318 research outputs found
Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra
We study the partially asymmetric exclusion process with open boundaries. We
generalise the matrix approach previously used to solve the special case of
total asymmetry and derive exact expressions for the partition sum and currents
valid for all values of the asymmetry parameter q. Due to the relationship
between the matrix algebra and the q-deformed quantum harmonic oscillator
algebra we find that q-Hermite polynomials, along with their orthogonality
properties and generating functions, are of great utility. We employ two
distinct sets of q-Hermite polynomials, one for q1. It
turns out that these correspond to two distinct regimes: the previously studied
case of forward bias (q1) where the
boundaries support a current opposite in direction to the bulk bias. For the
forward bias case we confirm the previously proposed phase diagram whereas the
case of reverse bias produces a new phase in which the current decreases
exponentially with system size.Comment: 27 pages LaTeX2e, 3 figures, includes new references and further
comparison with related work. To appear in J. Phys.
Criterion for phase separation in one-dimensional driven systems
A general criterion for the existence of phase separation in driven
one-dimensional systems is proposed. It is suggested that phase separation is
related to the size dependence of the steady-state currents of domains in the
system. A quantitative criterion for the existence of phase separation is
conjectured using a correspondence made between driven diffusive models and
zero-range processes. Several driven diffusive models are discussed in light of
the conjecture
Construction of a matrix product stationary state from solutions of finite size system
Stationary states of stochastic models, which have states per site, in
matrix product form are considered. First we give a necessary condition for the
existence of a finite -dimensional matrix product state for any .
Second, we give a method to construct the matrices from the stationary states
of small size system when the above condition and are satisfied.
Third, the method by which one can check that the obtained matrices are valid
for any system size is presented for the case where is satisfied. The
application of our methods is explained using three examples: the asymmetric
exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen.
36 (2003) 7497] and a hybrid of both of the models.Comment: 22 pages, no figure. Major changes: sec.3 was shortened; the list of
references were changed. This is the final version, which will appear in
J.Phys.
Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
The studies of fluctuations of the one-dimensional Kardar-Parisi-Zhang
universality class using the techniques from random matrix theory are reviewed
from the point of view of the asymmetric simple exclusion process. We explain
the basics of random matrix techniques, the connections to the polynuclear
growth models and a method using the Green's function.Comment: 41 pages, 10 figures, minor corrections, references adde
One-Dimensional Partially Asymmetric Simple Exclusion Process on a Ring with a Defect Particle
The effect of a moving defect particle for the one-dimensional partially
asymmetric simple exclusion process on a ring is considered. The current of the
ordinary particles, the speed of the defect particle and the density profile of
the ordinary particles are calculated exactly. The phase diagram for the
correlation length is identified. As a byproduct, the average and the variance
of the particle density of the one-dimensional partially asymmetric simple
exclusion process with open boundaries are also computed.Comment: 23 pages, 1 figur
Bethe ansatz solution of zero-range process with nonuniform stationary state
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 -numbers . This is a generalization of the rates
of hopping of noninteracting particles equal to the occupation number of a
site of departure. The noninteracting case can be restored in the limit . The limiting cases of the model for 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 travelled by particles at large time
in the scaling limit. In case of non-zero interaction, , the
generating function has the universal scaling form specific for the
Kardar-Parizi-Zhang universality class.Comment: 7 pages, Revtex4, mistypes correcte
Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries
The one-dimensional partially asymmetric simple exclusion process with open
boundaries is considered. The stationary state, which is known to be
constructed in a matrix product form, is studied by applying the theory of
q-orthogonal polynomials. Using a formula of the q-Hermite polynomials, the
average density profile is computed in the thermodynamic limit. The phase
diagram for the correlation length, which was conjectured in the previous
work[J. Phys. A {\bf 32} (1999) 7109], is confirmed.Comment: 24 pages, 6 figure
Will jams get worse when slow cars move over?
Motivated by an analogy with traffic, we simulate two species of particles
(`vehicles'), moving stochastically in opposite directions on a two-lane ring
road. Each species prefers one lane over the other, controlled by a parameter
such that corresponds to random lane choice and
to perfect `laning'. We find that the system displays one large cluster (`jam')
whose size increases with , contrary to intuition. Even more remarkably, the
lane `charge' (a measure for the number of particles in their preferred lane)
exhibits a region of negative response: even though vehicles experience a
stronger preference for the `right' lane, more of them find themselves in the
`wrong' one! For very close to 1, a sharp transition restores a homogeneous
state. Various characteristics of the system are computed analytically, in good
agreement with simulation data.Comment: 7 pages, 3 figures; to appear in Europhysics Letters (2005
Nonequilibrium stationary states and equilibrium models with long range interactions
It was recently suggested by Blythe and Evans that a properly defined steady
state normalisation factor can be seen as a partition function of a fictitious
statistical ensemble in which the transition rates of the stochastic process
play the role of fugacities. In analogy with the Lee-Yang description of phase
transition of equilibrium systems, they studied the zeroes in the complex plane
of the normalisation factor in order to find phase transitions in
nonequilibrium steady states. We show that like for equilibrium systems, the
``densities'' associated to the rates are non-decreasing functions of the rates
and therefore one can obtain the location and nature of phase transitions
directly from the analytical properties of the ``densities''. We illustrate
this phenomenon for the asymmetric exclusion process. We actually show that its
normalisation factor coincides with an equilibrium partition function of a walk
model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure
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