18 research outputs found
Current fluctuations in stochastic systems with long-range memory
We propose a method to calculate the large deviations of current fluctuations
in a class of stochastic particle systems with history-dependent rates.
Long-range temporal correlations are seen to alter the speed of the large
deviation function in analogy with long-range spatial correlations in
equilibrium systems. We give some illuminating examples and discuss the
applicability of the Gallavotti-Cohen fluctuation theorem.Comment: 10 pages, 1 figure. v2: Minor alterations. v3: Very minor alterations
for consistency with published version appearing at
http://stacks.iop.org/1751-8121/42/34200
Entropy production and fluctuation relations for a KPZ interface
We study entropy production and fluctuation relations in the restricted
solid-on-solid growth model, which is a microscopic realization of the KPZ
equation. Solving the one dimensional model exactly on a particular line of the
phase diagram we demonstrate that entropy production quantifies the distance
from equilibrium. Moreover, as an example of a physically relevant current
different from the entropy, we study the symmetry of the large deviation
function associated with the interface height. In a special case of a system of
length L=4 we find that the probability distribution of the variation of height
has a symmetric large deviation function, displaying a symmetry different from
the Gallavotti-Cohen symmetry.Comment: 21 pages, 5 figure
On the range of validity of the fluctuation theorem for stochastic Markovian dynamics
We consider the fluctuations of generalized currents in stochastic Markovian
dynamics. The large deviations of current fluctuations are shown to obey a
Gallavotti-Cohen (GC) type symmetry in systems with a finite state space.
However, this symmetry is not guaranteed to hold in systems with an infinite
state space. A simple example of such a case is the Zero-Range Process (ZRP).
Here we discuss in more detail the already reported breakdown of the GC
symmetry in the context of the ZRP with open boundaries and we give a physical
interpretation of the phases that appear. Furthermore, the earlier analytical
results for the single-site case are extended to cover multiple-site systems.
We also use our exact results to test an efficient numerical algorithm of
Giardina, Kurchan and Peliti, which was developed to measure the current large
deviation function directly. We find that this method breaks down in some
phases which we associate with the gapless spectrum of an effective
Hamiltonian.Comment: 37 pages, 10 figures. Minor alterations, fixed typos (as appeared in
JSTAT
The statistics of diffusive flux
We calculate the explicit probability distribution function for the flux
between sites in a simple discrete time diffusive system composed of
independent random walkers. We highlight some of the features of the
distribution and we discuss its relation to the local instantaneous entropy
production in the system. Our results are applicable both to equilibrium and
non-equilibrium steady states as well as for certain time dependent situations.Comment: 12 pages, 1 figur
Relevance of initial and final conditions for the Fluctuation Relation in Markov processes
Numerical observations on a Markov chain and on the continuous Markov process
performed by a granular tracer show that the ``usual'' fluctuation relation for
a given observable is not verified for finite (but arbitrarily large) times.
This suggests that some terms which are usually expected to be negligible, i.e.
``border terms'' dependent only on initial and final states, in fact cannot be
neglected. Furthermore, the Markov chain and the granular tracer behave in a
quite similar fashion.Comment: 23 pages, 5 figures, submitted to JSTA
Fluctuation theorems for stochastic dynamics
Fluctuation theorems make use of time reversal to make predictions about
entropy production in many-body systems far from thermal equilibrium. Here we
review the wide variety of distinct, but interconnected, relations that have
been derived and investigated theoretically and experimentally. Significantly,
we demonstrate, in the context of Markovian stochastic dynamics, how these
different fluctuation theorems arise from a simple fundamental time-reversal
symmetry of a certain class of observables. Appealing to the notion of Gibbs
entropy allows for a microscopic definition of entropy production in terms of
these observables. We work with the master equation approach, which leads to a
mathematically straightforward proof and provides direct insight into the
probabilistic meaning of the quantities involved. Finally, we point to some
experiments that elucidate the practical significance of fluctuation relations.Comment: 48 pages, 2 figures. v2: minor changes for consistency with published
versio
Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges
We study stochastic processes in which the trajectories are constrained so
that the process realises a large deviation of the unconstrained process. In
particular we consider stochastic bridges and the question of inequivalence of
path ensembles between the microcanonical ensemble, in which the end points of
the trajectory are constrained, and the canonical or s ensemble in which a bias
or tilt is introduced into the process. We show how ensemble inequivalence can
be manifested by the phenomenon of temporal condensation in which the large
deviation is realised in a vanishing fraction of the duration (for long
durations). For diffusion processes we find that condensation happens whenever
the process is subject to a confining potential, such as for the
Ornstein-Uhlenbeck process, but not in the borderline case of dry friction in
which there is partial ensemble equivalence. We also discuss continuous-space,
discrete-time random walks for which in the case of a heavy tailed step-size
distribution it is known that the large deviation may be achieved in a single
step of the walk. Finally we consider possible effects of several constraints
on the process and in particular give an alternative explanation of the
interaction-driven condensation in terms of constrained Brownian excursions.Comment: 22 pages, 7 figures, minor revisio
Current Fluctuations and Statistics During a Large Deviation Event in an Exactly-Solvable Transport Model
We study the distribution of the time-integrated current in an
exactly-solvable toy model of heat conduction, both analytically and
numerically. The simplicity of the model allows us to derive the full current
large deviation function and the system statistics during a large deviation
event. In this way we unveil a relation between system statistics at the end of
a large deviation event and for intermediate times. Midtime statistics is
independent of the sign of the current, a reflection of the time-reversal
symmetry of microscopic dynamics, while endtime statistics do depend on the
current sign, and also on its microscopic definition. We compare our exact
results with simulations based on the direct evaluation of large deviation
functions, analyzing the finite-size corrections of this simulation method and
deriving detailed bounds for its applicability. We also show how the
Gallavotti-Cohen fluctuation theorem can be used to determine the range of
validity of simulation results.Comment: 13 pages, 7 figures, published versio
Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition
For the symmetric simple exclusion process on an infinite line, we calculate
exactly the fluctuations of the integrated current during time
through the origin when, in the initial condition, the sites are occupied with
density on the negative axis and with density on the positive
axis. All the cumulants of grow like . In the range where , the decay of the distribution of is
non-Gaussian. Our results are obtained using the Bethe ansatz and several
identities recently derived by Tracy and Widom for exclusion processes on the
infinite line.Comment: 2 figure