On the time-dependent analysis of Gamow decay

Gamow's explanation of the exponential decay law uses complex 'eigenvalues' and exponentially growing 'eigenfunctions'. This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wavefunctions. Observing that the time evolution of any wavefunction is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover, the presentation can well be used in physics lectures to students.

Time-dependent flows in the coupled solar wind-magnetosphere-ionosphere system

Concepts of time-dependent flow in the coupled solar wind-magnetosphere-ionosphere system are discussed and compared with the frequently-adopted steady-state paradigm. Flows are viewed as resulting from departures of the system from equilibrium excited by dayside and nightside reconnection processes, with the flows then taking the system back towards a new equilibrium configuration. The response of the system to reconnection impulses, continuous but unbalanced reconnection and balanced steady-state reconnection are discussed in these terms. It is emphasized that in the time-dependent case the ionospheric and interplanetary electric fields are generally inductively decoupled from each other; a simple mapping of the interplanetary electric field along equipotential field lines into the ionosphere occurs only in the electrostatic steady-state case.

Unrestricted Skyrme-tensor time-dependent Hartree-Fock model and its application to the nuclear response from spherical to triaxial nuclei

The nuclear time-dependent Hartree-Fock model formulated in three-dimensional space, based on the full standard Skyrme energy density functional complemented with the tensor force, is presented. Full self-consistency is achieved by the model. The application to the isovector giant dipole resonance is discussed in the linear limit, ranging from spherical nuclei (16O and 120Sn) to systems displaying axial or triaxial deformation (24Mg, 28Si, 178Os, 190W and 238U). Particular attention is paid to the spin-dependent terms from the central sector of the functional, recently included together with the tensor. They turn out to be capable of producing a qualitative change on the strength distribution in this channel. The effect on the deformation properties is also discussed. The quantitative effects on the linear response are small and, overall, the giant dipole energy remains unaffected. Calculations are compared to predictions from the (quasi)-particle random-phase approximation and experimental data where available, finding good agreement

Time-dependent gel to gel transformation of a peptide based supramolecular gelator

A dipeptide with a long fatty acid chain at its N-terminus gives hydrogels in phosphate buffer in the pH range 7.0–8.5. The hydrogel with a gelator concentration of 0.45% (w/v) at pH 7.46 (physiological pH) provides a very good platform to study dynamic changes within a supramolecular framework as it exhibits remarkable change in its appearance with time. Interestingly, the first formed transparent hydrogel gradually transforms into a turbid gel within 2 days. These two forms of the hydrogel have been thoroughly investigated by using small angle X-ray scattering (SAXS), powder X-ray diffraction (PXRD), field emission scanning electron microscopic (FE-SEM) and high-resolution transmission electron microscopic (HR-TEM) imaging, FT-IR and rheometric analyses. The SAXS and low angle PXRD studies substantiate different packing arrangements for the gelator molecules for these two different gel states (the freshly prepared and the aged hydrogel). Moreover, rheological studies of these two gels reveal that the aged gel is stiffer than the freshly prepared gel.

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) J. Math. Biol. 45(2), 125-152. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.

A time-domain probe method for three-dimensional rough surface reconstructions

The task of this paper is to develop a Time-Domain Probe Method for the reconstruction of impenetrable scatterers. The basic idea of the method is to use pulses in the time domain and the time-dependent response of the scatterer to reconstruct its location and shape. The method is based on the basic causality principle of timedependent scattering. The method is independent of the boundary condition and is applicable for limited aperture scattering data. In particular, we discuss the reconstruction of the shape of a rough surface in three dimensions from time-domain measurements of the scattered field. In practise, measurement data is collected where the incident field is given by a pulse. We formulate the time-domain fieeld reconstruction problem equivalently via frequency-domain integral equations or via a retarded boundary integral equation based on results of Bamberger, Ha-Duong, Lubich. In contrast to pure frequency domain methods here we use a time-domain characterization of the unknown shape for its reconstruction. Our paper will describe the Time-Domain Probe Method and relate it to previous frequency-domain approaches on sampling and probe methods by Colton, Kirsch, Ikehata, Potthast, Luke, Sylvester et al. The approach significantly extends recent work of Chandler-Wilde and Lines (2005) and Luke and Potthast (2006) on the timedomain point source method. We provide a complete convergence analysis for the method for the rough surface scattering case and provide numerical simulations and examples.

The point source method for inverse scattering in the time domain

Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time-harmonic scattering, or nearly time-harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non-time-harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time-dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi-frequency data. Copyright (C) 2006 John Wiley & Sons, Ltd.

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

Modeling entangled dynamics: comparison between stochastic single-chain and multichain models

To test the effectiveness of stochastic single-chain models in describing the dynamics of entangled polymers, we systematically compare one such model; the slip-spring model; to a multichain model solved using stochastic molecular dynamics(MD) simulations (the Kremer-Grest model). The comparison involves investigating if the single-chain model can adequately describe both a microscopic dynamical and a macroscopic rheological quantity for a range of chain lengths. Choosing a particular chain length in the slip-spring model, the parameter values that best reproduce the mean-square displacement of a group of monomers is determined by fitting toMDdata. Using the same set of parameters we then test if the predictions of the mean-square displacements for other chain lengths agree with the MD calculations. We followed this by a comparison of the time dependent stress relaxation moduli obtained from the two models for a range of chain lengths. After identifying a limitation of the original slip-spring model in describing the static structure of the polymer chain as seen in MD, we remedy this by introducing a pairwise repulsive potential between the monomers in the chains. Poor agreement of the mean-square monomer displacements at short times can be rectified by the use of generalized Langevin equations for the dynamics and resulted in significantly improved agreement.

