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Condensation in the zero-range process: stationary and dynamical properties

Stefan Großkinsky, Gunter Markus Schutz, Herbert E. Spohn

Published Jan 1, 2003• 165 citations
OpenAlex

Abstract

The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites. Key words: zero range process; nonequilibrium phase transition; equivalence of ensembles; relative entropy. 1

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We study a disordered, driven zero range process which models a closed system of attractive particles that hop with site-dependent rates and whose steady state shows a condensation transition with increasing density. We characterize the dynamical properties of the mass fluctuations in the steady state in one dimension both analytically and numerically and show that there is a dynamic phase transition in the density-disorder plane. We also determine the form of the scaling function which describes the growth of the condensate as a function of time, starting from a uniform density distribution.

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The eigenfunctions and eigenvalues of the master equation for zero-range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the q numbers [n](q). This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number n of a site of departure. The noninteracting case can be restored in the limit q-->1. The limiting cases of the model for q=0, infinity correspond to the totally asymmetric exclusion process, and the drop-push model, respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance traveled by particles at large time in the scaling limit. In case of nonzero interaction, q not equal 1, the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.

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Criticality and condensation in a nonconserving zero-range process

A G Angel, M. R. Evans, E. Levine +1 more

Abstract. The Zero-Range Process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving Zero-Range Process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterised by mesocondensates each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived. PACS numbers: 89.75.-k, 05.70.Ln, 05.40.-aCriticality and Condensation in a Non-conserving Zero Range Process 2 1.

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Criticality and condensation in a non-conserving zero-range process

A G Angel, M. R. Evans, E Levine +1 more

The zero-range process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving zero-range process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterized by mesocondensates, each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived.

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Condensation in the inhomogeneous zero-range process: an interplay between interaction and diffusion disorder

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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 the condensate may be located at any site of a large hosting set of favored sites, whose size grows sub-extensively. The novel extended condensed phase thus interpolates continuously between the two scenarios associated so far with the condensation transition, namely spontaneous symmetry breaking and explicit symmetry breaking.

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Long-range random-field Ising model: Phase transition threshold and equivalence of short and long ranges

Luca Leuzzi, Giorgio Parisi A

The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range $D$-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.

arXiv:1303.6333•Physical Review B•Dec 19, 2013•23 citations
The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites. Key words: zero range process; nonequilibrium phase transition; equivalence of ensembles; relative entropy. 1
We study a disordered, driven zero range process which models a closed system of attractive particles that hop with site-dependent rates and whose steady state shows a condensation transition with increasing density. We characterize the dynamical properties of the mass fluctuations in the steady state in one dimension both analytically and numerically and show that there is a dynamic phase transition in the density-disorder plane. We also determine the form of the scaling function which describes the growth of the condensate as a function of time, starting from a uniform density distribution.
The eigenfunctions and eigenvalues of the master equation for zero-range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the q numbers [n](q). This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number n of a site of departure. The noninteracting case can be restored in the limit q-->1. The limiting cases of the model for q=0, infinity correspond to the totally asymmetric exclusion process, and the drop-push model, respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance traveled by particles at large time in the scaling limit. In case of nonzero interaction, q not equal 1, the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.
Abstract. The Zero-Range Process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving Zero-Range Process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterised by mesocondensates each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived. PACS numbers: 89.75.-k, 05.70.Ln, 05.40.-aCriticality and Condensation in a Non-conserving Zero Range Process 2 1.
The zero-range process, in which particles hop between sites on a lattice under conserving dynamics, is a prototypical model for studying real-space condensation. Within this model the system is critical only at the transition point. Here we consider a non-conserving zero-range process which is shown to exhibit generic critical phases which exist in a range of creation and annihilation parameters. The model also exhibits phases characterized by mesocondensates, each of which contains a subextensive number of particles. A detailed phase diagram, delineating the various phases, is derived.
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 the condensate may be located at any site of a large hosting set of favored sites, whose size grows sub-extensively. The novel extended condensed phase thus interpolates continuously between the two scenarios associated so far with the condensation transition, namely spontaneous symmetry breaking and explicit symmetry breaking.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.