Abstract
The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
REFERENCES
Y. Kafri, E. Levine, D. Mukamel, G. M. Schütz, and J. Török, Criterion for phase separation in one-dimensional driven systems, Phys. Rev. Lett. 89:035702(2002).
F. Spitzer, Interaction of markov processes, Adv. Math. 5:246–290 (1970).
E. D. Andjel, Invariant measures for the zero range process, Ann. Probab. 10:525–547 (1982).
C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Vol. 320 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, 1999).
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).
P. Bialas, Z. Burda, and D. Johnston, Condensation in the backgammon model, Nuclear Phys. B 493:505–516 (1997).
M. R. Evans, Phase transitions in one-dimensional nonequilibrium systems, Braz. J. Phys. 30:42–57 (2000).
V. Vinogradov, Refined Large Deviation Limit Theorems, Vol. 315 of Pitman Research Notes in Mathematics Series (Longman, Harlow, England, 1994).
A. Baltrunas and C. Klüppelberg, Subexponential distributions-large deviations with applications to insurance and queueing models, Aust. N. Z. J. Stat. (to appear March 2004).
C. Godrèche, Dynamics of condensation in zero-range processes, J. Phys. A: Math. Gen. 36:6313–6328 (2003).
G. M. Schütz, Exactly solvable models for many-body systems far from equilibrium, in Phase Transitions and Critical Phenomena, C. Domb and J. Lebowitz, eds., Vol. 19 (Academic Press, London, 2000), pp. 1–251.
D. Mukamel, Phase transitions in nonequilibrium systems, in Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow, M. E. Cates and M. R. Evans, eds. (Institute of Physics Publishing, Bristol, 2000).
Y. Kafri, E. Levine, D. Mukamel, G. M. Schütz, and R. D. Willmann, Novel phase-separation transition in one-dimensional driven models, cond-mat/0211269 (2002).
M. R. Evans, Y. Kafri, H. M. Koduvely, and D. Mukamel, Phase separation in one-dimensional driven diffusive systems, Phys. Rev. Lett. 80:425–429 (1998).
P. F. Arndt, T. Heinzel, and V. Rittenberg, Spontaneous breaking of translational invariance in one-dimensional stationary states on a ring, J. Phys. A 31:L45(1998).
B. Tóth and B. Valkó, Onsager relations and eulerian hydrodynamic limit for systems with several conservation laws, J. Stat. Phys. 112:497–521 (2003).
V. Popkov and G. M. Schütz, Shocks and excitation dynamics in a driven diffusive two-channel system, J. Stat. Phys. 112:523–540 (2003).
J. Krug, Boundary induced phase transitions in driven diffusive systems, Phys. Rev. Lett. 67:1882–1885 (1991).
A. B. Kolomeisky, G. M. Schütz, E. B. Kolomeisky, and J. P. Straley, Phase diagram of one-dimensional driven lattice gases with open boundaries, J. Phys. A: Math. Gen. 31:6911–6919 (1998).
V. Popkov and G. M. Schütz, Steady-state selection in driven diffusive systems with open boundaries, Europhys. Lett. 48:257–264 (1999).
M. R. Evans, D. P. Foster, C. Godrèche, and D. Mukamel, Spontaneous symmetry-breaking in a one-dimensional driven diffusive system, Phys. Rev. Lett. 74:208(1995).
A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50:221–260 (1978).
B. W. Gnedenko and A. N. Kolmogorov, Limit-Distributions for Sums of Independent Random Variables, Addison-Wesley Mathematics Series (Addison-Wesley, London, 1954).
C. M. Goldie and C. Klüppelberg, Subexponential distributions, in A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, R. Adler, R. Feldman, and M. S. Taqqu, eds. (Birkhäuser, Boston, 1998), pp. 435–459.
M. Abramowitz, Handbook of Mathematical Functions (Dover, New York, 1972).
F. Bardou, J. P. Bouchaud, A. Aspect, and C. Cohen-Tannoudi, Lévy Statistics and Laser Cooling, Chap. 4, (Cambridge University Press, Cambridge, 2002), pp. 42–59.
M. Barma and K. Jain, Locating the minimum: Approach to equilibrium in a disordered, symmetric zero range process, Pramana—J. Phys. 58:409–417 (2002).
K. P. N. Murthy and K. W. Kehr, Mean first-passage time of random walks on a random lattice, Phys. Rev. A 40:2082–2087 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Großkinsky, S., Schütz, G.M. & Spohn, H. Condensation in the Zero Range Process: Stationary and Dynamical Properties. Journal of Statistical Physics 113, 389–410 (2003). https://doi.org/10.1023/A:1026008532442
Issue Date:
DOI: https://doi.org/10.1023/A:1026008532442