17 research outputs found
Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current
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 intriguing
supercritical bulk region where condensates can grow. These results provide a
microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly
asymmetric case: It turns out that the predictions of MFT remain valid in the
non-rigorous limit of finite asymmetry. In addition, the microscopic results
yield the correct scaling factor for the asymmetry that MFT cannot predict.Comment: 26 pages, 4 figure
Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium
The equation which describes a particle diffusing in a logarithmic potential
arises in diverse physical problems such as momentum diffusion of atoms in
optical traps, condensation processes, and denaturation of DNA molecules. A
detailed study of the approach of such systems to equilibrium via a scaling
analysis is carried out, revealing three surprising features: (i) the solution
is given by two distinct scaling forms, corresponding to a diffusive (x ~
\sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii)
the scaling exponents and scaling functions corresponding to both regimes are
selected by the initial condition; and (iii) this dependence on the initial
condition manifests a "phase transition" from a regime in which the scaling
solution depends on the initial condition to a regime in which it is
independent of it. The selection mechanism which is found has many similarities
to the marginal stability mechanism which has been widely studied in the
context of fronts propagating into unstable states. The general scaling forms
are presented and their practical and theoretical applications are discussed.Comment: 42 page
Large deviations in the symmetric simple exclusion process with slow boundaries
We obtain the exact large deviation functions of the density profile and of
the current, in the non-equilibrium steady state of a one dimensional symmetric
simple exclusion process coupled to boundary reservoirs with slow rates.
Compared to earlier results, where rates at the boundaries are comparable to
the bulk ones, we show how macroscopic fluctuations are modified when the
boundary rates are slower by an order of inverse of the system length.Comment: 14 pages, 4 figure
Condensation in Temporally Correlated Zero-Range Dynamics
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 Markovian
ZRP, but with modified hopping rates which can affect condensation, and (2) for
nearest-neighbor hopping in one dimension, the condensate occupies two adjacent
sites on the lattice and drifts with a finite velocity. The validity of these
results in a more general context is discussed.Comment: 4 pages, 3 figure
Approach to equilibrium of diffusion in a logarithmic potential
The late-time distribution function P(x,t) of a particle diffusing in a
one-dimensional logarithmic potential is calculated for arbitrary initial
conditions. We find a scaling solution with three surprising features: (i) the
solution is given by two distinct scaling forms, corresponding to a diffusive
(x ~ t^(1/2)) and a subdiffusive (x ~ t^{\gamma} with a given {\gamma} < 1/2)
length scale, respectively, (ii) the overall scaling function is selected by
the initial condition, and (iii) depending on the tail of the initial
condition, the scaling exponent which characterizes the scaling function is
found to exhibit a transition from a continuously varying to a fixed value.Comment: 4 pages, 3 figures; Published versio
Motion of condensates in non-Markovian zero-range dynamics
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 steady
state of this model is found to be a product measure, and the condensate is
stationary.Comment: 31 pages, 9 figure
Emergent motion of condensates in mass-transport models
We examine the effect of spatial correlations on the phenomenon of real-space
condensation in driven mass-transport systems. We suggest that in a broad class
of models with a spatially correlated steady state, the condensate drifts with
a non-vanishing velocity. We present a robust mechanism leading to this
condensate drift. This is done within the framework of a generalized zero-range
process (ZRP) in which, unlike the usual ZRP, the steady state is not a product
measure. The validity of the mechanism in other mass-transport models is
discussed.Comment: 5 pages, 2 figures. For supplementary material see
http://www.weizmann.ac.il/weizsites/mukamel/sm/cond-drift