Equivalence of ensembles for two-species zero-range invariant measures

Großkinsky, Stefan

doi:10.1016/j.spa.2007.09.006

Cited by 19 publications

(41 citation statements)

References 31 publications

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“…This happens because the rare event is most likely realised by a large deviation of the maximum component. A similar result was proved by Ferrari, Landim and Sisko [7] when the number of sites is fixed while the particle number grows to infinity, and by Großkinsky [10] for systems with two particle species.…”

Section: Introductionsupporting

confidence: 78%

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Thermodynamic limit for the invariant measures in supercritical zero range processes

Armendáriz

Loulakis

2008

Probab. Theory Relat. Fields

112

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We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Großkinsky, Schütz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk.

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Section: Introductionsupporting

confidence: 78%

“…Furthermore, it has been proved [10][11][12] that when the density is supercritical a condensation phenomenon emerges. Precisely, if ρ > ρ c and ε > 0 then…”

Section: Notation and Resultsmentioning

confidence: 99%

Thermodynamic limit for the invariant measures in supercritical zero range processes

Armendáriz

Loulakis

2008

Probab. Theory Relat. Fields

112

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“…Furthermore, the phase diagram associated with the corresponding equality constraints, i n i = MU and i n i /µ i = MV , is dramatically different from that of the inequality constraints, with two phases only rather than seven. Note, that this two dimensional condensation is very different in nature from that considered in the context of two, distinct, conserved quantities [21,22,23]. The two-constraint problem, in its general form, is the focus of the main part of the paper.…”

Section: Informal Illustration Of the Resultsmentioning

confidence: 98%

Bose–Einstein condensation in large deviations with applications to information systems

Merhav¹,

Kafri²

2010

J. Stat. Mech.

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Abstract. We study the large deviations behavior of systems that admit a certain form of a product distribution, which is frequently encountered both in Physics and in various information system models. First, to fix ideas, we demonstrate a simple calculation of the large deviations rate function for a single constraint (event). Under certain conditions, the behavior of this function is shown to exhibit an analogue of Bose-Einstein condensation (BEC). More interestingly, we also study the large deviations rate function associated with two constraints (and the extension to any number of constraints is conceptually straightforward). The phase diagram of this rate function is shown to exhibit as many as seven phases, and it suggests a two-dimensional generalization of the notion of BEC (or more generally, a multidimensional BEC). While the results are illustrated for a simple model, the underlying principles are actually rather general. We also discuss several applications and implications pertaining to information system models.

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“…We briefly give the definition of two-species ZRPs as Markov jump processes (Section A.1.2 in [19]) and their equilibrium states. Main references on this preliminary material are [12,13]. We take the discrete d-dimensional N -torus Ì d N as underlying lattice.…”

Section: The Particle Modelmentioning

confidence: 99%

Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles

2017

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Two-species condensing zero range processes (ZRPs) are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub-critical densities. We prove the hydrodynamic limit of nearest neighbour mean zero two-species condensing ZRP with bounded local jump rate for sub-critical initial profiles, i.e., for initial profiles whose image is contained in the region of sub-critical densities. The proof is based on H.T. Yau's relative entropy method, which relies on the existence of sufficiently regular solutions to the hydrodynamic equation. In the particular case of the species-blind ZRP, we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times.

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Equivalence of ensembles for two-species zero-range invariant measures

Cited by 19 publications

References 31 publications

Thermodynamic limit for the invariant measures in supercritical zero range processes

Thermodynamic limit for the invariant measures in supercritical zero range processes

Bose–Einstein condensation in large deviations with applications to information systems

Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles

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