An introduction to phase transitions in stochastic dynamical systems

We give an introduction to phase transitions in the steady states of systems that evolve stochastically with equilibrium and nonequilibrium dynamics, the latter defined as those that do not possess a time-reversal symmetry. We try as much as possible to discuss both cases within the same conceptual framework, focussing on dynamically attractive `peaks' in state space. A quantitative characterisation of these peaks leads to expressions for the partition function and free energy that extend from equilibrium steady states to their nonequilibrium counterparts. We show that for certain classes of nonequilibrium systems that have been exactly solved, these expressions provide precise predictions of their macroscopic phase behaviour.Comment: Pedagogical talk contributed to the "Ageing and the Glass Transition" Summer School, Luxembourg, September 2005. 12 pages, 8 figures, uses the IOP 'jpconf' document clas

The Grand-Canonical Asymmetric Exclusion Process and the One-Transit Walk

The one-dimensional Asymmetric Exclusion Process (ASEP) is a paradigm for nonequilibrium dynamics, in particular driven diffusive processes. It is usually considered in a canonical ensemble in which the number of sites is fixed. We observe that the grand-canonical partition function for the ASEP is remarkably simple. It allows a simple direct derivation of the asymptotics of the canonical normalization in various phases and of the correspondence with One-Transit Walks recently observed by Brak et.al.Comment: Published versio

Dyck Paths, Motzkin Paths and Traffic Jams

It has recently been observed that the normalization of a one-dimensional out-of-equilibrium model, the Asymmetric Exclusion Process (ASEP) with random sequential dynamics, is exactly equivalent to the partition function of a two-dimensional lattice path model of one-transit walks, or equivalently Dyck paths. This explains the applicability of the Lee-Yang theory of partition function zeros to the ASEP normalization. In this paper we consider the exact solution of the parallel-update ASEP, a special case of the Nagel-Schreckenberg model for traffic flow, in which the ASEP phase transitions can be intepreted as jamming transitions, and find that Lee-Yang theory still applies. We show that the parallel-update ASEP normalization can be expressed as one of several equivalent two-dimensional lattice path problems involving weighted Dyck or Motzkin paths. We introduce the notion of thermodynamic equivalence for such paths and show that the robustness of the general form of the ASEP phase diagram under various update dynamics is a consequence of this thermodynamic equivalence.Comment: Version accepted for publicatio

Generation of two-photon EPR and Wstates

In this paper we present a scheme for generation of two-photon EPR and W states in the cavity QED context. The scheme requires only one three-level Rydberg atom and two or three cavities. The atom is sent to interact with cavities previously prepared in vacuum states, via two-photon process. An appropriate choice of the interaction times one obtains the mentioned state with maximized fidelities. These specific times and the values of success probability and fidelity are discussed.Comment: 4 pages, 5 figure

Spectral Analysis of a Four Mode Cluster State

We theoretically evaluate the squeezed joint operators produced in a single optical parametric oscillator which generates quadripartite entangled outputs, as demonstrated experimentally by Pysher et al. \cite{pysher}[Phys. Rev. Lett. 107, 030505 (2011)]. Using a linearized fluctuation analysis we calculate the squeezing of the joint quadrature operators below threshold for a range of local oscillator phases and frequencies. These results add to the existing theoretical understanding of this potentially important system.Comment: 4 pages, 6 figure

Dynamical Transition in the Open-boundary Totally Asymmetric Exclusion Process

We revisit the totally asymmetric simple exclusion process with open boundaries (TASEP), focussing on the recent discovery by de Gier and Essler that the model has a dynamical transition along a nontrivial line in the phase diagram. This line coincides neither with any change in the steady-state properties of the TASEP, nor the corresponding line predicted by domain wall theory. We provide numerical evidence that the TASEP indeed has a dynamical transition along the de Gier-Essler line, finding that the most convincing evidence was obtained from Density Matrix Renormalisation Group (DMRG) calculations. By contrast, we find that the dynamical transition is rather hard to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss in general terms scenarios that admit a distinction between static and dynamic phase behaviour.Comment: 27 pages, 18 figures. v2 to appear in J Phys A features minor corrections and better-quality figure

Relaxation rate of the reverse biased asymmetric exclusion process

We compute the exact relaxation rate of the partially asymmetric exclusion process with open boundaries, with boundary rates opposing the preferred direction of flow in the bulk. This reverse bias introduces a length scale in the system, at which we find a crossover between exponential and algebraic relaxation on the coexistence line. Our results follow from a careful analysis of the Bethe ansatz root structure.Comment: 22 pages, 12 figure

Utterance Selection Model of Language Change

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of population genetics, but there are significant differences. The continuous time formulation of the model is expressed in terms of a Fokker-Planck equation. This equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight. Whilst the stationary properties of this system have much in common with the single-speaker case, time-dependent properties are richer. In the particular case where linguistic forms can become extinct, we find that the presence of many speakers causes a two-stage relaxation, the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations.Comment: 21 pages, 17 figure