977 research outputs found
Generalized definition of time delay in scattering theory
We advocate for the systematic use of a symmetrized definition of time delay
in scattering theory. In two-body scattering processes, we show that the
symmetrized time delay exists for arbitrary dilated spatial regions symmetric
with respect to the origin. It is equal to the usual time delay plus a new
contribution, which vanishes in the case of spherical spatial regions. We also
prove that the symmetrized time delay is invariant under an appropriate mapping
of time reversal. These results are also discussed in the context of classical
scattering theory.Comment: 18 page
Uphill migration in coupled driven particle systems
In particle systems subject to a nonuniform drive, particle migration is
observed from the driven to the non--driven region and vice--versa, depending
on details of the hopping dynamics, leading to apparent violations of Fick's
law and of steady--state thermodynamics. We propose and discuss a very basic
model in the framework of independent random walkers on a pair of rings, one of
which features biased hopping rates, in which this phenomenon is observed and
fully explained.Comment: 8 pages, 10 figure
A class of CTRWs: Compound fractional Poisson processes
This chapter is an attempt to present a mathematical theory of compound
fractional Poisson processes. The chapter begins with the characterization of a
well-known L\'evy process: The compound Poisson process. The semi-Markov
extension of the compound Poisson process naturally leads to the compound
fractional Poisson process, where the Poisson counting process is replaced by
the Mittag-Leffler counting process also known as fractional Poisson process.
This process is no longer Markovian and L\'evy. However, several analytical
results are available and some of them are discussed here. The functional limit
of the compound Poisson process is an -stable L\'evy process, whereas
in the case of the compound fractional Poisson process, one gets an
-stable L\'evy process subordinated to the fractional Poisson process.Comment: 23 pages. To be published in a World Scientific book edited by Ralf
Metzle
On the dynamics of trap models in Z^d
We consider trap models on Z^d, namely continuous time Markov jump process on
Z^d with embedded chain given by a generic discrete time random walk, and whose
mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family
of positive iid random variables in the basin of attraction of an alpha-stable
law, 0<alpha<1. We may think of x as a trap, and tau_x as the depth of the trap
at x. We are interested in the trap process, namely the process that associates
to time t the depth of the currently visited trap. Our first result is the
convergence of the law of that process under suitable scaling. The limit
process is given by the jumps of a certain alpha-stable subordinator at the
inverse of another alpha-stable subordinator, correlated with the first
subordinator. For that result, the requirements for the embedded random walk
are a) the validity of a law of large numbers for its range, and b) the slow
variation at infinity of the tail of the distribution of its time of return to
the origin: they include all transient random walks as well as all planar
random walks, and also many one dimensional random walks. We then derive aging
results for the process, namely scaling limits for some two-time correlation
functions thereof, a strong form of which requires an assumption of transience,
stronger than a, b. The above mentioned scaling limit result is an averaged
result with respect to tau. Under an additional condition on the size of the
intersection of the ranges of two independent copies of the embeddded random
walk, roughly saying that it is small compared with the size of the range, we
derive a stronger scaling limit result, roughly stating that it holds in
probability with respect to tau. With that additional condition, we also
strengthen the aging results, from the averaged version mentioned above, to
convergence in probability with respect to tau.Comment: 36 pages, 5 figures. Replaces first version, with a correction
to/weakening of the statement of current Theorem 25, corrections to its proof
and that of Lemma 16, the addition of a subsection on integrated aging
results. Section 4, on convergence, was somewhat restructure
Subordination Pathways to Fractional Diffusion
The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and
under power law regime is splitted into three distinct random walks: (rw_1), a
random walk along the line of natural time, happening in operational time;
(rw_2), a random walk along the line of space, happening in operational
time;(rw_3), the inversion of (rw_1), namely a random walk along the line of
operational time, happening in natural time. Via the general integral equation
of CTRW and appropriate rescaling, the transition to the diffusion limit is
carried out for each of these three random walks. Combining the limits of
(rw_1) and (rw_2) we get the method of parametric subordination for generating
particle paths, whereas combination of (rw_2) and (rw_3) yields the
subordination integral for the sojourn probability density in space-time
fractional diffusion.Comment: 20 pages, 4 figure
Sojourn measures of Student and Fisher-Snedecor random fields
Limit theorems for the volumes of excursion sets of weakly and strongly
dependent heavy-tailed random fields are proved. Some generalizations to
sojourn measures above moving levels and for cross-correlated scenarios are
presented. Special attention is paid to Student and Fisher-Snedecor random
fields. Some simulation results are also presented.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ529 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Continuous time random walk and parametric subordination in fractional diffusion
The well-scaled transition to the diffusion limit in the framework of the
theory of continuous-time random walk (CTRW)is presented starting from its
representation as an infinite series that points out the subordinated character
of the CTRW itself. We treat the CTRW as a combination of a random walk on the
axis of physical time with a random walk in space, both walks happening in
discrete operational time. In the continuum limit we obtain a generally
non-Markovian diffusion process governed by a space-time fractional diffusion
equation. The essential assumption is that the probabilities for waiting times
and jump-widths behave asymptotically like powers with negative exponents
related to the orders of the fractional derivatives. By what we call parametric
subordination, applied to a combination of a Markov process with a positively
oriented L\'evy process, we generate and display sample paths for some special
cases.Comment: 28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'.
Denton, Texas, August 200
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