126 research outputs found
Fractal structures of normal and anomalous diffusion in nonlinear nonhyperbolic dynamical systems
A paradigmatic nonhyperbolic dynamical system exhibiting deterministic
diffusion is the smooth nonlinear climbing sine map. We find that this map
generates fractal hierarchies of normal and anomalous diffusive regions as
functions of the control parameter. The measure of these self-similar sets is
positive, parameter-dependent, and in case of normal diffusion it shows a
fractal diffusion coefficient. By using a Green-Kubo formula we link these
fractal structures to the nonlinear microscopic dynamics in terms of fractal
Takagi-like functions.Comment: 4 pages (revtex) with 4 figures (postscript
Sub-diffusion in External Potential: Anomalous hiding behind Normal
We propose a model of sub-diffusion in which an external force is acting on a
particle at all times not only at the moment of jump. The implication of this
assumption is the dependence of the random trapping time on the force with the
dramatic change of particles behavior compared to the standard continuous time
random walk model. Constant force leads to the transition from non-ergodic
sub-diffusion to seemingly ergodic diffusive behavior. However, we show it
remains anomalous in a sense that the diffusion coefficient depends on the
force and the anomalous exponent. For the quadratic potential we find that the
anomalous exponent defines not only the speed of convergence but also the
stationary distribution which is different from standard Boltzmann equilibrium.Comment: 6 pages, 3 figure
Emergence of L\'{e}vy Walks in Systems of Interacting Individuals
Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed
an intriguing behavior of swarming bacteria: they fundamentally change their
collective motion from simple diffusion into a superdiffusive L\'{e}vy walk
dynamics. We introduce a nonlinear non-Markovian persistent random walk model
that explains the emergence of superdiffusive L\'{e}vy walks. We show that the
alignment interaction between individuals can lead to the superdiffusive growth
of the mean squared displacement and the power law distribution of run length
with infinite variance. The main result is that the superdiffusive behavior
emerges as a nonlinear collective phenomenon, rather than due to the standard
assumption of the power law distribution of run distances from the inception.
At the same time, we find that the repulsion/collision effects lead to the
density dependent exponential tempering of power law distributions. This
qualitatively explains experimentally observed transition from superdiffusion
to the diffusion of mussels as their density increases (M. de Jager et al.,
Proc. R. Soc. B 281, 20132605 (2014))
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
Paradoxes of Subdiffusive Infiltration in Disordered Systems
Infiltration of diffusing particles from one material to another where the
diffusion mechanism is either normal or anomalous is a widely observed
phenomena. When the diffusion is anomalous we find interesting behaviors:
diffusion may lead to an averaged net drift from one material to another
even if all particles eventually flow in the opposite direction, or may lead to
a flow without drift. Starting with an underlying continuous time random walk
model we solve diffusion equations describing this problem. Similar drift
against flow is found in the quenched trap model. We argue that such a behavior
is a general feature of diffusion in disordered systems.Comment: 5 pages, 2 figure
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