Exponential Regime Research Articles

Analysis of the transmission process through single apertures surrounded by periodic corrugations

We have analyzed the transmission process through single subwavelength apertures surrounded by a set of periodic grooves in optically thick Ag films. On one hand, we found that the intensity of both single- and double-corrugated structures follows just one exponential regime as a function of the hole depth. On the other hand, it is shown that the transmission process can be separated into three independent steps: coupling in, transmission through the aperture and coupling out. This is in contrast with the transmission through hole arrays reported by previous studies where two transmission regimes were found. These findings are of relevance not only for further understanding the enhanced transmission but also for any applications based on this phenomenon.

Crossover of quantum Loschmidt echo from golden-rule decay to perturbation-independent decay.

We study the crossover of the quantum Loschmidt echo (or fidelity) from the golden-rule regime to the perturbation-independent exponential decay regime by using the kicked top model. It is shown that the deviation of the perturbation-independent decay of the averaged fidelity from the Lyapunov decay results from quantum fluctuations in individual fidelity, which are caused by the coherence in the initial coherent states. With an averaging procedure suppressing the quantum fluctuations effectively, the perturbation-independent decay is found to be close to the Lyapunov decay. We also show that the Fourier transform of the fidelity is determined directly by the initial state and the eigenstates of the Floquet operators of the two classically chaotic systems concerned. The absolute value part and the phase part of the Fourier transform of the fidelity are found to be divided into several correlated parts, which is a manifestation of the coherence of the initial coherent state. In the whole crossover region, some important properties of the fidelity, such as the exponent of its exponential decay and the short initial time within which the fidelity almost does not change, are found to be closely related to the properties of the central part of its Fourier transform.

Molecular basis for the explanation of the exponential growth of polyelectrolyte multilayers.

The structure of poly(l-lysine) (PLL)/hyaluronan (HA) polyelectrolyte multilayers formed by electrostatic self-assembly is studied by using confocal laser scanning microscopy, quartz crystal microbalance, and optical waveguide lightmode spectroscopy. These films exhibit an exponential growth regime where the thickness increases exponentially with the number of deposited layers, leading to micrometer thick films. Previously such a growth regime was suggested to result from an "in" and "out" diffusion of the PLL chains through the film during buildup, but direct evidence was lacking. The use of dye-conjugated polyelectrolytes now allows a direct three-dimensional visualization of the film construction by introducing fluorescent polyelectrolytes at different steps during the film buildup. We find that, as postulated, PLL diffuses throughout the film down into the substrate after each new PLL injection and out of the film after each PLL rinsing and further after each HA injection. As PLL reaches the outer layer of the film it interacts with the incoming HA, forming the new HA/PLL layer. The thickness of this new layer is thus proportional to the amount of PLL that diffuses out of the film during the buildup step, which explains the exponential growth regime. HA layers are also visualized but no diffusion is observed, leading to a stratified film structure. We believe that such a diffusion-based buildup mechanism explains most of the exponential-like growth processes of polyelectrolyte multilayers reported in the literature.

Fluctuating charge-state effects on energy spectra of fast ions in solids

We describe the evolution of the charge state of a fast ion in a thin solid foil by a non-stationary continuous-time Markov process which takes two discrete values, appropriate for H and He projectiles. General expression is presented for the ion energy distribution after traversing the foil and is used to derive the expectation values for the ion stopping power and the energy-loss straggling. A careful analysis of the time auto-correlation function for the ion charge state in a non-stationary regime reveals that, in addition to the familiar exponential transient regime, the initial charge effects exhibit a long-time algebraic decay, inversely proportional to the penetration time through the foil.

A simple subgrid scale stabilized method for the advection–diffusion-reaction equation

A subgrid scale method has been developed for the advection–diffusion-reaction equation with a simple intrinsic time-scale parameter. The proposed formulation combines very good accuracy and stability in all the physical regimes of the equation, that is, in the exponential regime for both, negative and positive source terms, and in the propagation regime. Numerical examples confirm the robustness and accuracy of the method.

