Abstract
The zero-range process offers yet another example of a system for which potential-theoretic methods can be used to describe metastable behaviour. The free energy landscape is of a different nature than what we encountered in the models treated so far. In particular, there is no temperature parameter, and the key quantity to control is entropy. This necessitates a different approach to the choice of test functions to estimates capacities, which is worthwhile to expose. Sections 21.1–21.2 define the model and state the main results. Sections 21.3–21.5 provide the proofs.
The shop seemed to be full of all manner of curious things—but the oddest part of it all was that, whenever she looked at any shelf, to make out exactly what it had on it, that particular shelf was always quite empty, though the others round it were crowded as full as they could hold. (Lewis Carroll, Through the Looking-Glass, and what Alice found there)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andjel, E.D.: Invariant measures for the zero range processes. Ann. Probab. 10, 525–547 (1982)
Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains II, the nonreversible case. J. Stat. Phys. 149, 598–618 (2012)
Bianchi, A., Bovier, A., Ioffe, D.: Pointwise estimates and exponential laws in metastable systems via coupling methods. Ann. Probab. 40, 339–379 (2012)
Bovier, A., Neukirch, R.: A note on metastable behaviour in the zero-range process. In: Griebel, M. (ed.) Singular Phenomena and Scaling in Mathematical Models, pp. 365–376. Springer, Berlin (2013)
Evans, M.: Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys. 30, 42–57 (2000)
Großkinsky, S., Schütz, G.M.: Discontinuous condensation transition and nonequivalence of ensembles in a zero-range process. J. Stat. Phys. 132, 77–108 (2008)
Großkinsky, S., Schütz, G.M., Spohn, H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys. 113, 389–410 (2003)
Großkinsky, S., Spohn, H.: Stationary measures and hydrodynamics of zero range processes with several species of particles. Bull. Braz. Math. Soc. (N.S.), 489–507 (2003)
Landim, C.: Metastability for a non-reversible dynamics: The evolution of the condensate in totally asymmetric zero range processes. Commun. Math. Phys. 330, 1–32 (2014)
Neukirch, R.: Metastability in the Zero-range Process. Diploma thesis, Bonn University (2011)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bovier, A., den Hollander, F. (2015). The Zero-Range Process. In: Metastability. Grundlehren der mathematischen Wissenschaften, vol 351. Springer, Cham. https://doi.org/10.1007/978-3-319-24777-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-24777-9_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24775-5
Online ISBN: 978-3-319-24777-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)