1,353 research outputs found
Numerical coalescence of chaotic trajectories
Pairs of numerically computed trajectories of a chaotic system may coalesce because of finite arithmetic precision. We analyse an example of this phenomenon, showing that it occurs surprisingly frequently. We argue that our model belongs to a universality class of chaotic systems where this numerical coincidence effect can be described by mapping it to a first-passage process. Our results are applicable to aggregation of small particles in random flows, as well as to numerical investigation of chaotic systems
Super-diffusion in optical realizations of Anderson localization
We discuss the dynamics of particles in one dimension in potentials that are
random both in space and in time. The results are applied to recent optics
experiments on Anderson localization, in which the transverse spreading of a
beam is suppressed by random fluctuations in the refractive index. If the
refractive index fluctuates along the direction of the paraxial propagation of
the beam, the localization is destroyed. We analyze this broken localization,
in terms of the spectral decomposition of the potential. When the potential has
a discrete spectrum, the spread is controlled by the overlap of Chirikov
resonances in phase space. As the number of Fourier components is increased,
the resonances merge into a continuum, which is described by a Fokker-Planck
equation. We express the diffusion coefficient in terms of the spectral
intensity of the potential. For a general class of potentials that are commonly
used in optics, the solutions of the Fokker-Planck equation exhibit anomalous
diffusion in phase space, implying that when Anderson localization is broken by
temporal fluctuations of the potential, the result is transport at a rate
similar to a ballistic one or even faster. For a class of potentials which
arise in some existing realizations of Anderson localization atypical behavior
is found.Comment: 11 pages, 2 figure
Magnetic Dipole Absorption of Radiation in Small Conducting Particles
We give a theoretical treatment of magnetic dipole absorption of
electromagnetic radiation in small conducting particles, at photon energies
which are large compared to the single particle level spacing, and small
compared to the plasma frequency. We discuss both diffusive and ballistic
electron dynamics for particles of arbitrary shape.
The conductivity becomes non-local when the frequency is smaller than the
frequency \omega_c characterising the transit of electrons from one side of the
particle to the other, but in the diffusive case \omega_c plays no role in
determining the absorption coefficient. In the ballistic case, the absorption
coefficient is proportional to \omega^2 for \omega << \omega_c, but is a
decreasing function of \omega for \omega >> \omega_c.Comment: 25 pages of plain TeX, 2 postscipt figure
Distribution of Oscillator Strengths for Recombination of Localised Excitons in Two Dimensions
We investigate the distribution of oscillator strengths for the recombination
of excitons in a two dimensional sample, trapped in local minima of the
confinement potential: the results are derived from a statistical topographic
model of the potential. The predicted distribution of oscillator strengths is
very different from the Porter-Thomas disribution which usually characterises
disordered systems, and is notable for the fact that small oscillator strengths
are extremely rare.Comment: Plain TeX, 11 pages, 2 of 3 Postscript figures, to appear in "Chaos,
Solitons and Fractals" special issue on Mesoscopic Physics, July 199
Kinematically Cold Populations at Large Radii in the Draco and Ursa Minor Dwarf Spheroidals
We present projected velocity dispersion profiles for the Draco and Ursa
Minor (UMi) dwarf spheroidal galaxies based on 207 and 162 discrete stellar
velocities, respectively. Both profiles show a sharp decline in the velocity
dispersion outside ~30 arcmin (Draco) and ~40 arcmin (UMi). New, deep
photometry of Draco reveals a break in the light profile at ~25 arcmin. These
data imply the existence of a kinematically cold population in the outer parts
of both galaxies. Possible explanations of both the photometric and kinematic
data in terms of both equilibrium and non-equilibrium models are discussed in
detail. We conclude that these data challenge the picture of dSphs as simple,
isolated stellar systems.Comment: 5 pages, accepted for publication in ApJ Letter
Convergent Chaos
Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we demonstrate that this criterion should be refined. Despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods, as revealed by our investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow). We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions. Power laws, indicative of scale-free features, characterize the distribution of particles in the system. We use large-deviation and extreme-value statistics to explain the effect. Our results show that the interpretation of the 'butterfly effect' needs to be carefully qualified. We argue that the combination of mixing and clustering processes makes our specific model relevant to understanding the evolution of simple organisms. Lastly, this notion of convergent chaos, which implies the existence of conditions for which uncertainties are unexpectedly small, may also be relevant to the valuation of insurance and futures contracts
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