Paul Chleboun

Condensation in Stochastic Particle Systems with Stationary Product Measures

We study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities. We review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles, providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit. This leads to strengthened results on weak convergence for subcritical systems, and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions, extending previous results which focused on attractive and finite systems. We use relative entropy techniques which provide simple proofs, making use of general versions of local limit theorems for independent random variables.

Finite size effects and metastability in zero-range condensation

We study zero-range processes which are known to exhibit a condensation transition, where above a critical density a non-zero fraction of all particles accumulates on a single lattice site. This phenomenon has been a subject of recent research interest and is well understood in the thermodynamic limit. The system shows large finite size effects, and we observe a switching between metastable fluid and condensed phases close to the critical point, in contrast to the continuous limiting behaviour of relevant observables. We describe the leading order finite size effects and establish a discontinuity near criticality in a rigorous scaling limit. We also characterise the metastable phases using a current matching argument and an extension of the fluid phase to supercritical densities. This constitutes an interesting example where the thermodynamic limit fails to capture essential parts of the dynamics, which are particularly relevant in applications with moderate system sizes such as traffic flow or granular clustering.

Instability of condensation in the zero-range process with random interaction.

The zero-range process is a stochastic interacting particle system that is known to exhibit a condensation transition. We present a detailed analysis of this transition in the presence of quenched disorder in the particle interactions. Using rigorous probabilistic arguments, we show that disorder changes the critical exponent in the interaction strength below which a condensation transition may occur. The local critical densities may exhibit large fluctuations, and their distribution shows an interesting crossover from exponential to algebraic behavior.

Dynamics of Condensation in the Totally Asymmetric Inclusion Process

We study the dynamics of condensation of the inclusion process on a one-dimensional periodic lattice in the thermodynamic limit, generalising recent results on finite lattices for symmetric dynamics. Our main focus is on totally asymmetric dynamics which have not been studied before, and which we also compare to exact solutions for symmetric systems. We identify all relevant dynamical regimes and corresponding time scales as a function of the system size, including a coarsening regime where clusters move on the lattice and exchange particles, leading to a growing average cluster size. The second moment of the occupation numbers is a suitable observable to characterise the transition, and exhibits a power law scaling in this regime before saturating to stationarity following an exponential decay depending on the system size. Our results are based on heuristic derivations and exact computations for symmetric systems, and are supported by detailed simulation data.

Relaxation to equilibrium of generalized east processes on Zd: Renormalization group analysis and energy-entropy competition

We consider a class of kinetically constrained interacting particle systems on Zd which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of Zd, to each occupation variable ηx∈{0,1} a new value is proposed by tossing a (1−q)-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For d=1, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On Z2, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit q↘0 of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on Zd scales as the 1/d-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex x out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the l1-norm of x but also on its direction.

A dynamical transition and metastability in a size-dependent zero-range process

We study a zero-range process with system-size dependent jump rates, which is known to exhibit a discontinuous condensation transition. Metastable homogeneous phases and condensed phases coexist in extended phase regions around the transition, which have been fully characterized in the context of the equivalence and non-equivalence of ensembles. In this communication we report rigorous results on the large deviation properties and the free energy landscape which determine the metastable dynamics of the system. Within the condensed phase region we identify a new dynamic transition line which separates two distinct mechanism of motion of the condensate, and provide a complete discussion of all relevant timescales. Our results are directly related to recent interest in metastable dynamics of condensing particle systems. Our approach applies to more general condensing particle systems, which exhibit the dynamical transition as a finite size effect.

Condensation in randomly perturbed zero-range processes

The zero-range process is a stochastic interacting particle system that exhibits a condensation transition under certain conditions on the dynamics. It has recently been found that a small perturbation of a generic class of jump rates leads to a drastic change of the phase diagram and prevents condensation in an extended parameter range. We complement this study with rigorous results on a finite critical density and quenched free energy in the thermodynamic limit as well as quantitative heuristic results for small and large noise which are supported by detailed simulation data. While our new results support the initial findings, they also shed new light on the actual (limited) relevance in large finite systems, which we discuss via fundamental diagrams obtained from exact numerics for finite systems.

Monotonicity and condensation in homogeneous stochastic particle systems

We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on fixed finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On fixed finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity

The influence of dimension on the relaxation process of East-like models : rigorous results

We consider the relaxation process and the out-of-equilibrium dynamics of natural generalizations to arbitrary dimensions of the well known one dimensional East process. These facilitated models are supposed to catch some of the main features of the complex dynamics of fragile glasses. We focus on the low temperature regime (small density

Lower current large deviations for zero-range processes on a ring

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