Power-law and exponential creep in class M materials: discrepancies in experimental observations and implications for creep modeling

This paper discusses our current understanding of the processes thought to be dominant in the exponential creep regime as well as the implications for creep modeling relating to both power-law and exponential creep regions. The significance and implications of creep controlled by vacancy diffusion along dislocation cores are discussed. It is pointed out that creep substructures, other than subgrains, have been reported in the literature, and a bifurcation diagram is presented to demonstrate how this evolution can occur from an initially homogeneous dislocation substructure. The use of nonlinear dislocation dynamics in creep modeling is advocated to rationalize the observed diversity in the creep substructures. It is demonstrated that the dislocation substructure evolution models can be coupled with a viscoplastic model through the volume fractions of the ‘hard’ and ‘soft’ phases. This coupling is shown to lead to the stress-subgrain size relationship in a simple and a natural way.

Stationarity coefficients and short-time deviations from exponential decay in atomic resonance states

By solving rigorously and accurately the time-dependent Schr\"odinger equation, we have obtained numerical results for the decay probability, $P(t),$ of real, multiparticle systems, in the time domain of $t\ensuremath{\approx}0.$ Three different types of atomic nonstationary states were examined, the ${\mathrm{He}}^{\ensuremath{-}}{1s2p}^{2}{}^{4}P,$ the Ca KLM $3d5p{}^{3}{F}^{o},$ and the ${\mathrm{He}}^{\mathrm{\ensuremath{-}}}$ $1s2s2p{}^{4}{P}_{5/2},$ the last one being metastable and decaying via spin-spin interactions. The main results are that there is a ${t}^{2}$ dependence of $P(t\ensuremath{\approx}0)$ and that a time-dependent short-time decay rate can be calculated. The computed coefficients of the ${t}^{2}$ term reflect the degree of stability of the state, (i.e., the degree of proximity to the notion of the standard stationary state of quantum mechanics), and are named the stationarity coefficients. These, together with the conventional quantity of the lifetime, corresponding to the exponential decay regime, constitute intrinsic properties of each real unstable state. For the herein studied metastable state the onset of exponential decay occurs after about $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\mathrm{s},$ i.e., after a duration which is achievable experimentally with laser pulses.

Signatures of the discrete level spectrum in temperature-dependent transport through open quantum-dot arrays

Temperature-dependent studies of the resistance of open quantum-dot arrays reveal a regime of intermediate temperature (\ensuremath{\sim}1--5 K), over which the resistance increases exponentially with decreasing temperature. In this Brief Report, we explore the origins of this unexpected localization by studying its correlation to the temperature-dependent variation of the magnetoconductance. Based on these studies, we suggest that the exponential regime corresponds to that over which we transition from strongly broadened to energetically resolved levels in the dots. In order to provide further support for this interpretation, we perform numerical studies of temperature-dependent transport through the quantum dots, and discuss the role that many-body effects may play in giving rise to the behavior found in experiment.

Self-amplified spontaneous emission in Smith–Purcell free-electron lasers

We present an analysis of a Smith–Purcell system in which a thin current sheet of electrons moves above a grating surface in the direction perpendicular to the grating grooves. We develop a consistent theory for evolution of the electromagnetic field and electron distribution in the exponential growth regime starting from the initial electron noise and the incoming amplitude. The dispersion relation for the complex growth rate for this system is a quadratic equation.

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents $\alpha$ of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Levy regime $\alpha < 2$, the power law decay with large exponent ($\alpha > 2$), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement.

Ultrafast Coulomb-induced dynamics of quantum well magnetoexcitons

We study theoretically the ultrafast nonlinear optical response of quantum well excitons in a perpendicular magnetic field. We show that for magnetoexcitons confined to the lowest Landau levels, the third-order four-wave-mixing (FWM) polarization is dominated by the exciton-exciton interaction effects. For repulsive interactions, we identify two regimes in the time-evolution of the optical polarization characterized by exponential and {\em power law} decay of the FWM signal. We describe these regimes by deriving an analytical solution for the memory kernel of the two-exciton wave-function in strong magnetic field. For strong exciton-exciton interactions, the decay of the FWM signal is governed by an antibound resonance with an interaction-dependent decay rate. For weak interactions, the continuum of exciton-exciton scattering states leads to a long tail of the time-integrated FWM signal for negative time delays, which is described by the product of a power law and a logarithmic factor. By combining this analytic solution with numerical calculations, we study the crossover between the exponential and non-exponential regimes as a function of magnetic field. For attractive exciton-exciton interaction, we show that the time-evolution of the FWM signal is dominated by the biexcitonic effects.

