We numerically survey predictions on the shapes and scaling laws of particle condensates that emerge as a result of spontaneous symmetry breaking in pairfactorized steady states of a stochastic transport process. The specific model consists of indistinguishable particles that stochastically hop between sites controlled by a tunable potential. We identify the different condensate shapes within their respective parameter regimes as well as determine precisely the condensate width scaling.

No abstract

Using numerical methods we discuss the effects of open boundary conditions on condensation phenomena in the zero-range process (ZRP) and transport processes with pair-factorized steady states (PFSS), an extended model of the ZRP with nearest-neighbor interaction. For the zero-range process we compare to analytical results in the literature with respect to criticality and condensation. For the extended model we find a similar phase structure, but observe supercritical phases with droplet formation for strong boundary drives.

Abstract. From a coarse-grained perspective the motif of a self-activating species, activating a second species which acts as its own repressor, is widely found in biological systems, in particular in genetic systems with inherent oscillatory behavior. Here we consider a specific realization of this motif as a genetic circuit, in which genes are described as directly producing proteins, leaving out the intermediate step of mRNA production. We focus on the effect that inherent time scales on the underlying finegrained scale can have on the bifurcation patterns on a coarser scale in time. Time scales are set by the binding and unbinding rates of the transcription factors to the promoter regions of the genes. Depending on the ratio of these rates to the decay times of the proteins, the appropriate averaging procedure for obtaining a coarse-grained description changes and leads to sets of deterministic equations, which differ in their bifurcation structure. In particular the desired intermediate range of regular limit cycles fades away when the binding rates of genes are of the same order or less than the decay time of at least one of the proteins. Our analysis illustrates that the common topology of the widely found motif alone does not necessarily imply universal features in the dynamics.

-We discuss the effects of particle exchange through open boundaries and the induced drive on the phase structure and condensation phenomena of a stochastic transport process with tunable short-range interactions featuring pair-factorized steady states (PFSS) in the closed system. In this model, the steady state of the particle hopping process can be tuned to yield properties from the zero-range process (ZRP) condensation model to those of models with spatially extended condensates. By varying the particle exchange rates as well as the presence of a global drift, we observe a phase transition from a free particle gas to a phase with condensates aggregated to the boundaries. While this transition is similar to previous results for the ZRP, we find that the mechanism is different as the presence of the boundary actually influences the interaction due to the non-zero interaction range.Condensation phenomena are observed in a broad range of physical processes. While they are originally associated with phase transitions of matter from the gas state to some liquid or solid state, they are also closely related to nucleation and coarsening phenomena. Examples of condensation appear in processes such as the formation of breath figures [1], Bose-Einstein condensation [2], polymer aggregation [3], but in a wider sense also in more generic systems like networks as the formation of clusters [4] through the accumulation of links on sites.For many such systems, the involved condensation process can be modeled as a stochastic transport process with a set of particles occupying a number of discrete sites. With particles representing microscopic to macroscopic objects and appropriate dynamics a wide spectrum of physical processes has been studied. Examples include refs. [2, 4] mentioned above, but also processes such as wealth condensation [5] or traffic flow [6].The zero-range process (ZRP) with condensation dynamics [7,8] is a well known paradigm of such transport processes. While it has a fully symmetric steady state, above some critical density ρ c the symmetry breaks spon-(a) E-mail: hannes.nagel@itp.uni-leipzig.de (b) E-mail: h.ortmanns@jacobs-university.de (c) E-mail: wolfhard.janke@itp.uni-leipzig.de taneously and a particle condensate emerges at a single site such that the density at the remaining sites stays critical. When short-range interactions are introduced, a similar condensation process can be observed with the main difference, that condensates can be spatially extended [9][10][11][12]. In this work, however, we shall consider condensation not as an effect in the steady state, but as a signature of a boundary induced phase transition. Such transitions can occur in driven systems, where the drive is implemented in terms of the interaction at the boundary of the system [13]. A well known transport process with such a transition is the totally asymmetric simple exclusion process (TASEP) [14,15], where a high-density, a lowdensity and a maximal-current phase exist [16]. For the ZRP, such effects of open boun...