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Generalized space-time fractional diffusion equation with composite fractional time derivative
(2012)
We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grunwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of the form delta(x). t-beta/Gamma(1-beta) (beta > 0).
Black hole initial data are usually produced using Bowen-York-type puncture initial data or by applying an excision boundary condition. The benefits of the Bowen-York initial data are the ability to specify the spin and momentum of the system as parameters of the initial data. In an attempt to extend these benefits to other formulations of the Einstein constraints, the puncture method is reformulated using distributions as source terms. It is shown how the Bowen-York puncture black hole initial data and the trumpet variation are generated by distributional sources. A heuristic argument is presented to argue that these sources are the general sources of spin and momentum. In order to clarify the meaning of other distributional sources, an exact family of initial data with generalized sources to the Hamiltonian constraint are studied; spinning trumpet black hole initial data and black hole initial data with higher order momentum sources are also studied.
The problem of how complex quantum systems eventually come to rest lies at the heart of statistical mechanics. The maximum-entropy principle describes which quantum states can be expected in equilibrium, but not how closed quantum many-body systems dynamically equilibrate. Here, we report the experimental observation of the non-equilibrium dynamics of a density wave of ultracold bosonic atoms in an optical lattice in the regime of strong correlations. Using an optical superlattice, we follow its dynamics in terms of quasi-local densities, currents and coherences-all showing a fast relaxation towards equilibrium values. Numerical calculations based on matrix-product states are in an excellent quantitative agreement with the experimental data. The system fulfills the promise of being a dynamical quantum simulator, in that the controlled dynamics runs for longer times than present classical algorithms can keep track of.
The electropolymerization of 3,4-(2,2-dibutylpropylenedioxy)thiophene (ProDOT-Bu-2) onto single carbon fiber microelectrode (SCFME) was conducted in acetonitrile (ACN) containing sodium perchlorate (NaClO4) as electrolyte and investigated by cyclic voltammetry (CV). The nanostructured films of poly[3,4-(2,2-dibutyl-propyleneclioxy)thiophene] (PProDOT-Bu-2) which were depositing showed complete reversible redox behavior in monomer-free electrolyte solution.
The capacitive behavior of the films was investigated by electrochemical impedance spectroscopy (EIS) at applied potentials from 0.1 V to 1.3 V. The analysis by equivalent circuit modeling revealed an applied potential around 0.4V to be most suitable for the system PProDOT-Bu-2/SCFME as a double layer supercapacitor component inducing a double layer capacitance C-d, value of 62 mFcm(-2).
Relative magnetic helicity, as a conserved quantity of ideal magnetohydrodynamics, has been highlighted as an important quantity to study in plasma physics. Due to its nonlocal nature, its estimation is not straightforward in both observational and numerical data. In this study we derive expressions for the practical computation of the gauge-independent relative magnetic helicity in three-dimensional finite domains. The derived expressions are easy to implement and rapid to compute. They are derived in Cartesian coordinates, but can be easily written in other coordinate systems. We apply our method to a numerical model of a force-free equilibrium containing a flux rope, and compare the results with those obtained employing known half-space equations. We find that our method requires a much smaller volume than half-space expressions to derive the full helicity content. We also prove that values of relative magnetic helicity of different magnetic fields can be compared with each other in the same sense as free-energy values can. Therefore, relative magnetic helicity can be meaningfully and directly compared between different datasets, such as those from different active regions, but also within the same dataset at different times. Typical applications of our formulae include the helicity computation in three-dimensional models of the solar atmosphere, e.g., coronal-field reconstructions by force-free extrapolation and discretized magnetic fields of numerical simulations.
We study the flux emergence process in NOAA active region 11024, between 29 June and 7 July 2009, by means of multi-wavelength observations and nonlinear force-free extrapolation. The main aim is to extend previous investigations by combining, as much as possible, high spatial resolution observations to test our present understanding of small-scale (undulatory) flux emergence, whilst putting these small-scale events in the context of the global evolution of the active region. The combination of these techniques allows us to follow the whole process, from the first appearance of the bipolar axial field on the east limb, until the buoyancy instability could set in and raise the main body of the twisted flux tube through the photosphere, forming magnetic tongues and signatures of serpentine field, until the simplification of the magnetic structure into a main bipole by the time the active region reaches the west limb. At the crucial time of the main emergence phase high spatial resolution spectropolarimetric measurements of the photospheric field are employed to reconstruct the three-dimensional structure of the nonlinear force-free coronal field, which is then used to test the current understanding of flux emergence processes. In particular, knowledge of the coronal connectivity confirms the identity of the magnetic tongues as seen in their photospheric signatures, and it exemplifies how the twisted flux, which is emerging on small scales in the form of a sea-serpent, is subsequently rearranged by reconnection into the large-scale field of the active region. In this way, the multi-wavelength observations combined with a nonlinear force-free extrapolation provide a coherent picture of the emergence process of small-scale magnetic bipoles, which subsequently reconnect to form a large-scale structure in the corona.
