12 research outputs found
Equilibrium fluctuations of additive functionals of zero-range models
For mean-zero and asymmetric zero-range processes on , the fluctuations of additive functionals starting from an invariant measure are considered. Under certain assumptions, we establish when the fluctuations are diffusive and satisfy functional central limit theorems. These results complement those for symmetric zero-range systems and also those for simple exclusion models already in the literature.FC
On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries
We investigate the structure of the nonequilibrium stationary state (NESS) of
a system of first and second class particles, as well as vacancies (holes), on
L sites of a one-dimensional lattice in contact with first class particle
reservoirs at the boundary sites; these particles can enter at site 1, when it
is vacant, with rate alpha, and exit from site L with rate beta. Second class
particles can neither enter nor leave the system, so the boundaries are
semi-permeable. The internal dynamics are described by the usual totally
asymmetric exclusion process (TASEP) with second class particles. An exact
solution of the NESS was found by Arita. Here we describe two consequences of
the fact that the flux of second class particles is zero. First, there exist
(pinned and unpinned) fat shocks which determine the general structure of the
phase diagram and of the local measures; the latter describe the microscopic
structure of the system at different macroscopic points (in the limit L going
to infinity in terms of superpositions of extremal measures of the infinite
system. Second, the distribution of second class particles is given by an
equilibrium ensemble in fixed volume, or equivalently but more simply by a
pressure ensemble, in which the pair potential between neighboring particles
grows logarithmically with distance. We also point out an unexpected feature in
the microscopic structure of the NESS for finite L: if there are n second class
particles in the system then the distribution of first class particles
(respectively holes) on the first (respectively last) n sites is exchangeable.Comment: 28 pages, 4 figures. Changed title and introduction for clarity,
added reference
Zero-range process with open boundaries
We calculate the exact stationary distribution of the one-dimensional
zero-range process with open boundaries for arbitrary bulk and boundary hopping
rates. When such a distribution exists, the steady state has no correlations
between sites and is uniquely characterized by a space-dependent fugacity which
is a function of the boundary rates and the hopping asymmetry. For strong
boundary drive the system has no stationary distribution. In systems which on a
ring geometry allow for a condensation transition, a condensate develops at one
or both boundary sites. On all other sites the particle distribution approaches
a product measure with the finite critical density \rho_c. In systems which do
not support condensation on a ring, strong boundary drive leads to a condensate
at the boundary. However, in this case the local particle density in the
interior exhibits a complex algebraic growth in time. We calculate the bulk and
boundary growth exponents as a function of the system parameters
Boundary driven zero-range processes in random media
The stationary states of boundary driven zero-range processes in random media
with quenched disorder are examined, and the motion of a tagged particle is
analyzed. For symmetric transition rates, also known as the random barrier
model, the stationary state is found to be trivial in absence of boundary
drive. Out of equilibrium, two further cases are distinguished according to the
tail of the disorder distribution. For strong disorder, the fugacity profiles
are found to be governed by the paths of normalized -stable
subordinators. The expectations of integrated functions of the tagged particle
position are calculated for three types of routes.Comment: 23 page
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