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Free zero-range processes on networks

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Original languageEnglish
Title of host publicationNoise and Stochastics in Complex Systems and Finance
EditorsJ Kertesz, S Bornholdt, RN Mantegna
Place of PublicationBELLINGHAM
PublisherSPIE
PagesU224-U234
Number of pages11
ISBN (Print)978-0-8194-6738-6
DOIs
Publication statusPublished - 2007
EventConference on Noise and Stochastics in Complex Systems and Finance - Florence
Duration: 21 May 200724 May 2007

Conference

ConferenceConference on Noise and Stochastics in Complex Systems and Finance
CityFlorence
Period21/05/0724/05/07

Abstract

A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a) FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes during the process in a way which depends on the current distribution of particles. The case (a) provides a very simple realization of the phenomenon of condensation which manifests as the appearance of a condensate of particles on the node with maximal degree. A particularly interesting example is the condensation on scale-free networks. Here we will model it by introducing a single-site inhomogeneity to a k-regular network. This simplified situation can be easily treated analytically and, on the other hand, shows quantitatively the same behavior as in the case of scale-free networks. The case (b) is very interesting since the averaging over typical ensembles of graphs acts as a kind of homogenization of the system which makes all nodes identical from the point of view of the FZRP. In effect, the partition function of the steady state becomes invariant with respect to the permutations of the particle occupation numbers. This type of symmetric systems has been intensively studied in the literature. In particular, they undergo a phase transition to the condensed phase, which is caused by a mechanism of spontaneous symmetry breaking. In the case (c), the distribution of particles and the dynamics of network are coupled to each other. The strength of this coupling depends on the ratio of two time scales: for changes of the topology and of the FZRP. We will discuss a specific example of that type of interaction and show that it leads to an interesting phase diagram. The case (b) mentioned above can be viewed as a limiting case where the typical time scale of topology fluctuations is much larger than that of the FZRP.

    Research areas

  • balls-in-boxes model, zero-range process, condensation, dynamical rewirings, FACTORIZED STEADY-STATES, STATISTICAL-MECHANICS, CONDENSATION, MODEL

Event

Conference on Noise and Stochastics in Complex Systems and Finance

21/05/0724/05/07

Florence

Event: Conference

ID: 1216971