Social networks: evolving graphs with memory dependent edges

The plethora, and mass take up, of digital communication tech- nologies has resulted in a wealth of interest in social network data collection and analysis in recent years. Within many such networks the interactions are transient: thus those networks evolve over time. In this paper we introduce a class of models for such networks using evolving graphs with memory dependent edges, which may appear and disappear according to their recent history. We consider time discrete and time continuous variants of the model. We consider the long term asymptotic behaviour as a function of parameters controlling the memory dependence. In particular we show that such networks may continue evolving forever, or else may quench and become static (containing immortal and/or extinct edges). This depends on the ex- istence or otherwise of certain infinite products and series involving age dependent model parameters. To test these ideas we show how model parameters may be calibrated based on limited samples of time dependent data, and we apply these concepts to three real networks: summary data on mobile phone use from a developing region; online social-business network data from China; and disaggregated mobile phone communications data from a reality mining experiment in the US. In each case we show that there is evidence for memory dependent dynamics, such as that embodied within the class of models proposed here.

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.

Modelling the time-evolution of phytoplankton size spectra from satellite remote sensing

A dynamic size-structured model is developed for phytoplankton and nutrients in the oceanic mixed layer and applied to extract phytoplankton biomass at discrete size fractions from remotely sensed, ocean-colour data. General relationships between cell size and biophysical processes (such as sinking, grazing, and primary production) of phytoplankton were included in the model through a bottom–up approach. Time-dependent, mixed-layer depth was used as a forcing variable, and a sequential data-assimilation scheme was implemented to derive model trajectories. From a given time-series, the method produces estimates of size-structured biomass at every observation, so estimates seasonal succession of individual phytoplankton size, derived here from remote sensing for the first time. From these estimates, normalized phytoplankton biomass size spectra over a period of 9 years were calculated for one location in the North Atlantic. Further analysis demonstrated that strong relationships exist between the seasonal trends of the estimated size spectra and the mixed-layer depth, nutrient biomass, and total chlorophyll. The results contain useful information on the time-dependent biomass flux in the pelagic ecosystem.

Solvent-Dependent Formation of Os(0) Complexes by Electrochemical Reduction of [Os(CO)(2,2′-bipyridine)(L)Cl2]; L = Cl−,PrCN

Cyclic voltammetry and ultraviolet−visible/infrared (UV−vis/IR) spectroelectrochemistry were used to study the cathodic electrochemical behavior of the osmium complexes mer-[OsIII(CO) (bpy)Cl3] (bpy = 2,2′-bipyridine) and trans(Cl)-[OsII(CO) (PrCN)(bpy)Cl2] at variable temperature in different solvents (tetrahydrofuran (THF), butyronitrile (PrCN), acetonitrile (MeCN)) and electrolytes (Bu4NPF6, Bu4NCl). The precursors can be reduced to mer-[OsII(CO) (bpy•−)Cl3]2− and trans(Cl)-[OsII(CO)(PrCN) (bpy•−)Cl2]−, respectively, which react rapidly at room temperature, losing the chloride ligands and forming Os(0) species. mer-[OsIII(CO) (bpy)Cl3] is reduced in THF to give ultimately an Os−Os-bonded polymer, probably [Os0(CO) (THF)-(bpy)]n, whereas in PrCN the well-soluble, probably mononuclear [Os0(CO) (PrCN)(bpy)], species is formed. The same products were observed for the 2 electron reduction of trans(Cl)-[OsII(CO)(PrCN) (bpy)Cl2] in both solvents. In MeCN, similar to THF, the[Os0(CO) (MeCN)(bpy)]n polymer is produced. It is noteworthy that the bpy ligand in mononuclear [Os0(CO) (PrCN)(bpy)] is reduced to the corresponding radical anion at a significantly less negative potential than it is in polymeric [Os0(CO) (THF)(bpy)]n: ΔE1/2 = 0.67 V. Major differences also exist in the IR spectra of the Os(0) species: the polymer shows a broad ν(CO) band at much smaller wavenumbers compared to the soluble Os(0) monomer that exhibits a characteristic ν(Pr-CN) band below 2200 cm−1 in addition to the intense and narrow ν(CO) absorption band. For the first time, in this work the M0-bpy(M = Ru, Os) mono- and dicarbonyl species soluble in PrCN have been formulated as a mononuclear complex. Density functional theory (DFT) and time-dependent-DFT calculations confirm the Os(0) oxidation state and suggest that [Os0(CO)(PrCN)(bpy)] is a square planar moiety. The reversible bpy-based reduction of [Os0(CO) (PrCN)(bpy)] triggers catalytic reduction of CO2 to CO and HCOO−.