Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems

We consider a dynamical system with state space $M$, a smooth, compact subset of some ${\Bbb R}^n$, and evolution given by $T_t$, $x_t = T_t x$, $x \in M$; $T_t$ is invertible and the time $t$ may be discrete, $t \in {\Bbb Z}$, $T_t = T^t$, or continuous, $t \in {\Bbb R}$. Here we show that starting with a continuous positive initial probability density $\rho(x,0) > 0$, with respect to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb R}^n$, the expectation value of $\log \rho(x,t)$, with respect to any stationary (i.e. time invariant) measure $\nu(dx)$, is linear in $t$, $\nu(\log \rho(x,t)) = \nu(\log \rho(x,0)) + Kt$. $K$ depends only on $\nu$ and vanishes when $\nu$ is absolutely continuous wrt $dx$.Comment: 7 pages, plain TeX; [email protected], [email protected], [email protected], to appear in Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 8, Issue

Scaling of the electron dissipation range of solar wind turbulence

Electron scale solar wind turbulence has attracted great interest in recent years. Clear evidences have been given from the Cluster data that turbulence is not fully dissipated near the proton scale but continues cascading down to the electron scales. However, the scaling of the energy spectra as well as the nature of the plasma modes involved at those small scales are still not fully determined. Here we survey 10 years of the Cluster search-coil magnetometer (SCM) waveforms measured in the solar wind and perform a statistical study of the magnetic energy spectra in the frequency range [$1, 180$]Hz. We show that a large fraction of the spectra exhibit clear breakpoints near the electon gyroscale $\rho_e$, followed by steeper power-law like spectra. We show that the scaling below the electron breakpoint cannot be determined unambiguously due to instrumental limitations that will be discussed in detail. We compare our results to recent ones reported in other studies and discuss their implication on the physical mechanisms and the theoretical modeling of energy dissipation in the SW.Comment: 10 pages, submitte

Spin-orbit induced interference in polygon-structures

We investigate the spin-orbit induced spin-interference pattern of ballistic electrons travelling along any regular polygon. It is found that the spin-interference depends strongly on the Rashba and Dresselhaus spin-orbit constants as well as on the sidelength and alignment of the polygon. We derive the analytical formulae for the limiting cases of either zero Dresselhaus or zero Rashba spin-orbit coupling, including the result obtained for a circle. We calculate the nonzero Dresselhaus and Rashba case numerically for the square, triangle, hexagon, and circle and discuss the observability of the spin-interference which can potentially be used to measure the Rashba and Dresselhaus coefficients.Comment: 17 pages, 4 figure

Axiomatic geometrical optics, Abraham-Minkowski controversy, and photon properties derived classically

By restating geometrical optics within the field-theoretical approach, the classical concept of a photon (and, more generally, any elementary excitation) in arbitrary dispersive medium is introduced, and photon properties are calculated unambiguously. In particular, the canonical and kinetic momenta carried by a photon, as well as the two corresponding energy-momentum tensors of a wave, are derived from first principles of Lagrangian mechanics. As an example application of this formalism, the Abraham-Minkowski controversy pertaining to the definitions of these quantities is resolved for linear waves of arbitrary nature, and corrections to the traditional formulas for the photon kinetic energy-momentum are found. Several other applications of axiomatic geometrical optics to electromagnetic waves are also presented

Nonclassical correlations of phase noise and photon number in quantum nondemolition measurements

The continuous transition from a low resolution quantum nondemolition measurement of light field intensity to a precise measurement of photon number is described using a generalized measurement postulate. In the intermediate regime, quantization appears as a weak modulation of measurement probability. In this regime, the measurement result is strongly correlated with the amount of phase decoherence introduced by the measurement interaction. In particular, the accidental observation of half integer photon numbers preserves phase coherence in the light field, while the accidental observation of quantized values increases decoherence. The quantum mechanical nature of this correlation is discussed and the implications for the general interpretation of quantization are considered.Comment: 16 pages, 5 figures, final version to be published in Phys. Rev. A, Clarifications of the nature of the measurement result and the noise added in section I

Challenges in modelling the random structure correctly in growth mixture models and the impact this has on model mixtures

Lifecourse trajectories of clinical or anthropological attributes are useful for identifying how our early-life experiences influence later-life morbidity and mortality. Researchers often use growth mixture models (GMMs) to estimate such phenomena. It is common to place constrains on the random part of the GMM to improve parsimony or to aid convergence, but this can lead to an autoregressive structure that distorts the nature of the mixtures and subsequent model interpretation. This is especially true if changes in the outcome within individuals are gradual compared with the magnitude of differences between individuals. This is not widely appreciated, nor is its impact well understood. Using repeat measures of body mass index (BMI) for 1528 US adolescents, we estimated GMMs that required variance-covariance constraints to attain convergence. We contrasted constrained models with and without an autocorrelation structure to assess the impact this had on the ideal number of latent classes, their size and composition. We also contrasted model options using simulations. When the GMM variance-covariance structure was constrained, a within-class autocorrelation structure emerged. When not modelled explicitly, this led to poorer model fit and models that differed substantially in the ideal number of latent classes, as well as class size and composition. Failure to carefully consider the random structure of data within a GMM framework may lead to erroneous model inferences, especially for outcomes with greater within-person than between-person homogeneity, such as BMI. It is crucial to reflect on the underlying data generation processes when building such models

Theory of four-wave mixing of matter waves from a Bose-Einstein condensate

A recent experiment [Deng et al., Nature 398, 218(1999)] demonstrated four-wave mixing of matter wavepackets created from a Bose-Einstein condensate. The experiment utilized light pulses to create two high-momentum wavepackets via Bragg diffraction from a stationary Bose-Einstein condensate. The high-momentum components and the initial low momentum condensate interact to form a new momentum component due to the nonlinear self-interaction of the bosonic atoms. We develop a three-dimensional quantum mechanical description, based on the slowly-varying-envelope approximation, for four-wave mixing in Bose-Einstein condensates using the time-dependent Gross-Pitaevskii equation. We apply this description to describe the experimental observations and to make predictions. We examine the role of phase-modulation, momentum and energy conservation (i.e., phase-matching), and particle number conservation in four-wave mixing of matter waves, and develop simple models for understanding our numerical results.Comment: 18 pages Revtex preprint form, 13 eps figure