The Annals of Probability

Hydrodynamical limit for space inhomogeneous one-dimensional totally asymmetric zero-range processes

C. Landim

Abstract

We consider totally asymmetric attractive zero-range processes with bounded jump rates on Z. In order to obtain a lower bound for the large deviations from the hydrodynamical limit of the empirical measure, we perturb the process in two ways. We first choose a finite number of sites and slow down the jump rate at these sites. We prove a hydrodynamical limit for this perturbed process and show the appearance of Dirac measures on the sites where the rates are slowed down. The second type of perturbation consists of choosing a finite number of particles and making them jump at a slower rate. In these cases the hydrodynamical limit is described by nonentropy weak solutions of quasilinear first-order hyperbolic equations. These two results prove that the large deviations for asymmetric processes with bounded jump rates are of order at least $e^{-CN}$. All these results can be translated to the context of totally asymmetric simple exclusion processes where a finite number of particles or a finite number of holes jump at a slower rate.

Article information

Source
Ann. Probab., Volume 24, Number 2 (1996), 599-638.

Dates
First available in Project Euclid: 11 December 2002

https://projecteuclid.org/euclid.aop/1039639356

Digital Object Identifier
doi:10.1214/aop/1039639356

Mathematical Reviews number (MathSciNet)
MR1404522

Zentralblatt MATH identifier
0862.60095

Citation

Landim, C. Hydrodynamical limit for space inhomogeneous one-dimensional totally asymmetric zero-range processes. Ann. Probab. 24 (1996), no. 2, 599--638. doi:10.1214/aop/1039639356. https://projecteuclid.org/euclid.aop/1039639356

References

• [1] Andjel, E. D. (1982). Invariant measures for the zero range process. Ann. Probab. 10 525-547.
• [2] DiPerna, R. J. (1985). Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 223-270.
• [3] Guo, M. Z., Papanicolaou, G. C. and Varadhan, S. R. S. (1988). Nonlinear diffusion limit for a sy stem with nearest neighbor interactions. Comm. Math. Phy s. 118 31-59.
• [4] Kipnis, C. and L´eonard, C. (1991). Private communication.
• [5] Kru zkov, J. N. (1970). First order quasi-linear equations with several space variables. Math. USSR-Sb. 10 217-243.
• [6] Landim, C. (1993). Conservation of local equilibrium for asy mmetric attractive particle sy stems on Zd. Ann. Probab. 21 1782-1808.
• [7] Rezakhanlou, F. (1990). Hy drody namic limit for attractive particle sy stems on Zd. Comm. Math. Phy s. 140 417-448.
• [8] Wick, D. (1985). A dy namical phase transition in an infinite particle sy stem. J. Statist. Phy s. 38 1015-1025.