Grosskinsky, Stefan and Spohn, Herbert, 1946-. (2003) Stationary measures and hydrodynamics of zero range processes with several species of particles. Bulletin of the Brazilian Mathematical Society, Vol.34 (No.3). pp. 489-507. ISSN 1678-7714

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Abstract

We study general zero range processes with different types of particles on a ddimensional lattice with periodic boundary conditions. A necessary and sufficient condition on the jump rates for the existence of stationary product measures is established. For translation invariant jump rates we prove the hydrodynamic limit on the Euler scale using Yau’s relative entropy method. The limit equation is a system of conservation laws, which are hyperbolic and have a globally convex entropy. We analyze this system in terms of entropy variables. In addition we obtain stationary density profiles for open boundaries.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Particles -- Mathematical models, Stochastic systems
Journal or Publication Title:	Bulletin of the Brazilian Mathematical Society
Publisher:	Springer
ISSN:	1678-7714
Date:	2003
Volume:	Vol.34
Number:	No.3
Page Range:	pp. 489-507
Identification Number:	10.1007/s00574-003-0026-z
Status:	Peer Reviewed
Access rights to Published version:	Restricted or Subscription Access
Funder:	Institut Henri Poincaré, Deutscher Akademischer Austauschdienst (DAAD), Brazil. Coordenação do Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
References:	[1] F. Spitzer. Interaction of Markov processes. Adv. Math., 5:246–290, 1970. [2] E.D. Andjel. Invariant measures for the zero range process. Ann. Probability, 10:525– 547, 1982. [3] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems, Volume 320 of Grundlehren der mathematischen Wissenschaften. Springer Verlag, 1999. [4] T.M. Liggett and F. Spitzer. Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 56:443–468, 1981. [5] H.T. Yau. Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys, 22:63–80, 1991. [6] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on Zd. Commun. Math. Phys., 140:417–448, 1991. [7] J. Smoller. Shock Waves and Reaction Diffusion Equations. Springer, Berlin, 1994. [8] B. T´oth and B. Valk´o. Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws. to appear in J. Stat. Phys., math.PR/0210426, 2002. [9] J. Krug. Boundary induced phase transitions in driven diffusive systems. Phys. Rev. Lett., 67:1882–1885, 1991. [10] V. Popkov and G.M. Sch¨utz. Steady-state selection in driven diffusive systems with open boundaries. Europhys. Lett., 48:257–264, 1999. [11] M.R. Evans, D.P. Foster, C. Godr`eche, and D. Mukamel. Spontaneous symmetrybreaking in a one-dimensional driven diffusive system. Phys. Rev. Lett., 74:208, 1995. [12] M.R. Evans, Y. Kafri, H.M. Koduvely, and D. Mukamel. Phase separation in onedimensional driven diffusive systems. Phys. Rev. Lett., 80:425–429, 1998. [13] G.M. Sch¨utz. Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles. to appear in J. Phys. A, 2003. [14] V. Popkov and G.M. Sch¨utz. Shocks and excitation dynamics in a driven diffusive two-channel system. to appear in J. Stat. Phys., cond-mat/0211659, 2002. [15] B. T´oth and B. Valk´o. Between equilibrium fluctuations and Eulerian scaling. Perturbation of equilibrium for a class of deposition models. J. Stat. Phys., 109:177–205, 2002. [16] S. Ulbrich. Stabile Randbedingungen und implizite entropiedissipative numerische Verfahren f¨ur Anfangs-Randwert-Probleme mehrdimensionaler nichtlinearer Systeme von Erhaltungsgleichungen mit Entropie. PhD thesis, TU M¨unchen, 1996. [17] M.R. Evans. Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys., 30:42–57, 2000. [18] S. Großkinsky, G.M. Sch¨utz, and H. Spohn. Condensation in the zero range process: Stationary and dynamical properties. submitted to J. Stat. Phys., cond-mat/0302079, 2003. [19] H. Spohn. Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Berlin, 1991.
URI:	http://wrap.warwick.ac.uk/id/eprint/35913

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