Seismic-like scaling regime in impulse reflection from underwater sediment

A universal scaling law is known empirically to characterize the high-frequency decay of bandpassed seismic coda. An explanation advanced for this universality, based on multiple-Rayleigh scatter of nonuniform strength, is generic enough to suggest that similar scaling may apply to decay of the impulse response in other random media. A submerged sand layer is probed experimentally as a candidate for such a medium, and its incoherent acoustic impulse reflection is found to have an exponential decay regime similar to that in seismology.

The effect of inhomogeneous broadening on optical strong field spectroscopy

We show that a recent theory of strong field spectroscopy (SFS) [R. I. Cukier and M. Morillo, Phys. Rev. B 57, 6972 (1998), M. Morillo and R. I. Cukier, J. Chem. Phys. 110, 7966 (1999)] can be used to circumvent the effects of inhomogeneous broadening on this spectroscopy. In SFS, a strong external field is used to connect, with the transition dipole, two electronic states of a solute immersed in a medium. The electronic dephasing due to the medium is characterized via the power absorbed by the solute. The average absorbed power P̄(t) for resonant, strong fields exhibits an oscillatory decay in time, reflecting the finite change in the population difference of the electronic states and the dephasing arising from the coupling to the medium. The decay rate is characterized by d≡Δ2τc, where Δ and τc are, respectively, the strength and time constant of the correlation function characterizing the solute–medium coupling. The decay can be very rapid, on a 10–100 fs time scale, and this necessitates an indirect procedure to experimentally probe P̄(t) that we develop. For strong, off-resonance fields, P̄(t) returns to an exponential decay regime. The contrasting behavior of resonant and nonresonant strong fields can be used to avoid the loss of information about the homogeneous properties due to inhomogeneous broadening of the optical transition, when this broadening arises from inhomogeneity in the optical transition frequency.

Growth of transverse coherence in SASE FELs

We introduce the correlation function between the electric field at two different points in the transverse plane as a parameter to quantify the degree of transverse coherence. We also propose a more realistic model for the initialization of the radiation in computer codes used to study SASE FELs. We make these modifications in the code tda and use it to study the growth of transverse coherence as a function of electron beam size, beam current and transverse emittance. Our results show explicitly that the onset of full transverse coherence in SASE takes place much before the power saturates. With the more realistic model the onset of the exponential growth regime is delayed, and to get a given power from the FEL one needs a longer undulator than would be predicted by the original tda code.

Effects of bunch density gradient in high-gain free-electron lasers

We investigate effects of the bunch density gradient in self-amplified spontaneous emission, including the role of coherent spontaneous emission (CSE) in the evolution of the free-electron laser process. In the exponential gain regime, we solve the coupled Maxwell–Vlasov equations and extend the linear theory to a bunched beam with energy spread. A time-dependent, nonlinear simulation algorithm is used to study the CSE effect and the nonlinear evolution of the radiation pulse.

A phenomenological description of primary creep in class M materials

Observations of deformation microstructures in the primary creep region in class M materials show a remarkable similarity with those formed in the exponential creep regime. As a result, it is proposed that the constitutive creep law for normal primary creep is similar to that for the exponential creep regime. A phenomenological description is discussed to rationalize these microstructural observations in terms of a normalized strain rate versus stress plot. The implications of this plot in describing different testing procedures, steady-state flow, and on the observed deviations from the universal creep law are discussed. The plot is also extended to explain the observed similarities in the transient creep behavior in pre-strained materials and in stress change experiments.

