64 research outputs found
Linear and fractal diffusion coefficients in a family of one dimensional chaotic maps
We analyse deterministic diffusion in a simple, one-dimensional setting
consisting of a family of four parameter dependent, chaotic maps defined over
the real line. When iterated under these maps, a probability density function
spreads out and one can define a diffusion coefficient. We look at how the
diffusion coefficient varies across the family of maps and under parameter
variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated
in terms of generalised Takagi functions, we derive exact, fully analytical
expressions for the diffusion coefficients. Typically, for simple maps these
quantities are fractal functions of control parameters. However, our family of
four maps exhibits both fractal and linear behavior. We explain these different
structures by looking at the topology of the Markov partitions and the ergodic
properties of the maps.Comment: 21 pages, 19 figure
From normal to anomalous diffusion in simple dynamical systems (Research on the Theory of Random Dynamical Systems and Fractal Geometry)
We have introduced to the problem of chaotic diffusion generated by deterministic dynamical systems. As the simplest examples possible we chose piecewise linear one-dimensional maps defined on the whole real line. For these dynamical systems the parameter-dependent diffusion coefficient can be calculated exactly analytically. We outlined a straightforward method of how to do so, which is based on evaluating fractal generalised Takagi functions. Surprisingly, the resulting diffusion coefficient exhibits fractal structures under parameter variation, which can be explained with respect to non-trivial dynamical correlations on microscopic scales. For certain classes of such maps the Hausdorff and the box counting dimensions of the corresponding diffusion coefficient curves under parameter variation have been calculated rigorously mathematically. Interestingly, these curves belong to a very special type of fractals for which both dimensions are exactly equal to one. The fractal structure emerges from logarithmic corrections in continuity properties that display an intricate dependence on parameter variation [24]. These fractal diffusion coefficients are not an artefact specific to one-dimensional maps. A line of work demonstrated that they also appear in Hamiltonian particle billiards [l] and even for particles moving in soft potential landscapes [25]. The latter system relates to electronic transport in artificial graphene that can be studied experimentally. Yet another string of work investigated anomalous diffusion in intermittent maps of Pomeau-Manneville type, which also turned out to display fractal parameter dependencies of suitably generalised anomalous diffusion coefficients [l], In this brief review we only discussed a simple type of random dynamical system as a second example generating non-trivial diffusion. We showed that mixing normal diffusive with localised dynamics randomly in time yields a novel type of intermittent dynamics characterised by the emergence of subdiffusion. To which extent the recipe that we proposed for obtaining this kind of random diffusive dynamical system can generate other types of anomalous diffusion remains to be explored
Capturing correlations in chaotic diffusion by approximation methods
We investigate three different methods for systematically approximating the
diffusion coefficient of a deterministic random walk on the line which contains
dynamical correlations that change irregularly under parameter variation.
Capturing these correlations by incorporating higher order terms, all schemes
converge to the analytically exact result. Two of these methods are based on
expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method
approximates Markov partitions and transition matrices by using the escape rate
theory of chaotic diffusion. We check the practicability of the different
methods by working them out analytically and numerically for a simple
one-dimensional map, study their convergence and critically discuss their
usefulness in identifying a possible fractal instability of parameter-dependent
diffusion, in case of dynamics where exact results for the diffusion
coefficient are not available.Comment: 11 pages, 5 figure
Dependence of chaotic diffusion on the size and position of holes
A particle driven by deterministic chaos and moving in a spatially extended
environment can exhibit normal diffusion, with its mean square displacement
growing proportional to the time. Here we consider the dependence of the
diffusion coefficient on the size and the position of areas of phase space
linking spatial regions (`holes') in a class of simple one-dimensional,
periodically lifted maps. The parameter dependent diffusion coefficient can be
obtained analytically via a Taylor-Green-Kubo formula in terms of a functional
recursion relation. We find that the diffusion coefficient varies
non-monotonically with the size of a hole and its position, which implies that
a diffusion coefficient can increase by making the hole smaller. We derive
analytic formulas for small holes in terms of periodic orbits covered by the
holes. The asymptotic regimes that we observe show deviations from the standard
stochastic random walk approximation. The escape rate of the corresponding open
system is also calculated. The resulting parameter dependencies are compared
with the ones for the diffusion coefficient and explained in terms of periodic
orbits.Comment: 12 pages, 5 figure
A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics
Analytically tractable dynamical systems exhibiting a whole range of normal
and anomalous deterministic diffusion are rare. Here we introduce a simple
non-chaotic model in terms of an interval exchange transformation suitably
lifted onto the whole real line which preserves distances except at a countable
set of points. This property, which leads to vanishing Lyapunov exponents, is
designed to mimic diffusion in non-chaotic polygonal billiards that give rise
to normal and anomalous diffusion in a fully deterministic setting. As these
billiards are typically too complicated to be analyzed from first principles,
simplified models are needed to identify the minimal ingredients generating the
different transport regimes. For our model, which we call the slicer map, we
calculate all its moments in position analytically under variation of a single
control parameter. We show that the slicer map exhibits a transition from
subdiffusion over normal diffusion to superdiffusion under parameter variation.
Our results may help to understand the delicate parameter dependence of the
type of diffusion generated by polygonal billiards. We argue that in different
parameter regions the transport properties of our simple model match to
different classes of known stochastic processes. This may shed light on
difficulties to match diffusion in polygonal billiards to a single anomalous
stochastic process.Comment: 15 pages, 3 figure
Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity
Stochastic processes play a key role for mathematically modeling a huge
variety of transport problems out of equilibrium. To formulate models of
stochastic dynamics the mainstream approach consists in superimposing random
fluctuations on a suitable deterministic evolution. These fluctuations are
sampled from probability distributions that are prescribed a priori, most
commonly as Gaussian or Levy. While these distributions are motivated by
(generalised) central limit theorems they are nevertheless unbounded. This
property implies the violation of fundamental physical principles such as
special relativity and may yield divergencies for basic physical quantities
like energy. It is thus clearly never valid in real-world systems by rendering
all these stochastic models ontologically unphysical. Here we solve the
fundamental problem of unbounded random fluctuations by constructing a
comprehensive theoretical framework of stochastic processes possessing finite
propagation velocity. Our approach is motivated by the theory of Levy walks,
which we embed into an extension of conventional Poisson-Kac processes. Our new
theory possesses an intrinsic flexibility that enables the modelling of many
different kinds of dynamical features, as we demonstrate by three examples. The
corresponding stochastic models capture the whole spectrum of diffusive
dynamics from normal to anomalous diffusion, including the striking Brownian
yet non Gaussian diffusion, and more sophisticated phenomena such as
senescence. Extended Poisson-Kac theory thus not only ensures by construction a
mathematical representation of physical reality that is ontologically valid at
all time and length scales. It also provides a toolbox of stochastic processes
that can be used to model potentially any kind of finite velocity dynamical
phenomena observed experimentally.Comment: 25 pages, 5 figure
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