862 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 $F$, 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 $(0,t_a)$, and the response to bias
switched on at time $t_a$. In particular in the dynamical phase where the
models exhibit aging we find finite correlations even in the asymptotic limit
$t_a \to \infty$, 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

### From the area under the Bessel excursion to anomalous diffusion of cold atoms

Levy flights are random walks in which the probability distribution of the
step sizes is fat-tailed. Levy spatial diffusion has been observed for a
collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice.
Using the semiclassical theory of Sisyphus cooling, we treat the problem as a
coupled Levy walk, with correlations between the length and duration of the
excursions. The problem is related to the area under Bessel excursions,
overdamped Langevin motions that start and end at the origin, constrained to
remain positive, in the presence of an external logarithmic potential. In the
limit of a weak potential, the Airy distribution describing the areal
distribution of the Brownian excursion is found. Three distinct phases of the
dynamics are studied: normal diffusion, Levy diffusion and, below a certain
critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the
paper is the analytical calculation of the joint probability density function
from a newly developed theory of the area under the Bessel excursion. The
latter describes the spatiotemporal correlations in the problem and is the
microscopic input needed to characterize the spatial diffusion of the atomic
cloud. A modified Montroll-Weiss (MW) equation for the density is obtained,
which depends on the statistics of velocity excursions and meanders. The
meander, a random walk in velocity space which starts at the origin and does
not cross it, describes the last jump event in the sequence. In the anomalous
phases, the statistics of meanders and excursions are essential for the
calculation of the mean square displacement, showing that our correction to the
MW equation is crucial, and points to the sensitivity of the transport on a
single jump event. Our work provides relations between the statistics of
velocity excursions and meanders and that of the diffusivity.Comment: Supersedes arXiv: 1305.008

### Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos

Weakly chaotic non-linear maps with marginal fixed points have an infinite
invariant measure. Time averages of integrable and non-integrable observables
remain random even in the long time limit. Temporal averages of integrable
observables are described by the Aaronson-Darling-Kac theorem. We find the
distribution of time averages of non-integrable observables, for example the
time average position of the particle. We show how this distribution is related
to the infinite invariant density. We establish four identities between
amplitude ratios controlling the statistics of the problem.Comment: 5 pages, 3 figure

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