114 research outputs found
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
Scale invariant Green-Kubo relation for time averaged diffusivity
In recent years it was shown both theoretically and experimentally that in
certain systems exhibiting anomalous diffusion the time and ensemble average
mean squared displacement are remarkably different. The ensemble average
diffusivity is obtained from a scaling Green-Kubo relation, which connects the
scale invariant non-stationary velocity correlation function with the transport
coefficient. Here we obtain the relation between time averaged diffusivity,
usually recorded in single particle tracking experiments, and the underlying
scale invariant velocity correlation function. The time averaged mean squared
displacement is given by where is the total measurement time and
the lag time. Here is the anomalous diffusion exponent
obtained from ensemble averaged measurements
while marks the growth or decline of the kinetic energy . Thus we establish a connection between exponents
which can be read off the asymptotic properties of the velocity correlation
function and similarly for the transport constant . We demonstrate our
results with non-stationary scale invariant stochastic and deterministic
models, thereby highlighting that systems with equivalent behavior in the
ensemble average can differ strongly in their time average. This is the case,
for example, if averaged kinetic energy is finite, i.e. , where
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