arXiv subjects

M. R. Evans

Publications and source records attributed to M. R. Evans.

At least 37 records · Page 2Linked to original sources

arXiv2008

Zero-Range Processes with Multiple Condensates: Statics and Dynamics

The steady-state distributions and dynamical behaviour of Zero Range Processes with hopping rates which are non-monotonic functions of the site occupation are studied. We consider two classes of non-monotonic hopping rates. The first results in a condensed phase containing a large (but subextensive) number of mesocondensates each containing a subextensive number of particles. The second results in a condensed phase containing a finite number of extensive condensates. We study the scaling behaviour of the peak in the distribution function corresponding to the condensates in both cases. In studying the dynamics of the condensate we identify two timescales: one for creation, the other for evaporation of condensates at a given site. The scaling behaviour of these timescales is studied within the Arrhenius law approach and by numerical simulations.

cond-mat.stat-mech↗

arXiv2007

A dynamically extending exclusion process

An extension of the totally asymmetric exclusion process, which incorporates a dynamically extending lattice is explored. Although originally inspired as a model for filamentous fungal growth, here the dynamically extending exclusion process (DEEP) is studied in its own right, as a nontrivial addition to the class of nonequilibrium exclusion process models. Here we discuss various mean-field approximation schemes and elucidate the steady state behaviour of the model and its associated phase diagram. Of particular note is that the dynamics of the extending lattice leads to a new region in the phase diagram in which a shock discontinuity in the density travels forward with a velocity that is lower than the velocity of the tip of the lattice. Thus in this region the shock recedes from both boundaries.

cond-mat.stat-mech↗

arXiv2007

Criticality and Condensation in a Non-Conserving Zero Range Process

The Zero-Range Process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving Zero-Range Process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterised by mesocondensates each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived.

cond-mat.stat-mech↗

arXiv2007

Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide

We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems - which includes exclusion processes - focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

cond-mat.stat-mech↗

arXiv2007

A model of hyphal tip growth involving microtubule-based transport

We propose a simple model for mass transport within a fungal hypha and its subsequent growth. Inspired by the role of microtubule-transported vesicles, we embody the internal dynamics of mass inside a hypha with mutually excluding particles progressing stochastically along a growing one-dimensional lattice. The connection between long range transport of materials for growth, and the resulting extension of the hyphal tip has not previously been addressed in the modelling literature. We derive and analyse mean-field equations for the model and present a phase diagram of its steady state behaviour, which we compare to simulations. We discuss our results in the context of the filamentous fungus, Neurospora crassa.

q-bio.SC↗

arXiv2006

Interaction driven real-space condensation

We study real-space condensation in a broad class of stochastic mass transport models. We show that the steady state of such models has a pair-factorised form which generalizes the standard factorized steady states. The condensation in this class of models is driven by interactions which give rise to a spatially extended condensate that differs fundamentally from the previously studied examples. We present numerical results as well as a theoretical analysis of the condensation transition and show that the criterion for condensation is related to the binding-unbinding transition of solid-on-solid interfaces.

cond-mat.stat-mech↗

arXiv2006

Factorised Steady States in Mass Transport Models on an Arbitrary Graph

We study a general mass transport model on an arbitrary graph consisting of $L$ nodes each carrying a continuous mass. The graph also has a set of directed links between pairs of nodes through which a stochastic portion of mass, chosen from a site-dependent distribution, is transported between the nodes at each time step. The dynamics conserves the total mass and the system eventually reaches a steady state. This general model includes as special cases various previously studied models such as the Zero-range process and the Asymmetric random average process. We derive a general condition on the stochastic mass transport rules, valid for arbitrary graph and for both parallel and random sequential dynamics, that is sufficient to guarantee that the steady state is factorisable. We demonstrate how this condition can be achieved in several examples. We show that our generalized result contains as a special case the recent results derived by Greenblatt and Lebowitz for $d$-dimensional hypercubic lattices with random sequential dynamics.

cond-mat.stat-mech↗

arXiv2005

Canonical Analysis of Condensation in Factorised Steady State

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as $L^{1/2}$ where $L$ is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities.

cond-mat.stat-mech↗

arXiv2005

Flocking Regimes in a Simple Lattice Model

We study a one-dimensional lattice flocking model incorporating all three of the flocking criteria proposed by Reynolds [Computer Graphics vol.21 4 (1987)]: alignment, centring and separation. The model generalises that introduced by O. J. O' Loan and M. R. Evans [J. Phys. A. vol. 32 L99 (1999)]. We motivate the dynamical rules by microscopic sampling considerations. The model exhibits various flocking regimes: the alternating flock, the homogeneous flock and dipole structures. We investigate these regimes numerically and within a continuum mean-field theory.

