In this thesis a simple, stochastic, interacting particle system – the zero-range process (ZRP) – is studied with various analytical and numerical methods. in particular, the application of the ZRP and some of its generalisations to complex networks is focused upon. The ZRP is a hopping particle model where particles hop between sites of a lattice under certain rules that depend only on the properties of the site from which the particles hop – hence the name zero-range. Through its simplicity the steady state of the ZRP can be solved, even for nonequilibrium dynamics, and yet despite its simplicity it can exhibit interesting phenomena such as condensation transitions, where a finite fraction of the total particles in the system will condense onto a single site of the lattice. Firstly, interesting finite-size effects surrounding the condensation transition in a one-dimensional, driven version of the ZRP are studied. These take the form of discrepancies in the current-density diagram between finite and infinite systems, with the finite behaviour resembling that seen in real traffic data. Following this, direct applications of the ZRP to complex networks, and interesting phenomena arising from the specifics of the applications, are studied. The ZRP is applied as a model of networks and is found capable of reproducing power-law degree distributions, as observed in many real networks, at the critical point of the condensation transition. The degree is the number of connections a component of the network has. This model is then generalised to include creation and annihilation of particles or links, and this is found to exhibit critical behaviour – namely power-law particle and degree distributions – in a region of the parameter space, rather than at a critical point. The full phase diagram of this system is investigated, revealing low density and high density phases as well as subdivisions of the critical phase.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:640541 |
| Date | January 2005 |
| Creators | Angel, Andrew George |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/11870 |
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