. (1999). Scaling Limits of Infinite Particle Systems . Springer, New York. · Zbl 0927.60002
[17] Kružkov, N. (1970). First order quasilinear equations in several independant variables. Math. USSR Sb. 10 217–243. · Zbl 0215.16203
[18] Landim, C. (1996). Hydrodynamical limit for space inhomogeneous one dimension totally asymmetric zero-range processes. Ann. Probab. 24 599–638. · Zbl 0862.60095
[19] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York. · Zbl 0559.60078
[20] Mallick, K. (1996). Shocks in the asymmetric exclusion model with an impurity. J. Phys. A 29 5375–5386. · Zbl 0905.60087
[21] Popkov, V., Krug, J. and Schütz, G. (2001). Minimal current phase and boundary layers in driven diffusive systems. Phys. Rev. E 63 56–110.
[22] Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on \(\mathbf Z^d\). Comm. Math. Phys. 140 417–448. · Zbl 0738.60098
[23] Rezakhanlou, F. (2001). Continuum limit for some growth models II. Ann. Probab. 29 1329–1372. · Zbl 1081.82016
[24] Rezakhanlou, F. (2002). Continuum limit for some growth models. Stochastic Process. Appl. 101 1–41. · Zbl 1075.82011
[25] Schütz, G. (1993). Generalized Bethe Ansatz solution of a one-dimensional asymmetric exclusion process on a ring with blockage. J. Statist. Phys. 71 471–505. · Zbl 0943.82558
[26] Seppäläinen, T. (1999). Existence of hydrodynamics for the \(K\)-exclusion process. Ann. Probab. 27 361–415. · Zbl 0947.60088
[27] Seppäläinen, T. (2000). A variational coupling for a totally asymmetric exclusion process with long jumps but no passing. In Hydrodynamic Limits and Related Topics (S. Feng, A. T. Lawniczak and S. R. S. Varadhan, eds.) 117–130. Amer. Math. Soc., Providence, RI. · Zbl 1060.82513
[28] Seppäläinen, T. (2001). Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond. J. Statist. Phys. 102 69–96. · Zbl 1036.82018
[29] Szepessy, A. (1989). An existence result for scalar conservation laws using measure-valued solutions. Comm. Partial Differential Equations 14 1329–1350. · Zbl 0704.35022
[30] Vol’pert, A. I. (1967). The spaces BV and quasilinear equations. Math. USSR Sb. 2 225–266. · 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.