## Hydrodynamic limit for attractive particle systems on $$\mathbb{Z}{}^ d$$.(English)Zbl 0738.60098

Summary: We study the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions. Under Euler scaling, a nonlinear conservation law is derived for the time evolution of the macroscopic particle density.

### MSC:

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

### References:

 [1] Andjel, E. D.: Invariant measures for the zero range process. Ann. Prob.10, 525–547 (1982) · Zbl 0492.60096 [2] Andjel, E. D., Kipnis, C.: Derivation of the hydrodynamical equation for the zero range interaction process. Ann. Prob.12, 325–334 (1984) · Zbl 0536.60097 [3] Andjel, E. D., Vares, M. E.: Hydrodynamic equation for attractive particle systems on $$\mathbb{Z}$$. J. Stat. Phys.47, 265–288 (1987) · Zbl 0685.58043 [4] Benassi, A., Fouque, J. P.: Hydrodynamical limit for the asymmetric simple exclusion process. Ann. Prob.15, 546–560 (1987) · Zbl 0623.60120 [5] Benassi, A., Fouque, J. P.: Hydrodynamical limit for the asymmetric zero range process. Ann. Inst. Henri Poincaré 189–200 (1988) · Zbl 0646.60038 [6] Cocozza, C. T.: Processus des misanthropes. Z. Wahrs. Verw. Gebiete70, 509–523 (1985) · Zbl 0554.60097 [7] Crandall, M.: The semigroup approach to first-order quasilinear equations in several space variables. Israel J. Math12, 108–132 (1972) · Zbl 0246.35018 [8] DeMasi, A., Ianiro, N., Pellegrinotti, A., Presutti, E.: A survey of the hydrodynamical behavior of many particle systems. In: Studies in Statistical Mechanics, Vol11, Non Equilibrium Phenomena II, from Stochastics to Hydrodynamics. Amsterdam: North-Holland (1984) [9] DiPerna, R. J.: Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal.88, 223–270 (1985) · Zbl 0616.35055 [10] Guo, M. Z., Papanicolaou, G. C., Varadhan, S. R. S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys.118, 31–59 (1988) · Zbl 0652.60107 [11] Kružkov, S. N.: First order quasilinear equations in several independent variables. Math. USSR-Sb.10, 217–243 (1970) · Zbl 0215.16203 [12] Landim, C.: Hydrodynamical equation for attractive particle systems in $$\mathbb{Z}$$ d , preprint · Zbl 0798.60084 [13] Liggett, T. M.: Interacting particle systems. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0559.60078 [14] Rost, H.: Non-equilibrium behavior of a many particle process: density profile and local equilibria. Z. Wahrs. Verw. Gebiete58, 41–43 (1981) · Zbl 0451.60097 [15] Varadhan, S. R. S.: Scaling limits for interacting diffusions, preprint · Zbl 0725.60085 [16] Wick, W. D.: Entropy arguments in the study of the hydrodynamical limit. CARR reports in Mathematical Physics 3/88, May 1988 [17] Benassi, A., Fouque, J. P., Saada, E., Vares, M. E.: Asymmetric attractive particle systems on Z: Hydrodynamical limit for monotone initial profiles, preprint
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.