We study the condensation phenomenon in a zero range process on weighted scale-free networks in order to show how the weighted transport influences the particle condensation. Instead of the approach of grand canonical ensemble which is generally used in a zero range process, we introduce an alternate approach of the mean field equations to study the dynamics of particle transport. We find that the condensation on scale-free network is easier to occur in the case of weighted transport than in the case of weight-free. In the weighted transport, especially, a dynamical condensation is even possible for the case of no interaction among particles, which is impossible in the case of weight-free.
We study a class of zero-range processes in which the real-space condensation phenomenon does not occur and is replaced by a saturated condensation: that is, an extensive number of finite-size "condensates" in the steady state. We determine the conditions under which this occurs, and investigate the dynamics of relaxation to the steady state. We identify two stages: a rapid initial growth of condensates followed by a slow process of activated evaporation and condensation. We analyze these nonequilibrium dynamics with a combination of meanfield approximations, first-passage time calculations an…
The dynamics of zero-range processes on complex networks is expected to be influenced by the topological structure of underlying networks. A real space complete condensation phase transition in the stationary state may occur. We have studied the finite density effects of the condensation transition in both the stationary and dynamical zero-range process on scale-free networks. By means of grand canonical ensemble method, we predict analytically the scaling laws of the average occupation number with respect to the finite density for the steady state. We further explore the relaxation dynamics o…
Condensation phenomena in non-equilibrium systems have been modeled by the zero-range process, which is a model of particles hopping between boxes with Markovian dynamics. In many cases, memory effects in the dynamics cannot be neglected. In an attempt to understand the possible impact of temporal correlations on the condensate, we introduce and study a process with non-Markovian zero-range dynamics. We find that memory effects have significant impact on the condensation scenario. Specifically, two main results are found: (1) In mean-field dynamics, the steady state corresponds to that of a Ma…
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 state 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…
We discuss statics and dynamics of condensation in a zero-range process with compartments of limited sizes. For the symmetric dynamics the stationary state has a factorized form. For the asymmetric dynamics the steady state factorizes only for special hopping rules which allow for overjumps of fully occupied compartments. In the limit of large system size the grand canonical analysis is exact also in a condensed phase, and for a broader class of hopping rates as compared to the previously studied systems with infinite compartments. The dynamics of condensation exhibits a dynamical self-blockin…
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 the set-up of stationary Zero Range models and discuss the onset of condensation induced by a local blockage on the lattice. We show that the introduction of a local feedback on the hopping rates allows to control the particle fraction in the condensed phase. This phenomenon results in a current vs. blockage parameter curve characterized by two non--analyticity points.
Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding or subtracting a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from zero to a positive value at a critical scale. Our results also include distributional limits for th…
We analyze the role of the interplay between on-site interaction and inhomogeneous diffusion on the phenomenon of condensation in the zero-range process. We predict a universal phase diagram in the plane of two exponents, respectively characterizing the interactions and the diffusion disorder. The most prominent outcome is the existence of an extended condensed phase. In the latter phase, which originates as a result of the combined effects of strong enough interaction and weak enough disorder, a typical high-density configuration has a unique condensate on top of a critical background, but th…
A simple model of Bose-Einstein condensation of interacting particles is proposed. It is shown that in the condensate state the dependence of thermodynamic quantities on the interaction constant does not allow an expansion in powers of the coupling constant. Therefore it is impossible to pass to the Einstein model of condensation in an ideal Bose gas by means of a limiting passage, setting the interaction constant to zero. The account for the interaction between particles eliminates difficulties in the description of condensation available in the model of an ideal gas, which are connected with…
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 …
We show that, at the critical temperature, there is a class of Lee-Yang zeros of the partition function in a general scalar field theory, which location scales with the size of the system with a characteristic exponent expressed in terms of the isothermal critical exponent $δ$. In the thermodynamic limit the zeros belonging to this class condense to the critical point ζ=1 on the real axis in the complex fugacity plane while the complementary set of zeros (with Re ζ < 1) covers uniformly the unit circle. Although the aforementioned class degenerates to a single point for an infinite system, whe…
In the present work we are studying a bosonic quantum field system at finite temperature, and at zero and non-zero chemical potential. For a simple spatial partition we derive the corresponding mutual information, a quantity that measures the total amount of information of one of the parts about the other. In order to find it, we first derive the geometric entropy corresponding to the specific partition and then we substract its extensive part which coincides with the thermal entropy of the system. In the case of non-zero chemical potential, we examine the influence of the underlying Bose-Eins…
Condensation transition in a non-Markovian zero-range process is studied in one and higher dimensions. In the mean-field approximation, corresponding to infinite range hopping, the model exhibits condensation with a stationary condensate, as in the Markovian case, but with a modified phase diagram. In the case of nearest-neighbor hopping, the condensate is found to drift by a "slinky" motion from one site to the next. The mechanism of the drift is explored numerically in detail. A modified model with nearest-neighbor hopping which allows exact calculation of the steady state is introduced. The…
We study the asymmetric zero-range process (ZRP) with L sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob h-transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the steady state of the conditioned dynamics, and, under generic conditions on the jump rates of the unconditioned ZRP, to an in…
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 discontinu…
A conserved generalized zero range process is considered in which two sites interact such that particles hop from the more populated site to the other with a probability $p$. The steady state particle distribution function $P(n)$ is obtained using both analytical and numerical methods. The system goes through several phases as $p$ is varied. In particular, a condensate phase appears for $p_l < p < p_c$, where the bounding values depend on the range of interaction, with $p_c < 0.5$ in general. Analysis of $P(n)$ in the condensate phase using a known scaling form shows there is universal behavio…
Condensation in stochastic lattice gases with size-dependent stationary weights
Institute of Mathematics, University of Augsburg（奥格斯堡大学数学研究所）；Department of Statistics, University of Warwick（华威大学统计系）；Department of Mathematics, Faculty of Science, Mahidol University（马希隆大学理学院数学系）
We establish an hitherto hidden connection between zero modes and instantons in the context of the Kraichanan model for passive scalar turbulent advection, that relies on the hypothesis that the production of strong gradients of the scalar is associated with Bose-like condensation of Lagrangian particles. It opens the way to the computation of scaling exponents of the N-th order structure functions of the scalar by techniques borrowed from many-body theory. To lowest order of approximation, scaling exponents are found to increase asymptotically as log N in two dimensions.
We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the co…
We study condensation in several particle systems related to the inclusion process. For an asymmetric one-dimensional version with closed boundary conditions and drift to the right, we show that all but a finite number of particles condense on the right-most site. This is extended to a general result for independent random variables with different tails, where condensation occurs for the index (site) with the heaviest tail, generalizing also previous results for zero-range processes. For inclusion processes with homogeneous stationary measures we establish condensation in the limit of vanishin…
