Several recent works have shown that the onedimensional fully asymmetric exclusion model, which describes a ystem of panicles hopping in a preferred direction with hard core interactions, can be solved exactb in the case of open boundaries. Here we present a new approach based on representing the weights of each mnfiguration in the steady state as a product of noncommuting matrices. Wth this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisQ ve'y simple algebraic NI= We obtain several explicit toms for these noncommuting matrices which are, in the general case. infinite-dimensional. Our approach allows exam expresions to be derived for the current and density profiles. Finally we discuss h'efly two possible generalizations of our results: the pmblem of panially asymmetric exclusion and the case of a mixture of two kinds of panicles.

We study simple diffusion where a particle stochastically resets to its initial position at a constant rate r. A finite resetting rate leads to a nonequilibrium stationary state with non-Gaussian fluctuations for the particle position. We also show that the mean time to find a stationary target by a diffusive searcher is finite and has a minimum value at an optimal resetting rate r * . Resetting also alters fundamentally the late time decay of the survival probability of a stationary target when there are multiple searchers: while the typical survival probability decays exponentially with time, the average decays as a power law with an exponent depending continuously on the density of searchers.PACS numbers: 87.23.Ge 'Stochastic resetting' is a rather common process in everyday life. Consider searching for some target such as, for example, a face in a crowd or one's misplaced keys at home. A natural tendency is, on having searched unsuccessfully for a while, to return to the starting point and recommence the search. In this Letter we explore the consequences of such resetting on perhaps the most simple and common process in nature, namely, the diffusion of a single or a multiparticle system. We show that a nonzero rate of resetting has a rather rich and dramatic effect on the diffusion process.The first major effect of resetting shows up in the position distribution of the diffusing particle. In the absence of resetting, it has the usual Gaussian distribution whose width grows diffusively ∼ √ t with time. Upon switching on a nonzero resetting rate r to its initial position, this time-dependent Gaussian distribution gives way to a globally current-carrying nonequilibrium stationary state (NESS) with non-Gaussian fluctuations, given in Eq. (2). The process of resetting manifestly violates detailed balance and thus provides an appealingly simple example of a NESS.Resetting also has a profound consequence on the firstpassage properties of a diffusing particle. The study of first-passage problems and survival probabilities of diffusing particles arises in diverse subjects such as in reactiondiffusion kinetics, predator-prey dynamics [1], as well as in persistence in nonequilibrium systems [2]. Such problems are fundamental to nonequilibrium statistical mechanics as they involve irreversible processes not obeying detailed balance. Related models are also relevant to the study of search strategies in ecology or sampling techniques for the characterisation of complex networks. For example, intermittent searches involve diffusive motion combined with long range movements of the searcher and mimic the scan and relocation phases of foraging animals [3][4][5][6][7][8]. A well-studied problem is the mean time for a stationary target at the origin to be absorbed by a single diffusing particle (trap) or a team of diffusing traps distributed with uniform density. Many significant results, such as the fact that the mean time to find the target by a single diffusing particle diverges and that the survival probabilty of the ta...

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.

Abstract. 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.

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

We consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r. We consider several generalisations of the model of M. R. Evans and S. N. Majumdar (2011), Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601: (i) a space dependent resetting rate r(x) ii) resetting to a random position z drawn from a resetting distribution P(z) iii) a spatial distribution for the absorbing target P T (x). As an example of (i) we show that the introduction of a non-resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target. We address the problem of optimal resetting, that is, minimising the mean time to absorption for a given target distribution. For an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows.

The phenomenon of phase transitions in one-dimensional systems is discussed. Equilibrium systems are reviewed and some properties of an energy function which m a y allow phase transitions and phase ordering in one dimension are identi ed. We then give a n o verview of the one-dimensional phase transitions which h a ve been studied in nonequilibrium systems. A particularly simple model, the zero-range process, for which the steady state is known exactly as a product measure, is discussed in some detail. Generalisations of the model, for which a product measure still holds, are also discussed. We analyse in detail a condensation phase transition in the model and show h o w conditions under which i t m a y occur may be related to the existence of an e ective long-range energy function. It is also shown that even when the conditions for condensation are not ful lled one can still observe v ery sharp crossover behaviour and apparent condensation in a nite system. Although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.

A disordered version of the one dimensional asymmetric exclusion model where the particle hopping rates are quenched random variables is studied. The steady state is solved exactly by use of a matrix product. It is shown how the phenomenon of Bose condensation whereby a finite fraction of the empty sites are condensed in front of the slowest particle may occur. Above a critical density of particles a phase transition occurs out of the low density phase (Bose condensate) to a high density phase. An exponent describing the decrease of the steady state velocity as the density of particles goes above the critical value is calculated analytically and shown to depend on the distribution of hopping rates. The relation to traffic flow models is discussed.