738 research outputs found
Ergodicity Breaking in a Deterministic Dynamical System
The concept of weak ergodicity breaking is defined and studied in the context
of deterministic dynamics. We show that weak ergodicity breaking describes a
weakly chaotic dynamical system: a nonlinear map which generates subdiffusion
deterministically. In the non-ergodic phase non-trivial distribution of the
fraction of occupation times is obtained. The visitation fraction remains
uniform even in the non-ergodic phase. In this sense the non-ergodicity is
quantified, leading to a statistical mechanical description of the system even
though it is not ergodic.Comment: 11 pages, 4 figure
Stochastic Ergodicity Breaking: a Random Walk Approach
The continuous time random walk (CTRW) model exhibits a non-ergodic phase
when the average waiting time diverges. Using an analytical approach for the
non-biased and the uniformly biased CTRWs, and numerical simulations for the
CTRW in a potential field, we obtain the non-ergodic properties of the random
walk which show strong deviations from Boltzmann--Gibbs theory. We derive the
distribution function of occupation times in a bounded region of space which,
in the ergodic phase recovers the Boltzmann--Gibbs theory, while in the
non-ergodic phase yields a generalized non-ergodic statistical law.Comment: 5 pages, 3 figure
Residence time statistics for blinking quantum dots and other stochastic processes
We present a study of residence time statistics for blinking quantum
dots. With numerical simulations and exact calculations we show sharp
transitions for a critical number of dots. In contrast to expectation the
fluctuations in the limit of are non-trivial. Besides quantum
dots our work describes residence time statistics in several other many
particle systems for example Brownian particles. Our work provides a
natural framework to detect non-ergodic kinetics from measurements of many
blinking chromophores, without the need to reach the single molecule limit
Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
We find a general formula for the distribution of time-averaged observables
for systems modeled according to the sub-diffusive continuous time random walk.
For Gaussian random walks coupled to a thermal bath we recover ergodicity and
Boltzmann's statistics, while for the anomalous subdiffusive case a weakly
non-ergodic statistical mechanical framework is constructed, which is based on
L\'evy's generalized central limit theorem. As an example we calculate the
distribution of : the time average of the position of the particle,
for unbiased and uniformly biased particles, and show that exhibits
large fluctuations compared with the ensemble average .Comment: 5 pages, 2 figure
Towards deterministic equations for Levy walks: the fractional material derivative
Levy walks are random processes with an underlying spatiotemporal coupling.
This coupling penalizes long jumps, and therefore Levy walks give a proper
stochastic description for a particle's motion with broad jump length
distribution. We derive a generalized dynamical formulation for Levy walks in
which the fractional equivalent of the material derivative occurs. Our approach
will be useful for the dynamical formulation of Levy walks in an external force
field or in phase space for which the description in terms of the continuous
time random walk or its corresponding generalized master equation are less well
suited
Analytical and Numerical Study of Internal Representations in Multilayer Neural Networks with Binary Weights
We study the weight space structure of the parity machine with binary weights
by deriving the distribution of volumes associated to the internal
representations of the learning examples. The learning behaviour and the
symmetry breaking transition are analyzed and the results are found to be in
very good agreement with extended numerical simulations.Comment: revtex, 20 pages + 9 figures, to appear in Phys. Rev.
Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
We investigate statistics of occupation times for an over-damped Brownian
particle in an external force field. A backward Fokker-Planck equation
introduced by
Majumdar and Comtet describing the distribution of occupation times is
solved. The solution gives a general relation between occupation time
statistics and probability currents which are found from solutions of the
corresponding problem of first passage time. This general relationship between
occupation times and first passage times, is valid for normal Markovian
diffusion and for non-Markovian sub-diffusion, the latter modeled using the
fractional Fokker-Planck equation. For binding potential fields we find in the
long time limit ergodic behavior for normal diffusion, while for the fractional
framework weak ergodicity breaking is found, in agreement with previous results
of Bel and Barkai on the continuous time random walk on a lattice. For
non-binding potential rich physical behaviors are obtained, and classification
of occupation time statistics is made possible according to whether or not the
underlying random walk is recurrent and the averaged first return time to the
origin is finite. Our work establishes a link between fractional calculus and
ergodicity breaking.Comment: 12 page
Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
We investigate the non-ergodic properties of blinking nano-crystals using a
stochastic approach. We calculate the distribution functions of the time
averaged intensity correlation function and show that these distributions are
not delta peaked on the ensemble average correlation function values; instead
they are W or U shaped. Beyond blinking nano-crystals our results describe
non-ergodicity in systems stochastically modeled using the Levy walk framework
for anomalous diffusion, for example certain types of chaotic dynamics,
currents in ion-channel, and single spin dynamics to name a few.Comment: 5 pages, 3 figure
A Random Walk to a Non-Ergodic Equilibrium Concept
Random walk models, such as the trap model, continuous time random walks, and
comb models exhibit weak ergodicity breaking, when the average waiting time is
infinite. The open question is: what statistical mechanical theory replaces the
canonical Boltzmann-Gibbs theory for such systems? In this manuscript a
non-ergodic equilibrium concept is investigated, for a continuous time random
walk model in a potential field. In particular we show that in the non-ergodic
phase the distribution of the occupation time of the particle on a given
lattice point, approaches U or W shaped distributions related to the arcsin
law. We show that when conditions of detailed balance are applied, these
distributions depend on the partition function of the problem, thus
establishing a relation between the non-ergodic dynamics and canonical
statistical mechanics. In the ergodic phase the distribution function of the
occupation times approaches a delta function centered on the value predicted
based on standard Boltzmann-Gibbs statistics. Relation of our work with single
molecule experiments is briefly discussed.Comment: 14 pages, 6 figure
Comment on "Mean First Passage Time for Anomalous Diffusion"
We correct a previously erroneous calculation [Phys. Rev. E 62, 6065 (2000)]
of the mean first passage time of a subdiffusive process to reach either end of
a finite interval in one dimension. The mean first passage time is in fact
infinite.Comment: To appear in Phys. Rev.
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