1,562 research outputs found
Dynamical Casimir effect in a periodically changing domain: A dynamical systems approach
We study the problem of the behavior of a quantum massless scalar field in
the space between two parallel infinite perfectly conducting plates, one of
them stationary, the other moving periodically. We reformulate the physical
problem into a problem about the asymptotic behavior of the iterates of a map
of the circle, and then apply results from theory of dynamical systems to study
the properties of the map. Many of the general mathematical properties of maps
of the circle translate into properties of the field in the cavity. For
example, we give a complete classification of the possible resonances in the
system, and show that small enough perturbations do not destroy the resonances.
We use some mathematical identities to give transparent physical interpretation
of the processes of creation and amplification of the quantum field due to the
motion of the boundary and to elucidate the similarities and the differences
between the classical and quantum fields in domains with moving boundaries.Comment: 19 pages, 7 figure
Energy Dissipation in Fractal-Forced Flow
The rate of energy dissipation in solutions of the body-forced 3-d
incompressible Navier-Stokes equations is rigorously estimated with a focus on
its dependence on the nature of the driving force. For square integrable body
forces the high Reynolds number (low viscosity) upper bound on the dissipation
is independent of the viscosity, consistent with the existence of a
conventional turbulent energy cascade. On the other hand when the body force is
not square integrable, i.e., when the Fourier spectrum of the force decays
sufficiently slowly at high wavenumbers, there is significant direct driving at
a broad range of spatial scales. Then the upper limit for the dissipation rate
may diverge at high Reynolds numbers, consistent with recent experimental and
computational studies of "fractal-forced'' turbulence.Comment: 14 page
Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity
Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. (C) 2008 American Institute of Physics.NSFMathematic
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Stratus cloud structure from MM-radar transects and satellite images: scaling properties and artifact detection with semi-discrete wavelet analysis
Spatial and/or temporal variabilities of clouds is of paramount importance for at least two in tensely researched sub-problems in global and regional climate modeling: (1) cloud-radiation interaction where correlations can trigger 3D radiative transfer effects; and (2) dynamical cloud modeling where the goal is to realistically reproduce the said correlations. We propose wavelets as a simple yet powerful way of quantifying cloud variability. More precisely, we use 'semi-discrete' wavelet transforms which, at least in the present statistical applications, have advantages over both its continuous and discrete counterparts found in the bulk of the wavelet literature. With the particular choice of normalization we adopt, the scale-dependence of the variance of the wavelet coefficients (i.e,, the wavelet energy spectrum) is always a better discriminator of transition from 'stationary' to 'nonstationary' behavior than conventional methods based on auto-correlation analysis, second-order structure function (a.k.a. the semi-variogram), or Fourier analysis. Indeed, the classic statistics go at best from monotonically scale- or wavenumber-dependent to flat at such a transition; by contrast, the wavelet spectrum changes the sign of its derivative with respect to scale. We apply 1D and 2D semi-discrete wavelet transforms to remote sensing data on cloud structure from two sources: (1) an upward-looking milli-meter cloud radar (MMCR) at DOE's climate observation site in Oklahoma deployed as part of the Atmospheric Radiation Measurement (ARM) Progrm; and (2) DOE's Multispectral Thermal Imager (MTI), a high-resolution space-borne instrument in sunsynchronous orbit that is described in sufficient detail for our present purposes by Weber et al. (1999). For each type of data, we have at least one theoretical prediction - with empirical validation already in existence - for a power-law relation for wavelet statistics with respect to scale. This is what is expected in physical (i.e., finite scaling range) fractal phenomena. In particular, we find long-range correlations in cloud structure coming from the important nonstationary regime. More surprisingly, we also uncover artifacts the data that are traceable either to instrumental noise (in the satellite data) or to smoothing assumptions (in the MMCR data processing). Finally, we discuss the potentially damaging ramifications the smoothing artifact can have on both cloud-radiation and cloud-modeling studies using MMCR data
Theory of Circle Maps and the Problem of One-Dimensional Optical Resonator with a Periodically Moving Wall
We consider the electromagnetic field in a cavity with a periodically
oscillating perfectly reflecting boundary and show that the mathematical theory
of circle maps leads to several physical predictions. Notably, well-known
results in the theory of circle maps (which we review briefly) imply that there
are intervals of parameters where the waves in the cavity get concentrated in
wave packets whose energy grows exponentially. Even if these intervals are
dense for typical motions of the reflecting boundary, in the complement there
is a positive measure set of parameters where the energy remains bounded.Comment: 34 pages LaTeX (revtex) with eps figures, PACS: 02.30.Jr, 42.15.-i,
42.60.Da, 42.65.Y
Regularity of critical invariant circles of the standard nontwist map
We study critical invariant circles of several noble rotation numbers at the edge of break-up for an area-preserving map of the cylinder, which violates the twist condition.These circles admit essentially unique parametrizations by rotational coordinates. We present a high accuracy computation of about 107 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and to show that, to the extent of numerical precision, it only depends on the tail of the continued fraction expansion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49075/2/non5_3_013.pd
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