cond-mat.stat-mech↗

arXiv2005

Condensation transitions in a model for a directed network with weighted links

An exactly solvable model for the rewiring dynamics of weighted, directed networks is introduced. Simulations indicate that the model exhibits two types of condensation: (i) a phase in which, for each node, a finite fraction of its total out-strength condenses onto a single link; (ii) a phase in which a finite fraction of the total weight in the system is directed into a single node. A virtue of the model is that its dynamics can be mapped onto those of a zero-range process with many species of interacting particles -- an exactly solvable model of particles hopping between the sites of a lattice. This mapping, which is described in detail, guides the analysis of the steady state of the network model and leads to theoretical predictions for the conditions under which the different types of condensation may be observed. A further advantage of the mapping is that, by exploiting what is known about exactly solvable generalisations of the zero-range process, one can infer a number of generalisations of the network model and dynamics which remain exactly solvable.

cond-mat.stat-mech↗

arXiv2005

Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics, also we discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorised form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarise recent progress in understanding the dynamics of condensation. We then turn to several generalisations which also, under certain specified conditions, share the property of a factorised steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.

cond-mat.stat-mech↗

arXiv2005

The Nature of the Condensate in Mass Transport Models

We study the phenomenon of real space condensation in the steady state of a class of one dimensional mass transport models. We derive the criterion for the occurrence of a condensation transition and analyse the precise nature of the shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as $L^{1/2}$ where $L$ is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. We interpret these results within the framework of sums of random variables.

cond-mat.stat-mech↗

arXiv2005

Critical phase in non-conserving zero-range processes and equilibrium networks

Zero-range processes, in which particles hop between sites on a lattice, are closely related to equilibrium networks, in which rewiring of links take place. Both systems exhibit a condensation transition for appropriate choices of the dynamical rules. The transition results in a macroscopically occupied site for zero-range processes and a macroscopically connected node for networks. Criticality, characterized by a scale-free distribution, is obtained only at the transition point. This is in contrast with the widespread scale-free real-life networks. Here we propose a generalization of these models whereby criticality is obtained throughout an entire phase, and the scale-free distribution does not depend on any fine-tuned parameter.

cond-mat.stat-mech↗

arXiv2004

Construction of the factorized steady state distribution in models of mass transport

For a class of one-dimensional mass transport models we present a simple and direct test on the chipping functions, which define the probabilities for mass to be transferred to neighbouring sites, to determine whether the stationary distribution is factorized. In cases where the answer is affirmative, we provide an explicit method for constructing the single-site weight function. As an illustration of the power of this approach, previously known results on the Zero-range process and Asymmetric random average process are recovered in a few lines. We also construct new models, namely a generalized Zero-range process and a binomial chipping model, which have factorized steady states.

cond-mat.stat-mech↗

arXiv2004

Factorised Steady States in Mass Transport Models

We study a class of mass transport models where mass is transported in a preferred direction around a one-dimensional periodic lattice and is globally conserved. The model encompasses both discrete and continuous masses and parallel and random sequential dynamics and includes models such as the Zero-range process and Asymmetric random average process as special cases. We derive a necessary and sufficient condition for the steady state to factorise, which takes a rather simple form.

cond-mat.stat-mech↗

arXiv2004

Disorder and non-conservation in a driven diffusive system

We consider a disordered asymmetric exclusion process in which randomly chosen sites do not conserve particle number. The model is motivated by features of many interacting molecular motors such as RNA polymerases. We solve the steady state exactly in the two limits of infinite and vanishing non-conserving rates. The first limit is used as an approximation to large but finite rates and allows the study of Griffiths singularities in a nonequilibrium steady state despite the absence of any transition in the pure model. The disorder is also shown to induce a stretched exponential decay of system density with stretching exponent ϕ= 2/5.

cond-mat.stat-mech↗

arXiv2004

Modelling one-dimensional driven diffusive systems by the Zero-Range Process

The recently introduced correspondence between one-dimensional two-species driven models and the Zero-Range Process is extended to study the case where the densities of the two species need not be equal. The correspondence is formulated through the length dependence of the current emitted from a particle domain. A direct numerical method for evaluating this current is introduced, and used to test the assumptions underlying this approach. In addition, a model for isolated domain dynamics is introduced, which provides a simple way to calculate the current also for the non-equal density case. This approach is demonstrated and applied to a particular two-species model, where a phase separation transition line is calculated.

cond-mat.stat-mech↗

arXiv2004

Condensation Transitions in Nonequilibrium systems

Systems driven out of equilibrium can often exhibit behaviour not seen in systems in thermal equilibrium- for example phase transitions in one-dimensional systems. In this talk I will review several `condensation' transitions that occur when a conserved quantity is driven through the system. Although the condensation is spatial, i.e. a finite fraction of the conserved quantity condenses into a small spatial region, useful comparison can be made with usual Bose-Einstein condensation. Amongst some one-dimensional examples I will discuss the `Bus Route Model' where the condensation corresponds to the clustering together of buses moving along a bus-route.

cond-mat.stat-mech↗