Global dynamics of cosmological expansion with a minimally coupled scalar field

We give a complete description of the asymptotic behavior of a Friedmann–Robertson–Walker Universe with `normal' matter and a minimally coupled scalar field. We classify the conditions under which the Universe is or is not accelerating. In particular, we show that only two types of large time behavior exist: an exponential regime, and a subexponential expansion with the logarithmic derivative of the scale factor tending to zero. In the case of the subexponetial expansion the Universe accelerates when the scalar field energy density is dominant and the potential behaves in a specified manner, or if matter violates the strong energy condition ρ+3 p>0. When the expansion is exponential the Universe accelerates, and the scalar field energy density is dominant. We find that for the Big Bang to occur at zero scale factor, the equation of state of matter needs to satisfy certain restrictions at large densities. Similarly, a never ending expansion of the Universe constrains the equation of state at small matter densities.

Growth, nutrient dynamics, and ectomycorrhizal development of container-grownPicea marianaseedlings in response to exponential nutrient loading

Containerized black spruce (Picea mariana (Mill.) BSP) seedlings fertilized conventionally (12.5 mg N/plant) or exponentially (12.5, 25, or 50 mg N/plant) and inoculated with Hebeloma crustuliniforme (Bull. Ex St-Amans) Quel. or Laccaria bicolor (R. Mre.) Orton were periodically monitored for a 20-week greenhouse rotation to assess growth dynamics, steady-state N and P nutrition, and ectomycorrhizal development. Growth and nutrient accumulation increased exponentially for the exponential regimes and more linearly for the conventional regime, although final biomass was similar except for the low-dose exponential addition. Shoot/root biomass ratios were relatively stable for most of the growing season, characterizing steady-state nutrient supply that benefits seedling outplanting performance and mycorrhizal colonization. Exponential fertilization also stimulated mycorrhiza formation even at high loading (25 or 50 mg N) rates that build up nutrient reserves in the seedlings without affecting seedling size. Plant nutrient uptake was more efficient under exponential fertilization and (or) fungal colonization, although efficiency dropped off at high loading levels. Vector nutrient diagnosis revealed marked nutrient dilution under conventional fertilization, but steady-state nutrition under exponential fertilization that coincided with satisfactory mycorrhiza development on seedlings. Dilution-free nutrient conditions for seedlings may provide stable carbohydrates for symbiosis and may develop enhanced tolerance to high fertilizer inputs.

Growth, nutrient dynamics, and ectomycorrhizal development of container-grown &lt;i&gt;Picea mariana&lt;/i&gt; seedlings in response to exponential nutrient loading

Containerized black spruce (Picea mariana (Mill.) BSP) seedlings fertilized conventionally (12.5 mg N/plant) or exponentially (12.5, 25, or 50 mg N/plant) and inoculated with Hebeloma crustuliniforme (Bull. Ex St-Amans) Quel. or Laccaria bicolor (R. Mre.) Orton were periodically monitored for a 20-week greenhouse rotation to assess growth dynamics, steady-state N and P nutrition, and ectomycorrhizal development. Growth and nutrient accumulation increased exponentially for the exponential regimes and more linearly for the conventional regime, although final biomass was similar except for the low-dose exponential addition. Shoot/root biomass ratios were relatively stable for most of the growing season, characterizing steady-state nutrient supply that benefits seedling outplanting performance and mycorrhizal colonization. Exponential fertilization also stimulated mycorrhiza formation even at high loading (25 or 50 mg N) rates that build up nutrient reserves in the seedlings without affecting seedling size. P...

Probing Confining Geometries with Molecular Diffusion: A Revisited Analysis of NMR-PGSE Experiments

AbstractWe present an interpretation of the self diffusion propagator for a molecular fluid confined in a porous medium. Breaking with conventional routes of interpretation, we focus in reciprocal space on the time dependence of the propagator at a fixed wave vector q. New theoretical results are reported, as well as NMR measurements on a water-saturated packing of glass beads and on a stack of plastic platelets with rough surfaces. It is shown that at least three time regimes may be distinguished, characteristic of new ways in which the propagator is affected by the geometry of the system: a short-time exponential regime of almost unrestricted diffusion, a pseudo-exponential regime probing the transport process across the material at the length scale λ=2π/q, and an algebraic regime at long times with exponent -d/2, where d is the dimensionality of connected parts of the pore space.