Non-equilibrium real-space condensation is a phenomenon in which a finite fraction of some conserved quantity (mass, particles, etc.) becomes spatially localised. We review two popular stochastic models of hopping particles that lead to condensation
and whose stationary states assume a factorized form: the zero-range process and the misanthrope process, and their various modifications. We also introduce a new model - a misanthrope process with parallel dynamics - that exhibits condensation and has a pair-factorized stationary state.
We consider an extension of the zero-range process to the case where the hop rate depends on the state of both departure and arrival sites. We recover the misanthrope and the target process as special cases for which the probability of the steady sta
te factorizes over sites. We discuss conditions which lead to the condensation of particles and show that although two different hop rates can lead to the same steady state, they do so with sharply contrasting dynamics. The first case resembles the dynamics of the zero-range process, whereas the second case, in which the hop rate increases with the occupation number of both sites, is similar to instantaneous gelation models. This new explosive condensation reveals surprisingly rich behaviour, in which the process of condensates formation goes through a series of collisions between clusters of particles moving through the system at increasing speed. We perform a detailed numerical and analytical study of the dynamics of condensation: we find the speed of the moving clusters, their scattering amplitude, and their growth time. We finally show that the time to reach steady state decreases with the size of the system.
We consider a large class of two-lane driven diffusive systems in contact with reservoirs at their boundaries and develop a stability analysis as a method to derive the phase diagrams of such systems. We illustrate the method by deriving phase diagra
ms for the asymmetric exclusion process coupled to various second lanes: a diffusive lane; an asymmetric exclusion process with advection in the same direction as the first lane, and an asymmetric exclusion process with advection in the opposite direction. The competing currents on the two lanes naturally lead to a very rich phenomenology and we find a variety of phase diagrams. It is shown that the stability analysis is equivalent to an `extremal current principle for the total current in the two lanes. We also point to classes of models where both the stability analysis and the extremal current principle fail.
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 coincid
es 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.
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 con
densed 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.