The Sun is surrounded by a 10^6 K hot atmosphere, the corona. The corona and the solar wind are fully ionized, and therefore in the plasma state. Magnetic fields play an important role in a plasma, since they bind electrically charged particles to their field lines. EUV spectroscopes, like the SUMER instrument on-board the SOHO spacecraft, reveal a preferred heating of coronal ions and strong temperature anisotropies. Velocity distributions of electrons can be measured directly in the solar wind, e.g. with the 3DPlasma instrument on-board the WIND satellite. They show a thermal core, an anisotropic suprathermal halo, and an anti-solar, magnetic-field-aligned, beam or "strahl". For an understanding of the physical processes in the corona, an adequate description of the plasma is needed. Magnetohydrodynamics (MHD) treats the plasma simply as an electrically conductive fluid. Multi-fluid models consider e.g. protons and electrons as separate fluids. They enable a description of many macroscopic plasma processes. However, fluid models are based on the assumption of a plasma near thermodynamic equilibrium. But the solar corona is far away from this. Furthermore, fluid models cannot describe processes like the interaction with electromagnetic waves on a microscopic scale. Kinetic models, which are based on particle velocity distributions, do not show these limitations, and are therefore well-suited for an explanation of the observations listed above. For the simplest kinetic models, the mirror force in the interplanetary magnetic field focuses solar wind electrons into an extremely narrow beam, which is contradicted by observations. Therefore, a scattering mechanism must exist that counteracts the mirror force. In this thesis, a kinetic model for electrons in the solar corona and wind is presented that provides electron scattering by resonant interaction with whistler waves. The kinetic model reproduces the observed components of solar wind electron distributions, i.e. core, halo, and a "strahl" with finite width. But the model is not only applicable on the quiet Sun. The propagation of energetic electrons from a solar flare is studied, and it is found that scattering in the direction of propagation and energy diffusion influence the arrival times of flare electrons at Earth approximately to the same degree. In the corona, the interaction of electrons with whistler waves does not only lead to scattering, but also to the formation of a suprathermal halo, as it is observed in interplanetary space. This effect is studied both for the solar wind as well as the closed volume of a coronal magnetic loop. The result is of fundamental importance for solar-stellar relations. The quiet solar corona always produces suprathermal electrons. This process is closely related to coronal heating, and can therefore be expected in any hot stellar corona. In the second part of this thesis it is detailed how to calculate growth or damping rates of plasma waves from electron velocity distributions. The emission and propagation of electron cyclotron waves in the quiet solar corona, and that of whistler waves during solar flares, is studied. The latter can be observed as so-called fiber bursts in dynamic radio spectra, and the results are in good agreement with observed bursts.
Thin films of ferroelectric beta-phase poly(vinylidene fluoride) (PVDF) were spin-coated from a solution that contained small amounts of the ionic liquid (IL) 1-ethyl-3-methylimidazolium nitrate. A remanent polarization of 60 mC/m(2) and a quasi-static pyroelectric coefficient of 19 mu C/m(2)K at 30 degrees C were observed in the films. It is suggested that the IL promotes the formation of the beta phase through dipolar interactions between PVDF chain-molecules and the IL. The dipolar interactions are identified as Coulomb attraction between hydrogen atoms in PVDF chains and anions in IL. The strong crystallinity increase is probably caused by the same dipolar interaction as well.
We compare the growth dynamics of the three n-alkanes C36H74, C40H82, and C44H90 on SiO2 using real-time and in situ energy-dispersive x-ray reflectivity. All molecules investigated align in an upright-standing orientation on the substrate and exhibit a transition from layer-by-layer growth to island growth after about 4 monolayers under the conditions employed. Simultaneous fits of the reflected intensity at five distinct points in reciprocal space show that films formed by longer n-alkanes roughen faster during growth. This behavior can be explained by a chain-length dependent height of the Ehrlich-Schwoebel barrier. Further x-ray diffraction measurements after growth indicate that films consisting of longer n-alkanes also incorporate more lying-down molecules in the top region. While the results reveal behavior typical for chain-like molecules, the findings can also be useful for the optimization of organic field effect transistors where smooth interlayers of n-alkanes without coexistence of two or more molecular orientations are required.
Context. The variation of the dimensionless fundamental physical constant mu = m(p)/m(e) - the proton to electron mass ratio - can be constrained via observation of Lyman and Werner lines of molecular hydrogen in the spectra of damped Lyman alpha systems (DLAs) in the line of sight to distant QSOs.
Aims. Our intention is to maximize the possible precision of quasar absorption spectroscopy with regard to the investigation of the variation of the proton-to-electron mass-ratio mu. The demand for precision requires an understanding of the errors involved and effective techniques to handle present systematic errors.
Methods. An analysis based on UVES high resolution data sets of QSO 0347-383 and its DLA is put forward and new approaches to some of the steps involved in the data analysis are introduced. We apply corrections for the observed offsets between discrete spectra and for the first time we find indications for inter-order distortions.
Results. Drawing on VLT-UVES observations of QSO 0347-383 in 2009 our analysis yields Delta mu/mu = (4.3 +/- 7.2) x 10(-6) at z(abs) = 3.025.
Conclusions. Current analyzes tend to underestimate the impact of systematic errors. Based on the scatter of the measured redshifts and the corresponding low significance of the redshift-sensitivity correlation we estimate the limit of accuracy of line position measurements to similar to 220 m s (1), consisting of roughly 150 m s (1) due to the uncertainty of the absorption line fit and about 150 m s (1) allocated to systematics related to instrumentation and calibration.