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.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

The Klein-Gordon equation in a domain with time-dependent boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.

Time series analysis of lava flux

We have applied time series analytical techniques to the flux of lava from an extrusive eruption. Tilt data acting as a proxy for flux are used in a case study of the May–August 1997 period of the eruption at Soufrière Hills Volcano, Montserrat. We justify the use of such a proxy by simple calibratory arguments. Three techniques of time series analysis are employed: spectral, spectrogram and wavelet methods. In addition to the well-known ~9-hour periodicity shown by these data, a previously unknown periodic flux variability is revealed by the wavelet analysis as a 3-day cycle of frequency modulation during June–July 1997, though the physical mechanism responsible is not clear. Such time series analysis has potential for other lava flux proxies at other types of volcanoes.

Integrable evolution equations in time-dependent domains

We discuss the implementation of a method of solving initial boundary value problems in the case of integrable evolution equations in a time-dependent domain. This method is applied to a dispersive linear evolution equation with spatial derivatives of arbitrary order and to the defocusing nonlinear Schrödinger equation, in the domain l(t)

An in situ time-dependent study of the photodimerisation of chloro-derivatives of trans-cinnamic acid using infrared microspectroscopy with a synchrotron radiation source

The photodimerisation of single crystals of substituted cinnamic acid has been monitored continuously by infrared microscopy using a synchrotron source. The beta-form of 2,4-dichloro-trans-cinnamic acid dimerises under ultraviolet irradiation to form the corresponding beta-truxinic acid derivative in a reaction which follows strictly first order kinetics. By contrast the corresponding reactions in single crystals of beta-2-chloro-trans-cinnamic acid and beta-4-chloro-trans-cinnamic acid deviate somewhat from first order kinetics as a result of solid-state effects. In all three cases the reactions proceed smoothly from monomer to dimer with no hint of any reaction intermediate.

A comparison of two time-domain analysis procedures in the determination of VO2 kinetics by pseudorandom binary sequence exercise testing

The purpose of this study was to apply and compare two time-domain analysis procedures in the determination of oxygen uptake (VO2) kinetics in response to a pseudorandom binary sequence (PRBS) exercise test. PRBS exercise tests have typically been analysed in the frequency domain. However, the complex interpretation of frequency responses may have limited the application of this procedure in both sporting and clinical contexts, where a single time measurement would facilitate subject comparison. The relative potential of both a mean response time (MRT) and a peak cross-correlation time (PCCT) was investigated. This study was divided into two parts: a test-retest reliability study (part A), in which 10 healthy male subjects completed two identical PRBS exercise tests, and a comparison of the VO2 kinetics of 12 elite endurance runners (ER) and 12 elite sprinters (SR; part B). In part A, 95% limits of agreement were calculated for comparison between MRT and PCCT. The results of part A showed no significant difference between test and retest as assessed by MRT [mean (SD) 42.2 (4.2) s and 43.8 (6.9) s] or by PCCT [21.8 (3.7) s and 22.7 (4.5) s]. Measurement error (%) was lower for MRT in comparison with PCCT (16% and 25%, respectively). In part B of the study, the VO2 kinetics of ER were significantly faster than those of SR, as assessed by MRT [33.4 (3.4) s and 39.9 (7.1) s, respectively; P<0.01] and PCCT [20.9 (3.8) s and 24.8 (4.5) s; P < 0.05]. It is possible that either analysis procedure could provide a single test measurement Of VO2 kinetics; however, the greater reliability of the MRT data suggests that this method has more potential for development in the assessment Of VO2 kinetics by PRBS exercise testing.

The effect of vessel diameter on time dependent gas hold-up variations in highly viscous impeller agitated liquids

Time dependent gas hold-up generated in the 0.3 and 0.6 m diameter vessels using high viscosity castor oil and carboxy methyl cellulose (CMC) solution was compared on the basis of impeller speed (N) and gas velocity (V-G). Two types of hold-up were distinguished-the hold-up due to tiny bubbles (epsilon(ft)) and total hold-up (epsilon(f)), which included large and tiny bubbles. It was noted that vessel diameter (i.e. the scale of operation) significantly influences (i) the trends and the values of epsilon(f) and epsilon(ft), and (ii) the values of tau (a constant reflecting the time dependency of hold-up). The results showed that a scale independent correlation for gas hold-up of the form epsilon(f) or epsilon(ft) = A(N or P-G/V)(a) (V-G)(b), where "a" and "b" are positive constants is not appropriate for viscous liquids. This warrants further investigations into the effect of vessel diameter on gas hold-up in impeller agitated high viscosity liquids (mu or mu(a) > 0.4 Pa s). (C) 2003 Elsevier B.V. All rights reserved.

Time-dependent orientation coupling in equilibrium polymer melts

The motion in concentrated polymer systems is described by either the Rouse or the reptation model, which both assume that the relaxation of each polymer chain is independent of the surrounding chains. This, however, is in contradiction with several experiments. In this Letter, we propose a universal description of orientation coupling in polymer melts in terms of the time-dependent coupling parameter κ(t). We use molecular dynamics simulations to show that the coupling parameter increases with time, reaching about 50% at long times, independently of the chain length or blend composition. This leads to predictions of component dynamics in mixtures of different molecular weights from the knowledge of monodisperse dynamics for unentangled melts. Finally, we demonstrate that entanglements do not play a significant role in the observed coupling. © 2010 The American Physical Society

Summer drought in northern midlatitudes in a time-dependent CO2 climate experiment

A time-dependent climate-change experiment with a coupled ocean–atmosphere general circulation model has been used to study changes in the occurrence of drought in summer in southern Europe and central North America. In both regions, precipitation and soil moisture are reduced in a climate of greater atmospheric carbon dioxide. A detailed investigation of the hydrology of the model shows that the drying of the soil comes about through an increase in evaporation in winter and spring, caused by higher temperatures and reduced snow cover, and a decrease in the net input of water in summer. Evaporation is reduced in summer because of the drier soil, but the reduction in precipitation is larger. Three extreme statistics are used to define drought, namely the frequency of low summer precipitation, the occurrence of long dry spells, and the probability of dry soil. The last of these is arguably of the greatest practical importance, but since it is based on soil moisture, of which there are very few observations, the authors’ simulation of it has the least confidence. Furthermore, long time series for daily observed precipitation are not readily available from a sufficient number of stations to enable a thorough evaluation of the model simulation, especially for the frequency of long dry spells, and this increases the systematic uncertainty of the model predictions. All three drought statistics show marked increases owing to the sensitivity of extreme statistics to changes in their distributions. However, the greater likelihood of long dry spells is caused by a tendency in the character of daily rainfall toward fewer events, rather than by the reduction in mean precipitation. The results should not be taken as firm predictions because extreme statistics for small regions cannot be calculated reliably from the output of the current generation of GCMs, but they point to the possibility of large increases in the severity of drought conditions as a consequence of climate change caused by increased CO2.