3,204 research outputs found
Strong Correlations Between Fluctuations and Response in Aging Transport
Once the problem of ensemble averaging is removed, correlations between the
response of a single molecule to an external driving field , with the
history of fluctuations of the particle, become detectable. Exact analytical
theory for the continuous time random walk and numerical simulations for the
quenched trap model give the behaviors of the correlation between fluctuations
of the displacement in the aging period , and the response to bias
switched on at time . In particular in the dynamical phase where the
models exhibit aging we find finite correlations even in the asymptotic limit
, while in the non-aging phase the correlations are zero in the
same limit. Linear response theory gives a simple relation between these
correlations and the fractional diffusion coefficient.Comment: 5 page
Stable Equilibrium Based on L\'evy Statistics: A Linear Boltzmann Equation Approach
To obtain further insight on possible power law generalizations of Boltzmann
equilibrium concepts, a stochastic collision model is investigated. We consider
the dynamics of a tracer particle of mass , undergoing elastic collisions
with ideal gas particles of mass , in the Rayleigh limit . The
probability density function (PDF) of the gas particle velocity is
. Assuming a uniform collision rate and molecular chaos, we
obtain the equilibrium distribution for the velocity of the tracer particle
. Depending on asymptotic properties of we find
that is either the Maxwell velocity distribution or a L\'evy
distribution. In particular our results yield a generalized Maxwell
distribution based on L\'evy statistics using two approaches. In the first a
thermodynamic argument is used, imposing on the dynamics the condition that
equilibrium properties of the heavy tracer particle be independent of the
coupling to the gas particles, similar to what is found for a
Brownian particle in a fluid. This approach leads to a generalized temperature
concept. In the second approach it is assumed that bath particles velocity PDF
scales with an energy scale, i.e. the (nearly) ordinary temperature, as found
in standard statistical mechanics. The two approaches yield different types of
L\'evy equilibrium which merge into a unique solution only for the
Maxwell--Boltzmann case. Thus, relation between thermodynamics and statistical
mechanics becomes non-trivial for the power law case. Finally, the relation of
the kinetic model to fractional Fokker--Planck equations is discussed
CTRW Pathways to the Fractional Diffusion Equation
The foundations of the fractional diffusion equation are investigated based
on coupled and decoupled continuous time random walks (CTRW). For this aim we
find an exact solution of the decoupled CTRW, in terms of an infinite sum of
stable probability densities. This exact solution is then used to understand
the meaning and domain of validity of the fractional diffusion equation. An
interesting behavior is discussed for coupled memories (i.e., L\'evy walks).
The moments of the random walk exhibit strong anomalous diffusion, indicating
(in a naive way) the breakdown of simple scaling behavior and hence of the
fractional approximation. Still the Green function is described well
by the fractional diffusion equation, in the long time limit.Comment: 11 pages, 4 figure
noise for scale-invariant processes: How long you wait matters
We study the power spectrum which is estimated from a nonstationary signal.
In particular we examine the case when the signal is observed in a measurement
time window , namely the observation started after a waiting
time , and is the measurement duration. We introduce a generalized
aging Wiener-Khinchin theorem which relates between the spectrum and the time-
and ensemble-averaged correlation function for arbitrary and .
Furthermore we provide a general relation between the non-analytical behavior
of the scale-invariant correlation function and the aging noise.
We illustrate our general results with two-state renewal models with sojourn
times' distributions having a broad tail
Nonergodisity of a time series obeying L\'evy statistics
Time-averaged autocorrelation functions of a dichotomous random process
switching between 1 and 0 and governed by wide power law sojourn time
distribution are studied. Such a process, called a L\'evy walk, describes
dynamical behaviors of many physical systems, fluorescence intermittency of
semiconductor nanocrystals under continuous laser illumination being one
example. When the mean sojourn time diverges the process is non-ergodic. In
that case, the time average autocorrelation function is not equal to the
ensemble averaged autocorrelation function, instead it remains random even in
the limit of long measurement time. Several approximations for the distribution
of this random autocorrelation function are obtained for different parameter
ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the
power spectrum of the process is briefly discussed, and a nonstationary
Wiener-Khintchine theorem, relating the correlation functions and the power
spectrum is presented. The considered situation is in full contrast to the
usual assumptions of ergodicity and stationarity.Comment: 15 pages, 10 figure
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