Exponential Decay Law Research Articles

Positive solutions of elliptic systems with bounded nonlinearities

SynopsisSemilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

A light transmittance probe for interfacial area measurements of dispersed two phase flows with small particles

An optical probe measuring interfacial area (Γ) by light attenuation has been designed with a special emphasis on flows with sub-millimetric particles. It permits measurements in liquid-liquid or gas-liquid dispersions without need of introducing empirical correcting factors for the standard exponential decay law of light intensity while keeping an extended application range. This probe was successfully tested with an air-glass particle flow, the parameters of which were carefully determined basically by hold-up methods. The volume fraction of the dispersed phase was varied between 0.05% and 5%, and the particle size between 10 μm and 300 μm.

A Monte Carlo calculation of secondary electrons emitted from Au, Ag and Cu

SUMMARYMonte Carlo calculation of slow secondary electrons (SE) has been performed with a hybrid model of the exponential decay law and cascade process. The contributions of both valence electron excitation and core electron excitation have been considered. The calculated yield, energy distribution, angular distribution and depth distribution for Au, Ag and Cu (28 keV, at normal incidence) are in agreement with the experimental results and better than those previously reported.

An analytical approach to discharge circuits in gas lasers

An analytical method for solving equations applied to circuits whose resistance varies according to an exponential decay law is proposed and applied. The solution is given in terms of generalized hypergeometric functions.

A Monte Carlo study of the excluded‐volume effect on the amylosic chain conformation

AbstractA Monte Carlo procedure was used to determine the effect of excluded volume on the conformational dimensions of amylosic chains. The excluded volume was introduced into the model by assuming that hard spheres, which cannot overlap each other, exist at the center of mass of each glucose unit in the chain sequence. Monte Carlo chains, which were generated to be distributed consistent with the potential energy of nonbonded nearest‐neighbor interactions, underwent self‐intersections, and the attrition rate in the generation of self‐avoiding chains was found to obey an exponential decay law with increasing chain length x. Thus, in order to generate effectively a large number of self‐avoiding chains with long sequences, we used the Wall–Erpenbeck s‐p method of chain enrichment [F. T. Wall and J. J. Erpenbeck (1959) J. Chem. Phys. 30, 634–637]. By examination of the radial distribution of the end‐to‐end distance and the chain‐length dependence of the quantity 〈r2〉/xl2 (〈r2〉 is the mean square end‐to‐end distance and l is the virtual bond length), it was found that unperturbed amylosic chains change in overall conformation from a non‐Gaussian chain having a helical character to Gaussian as x is increased, whereas perturbed chains do not show Gaussian behavior even at x = 500. For the perturbed chains, 〈r2〉 can be expressed by the equation 〈r2〉 = axb in the range of 100 ≤ x ≤ 500, where a and b > 1 are constants. From comparisons of the persistence vectors and perspective drawings of representative unperturbed and perturbed chains, we felt the local conformation of the amylosic chains, i.e., the local helical character, is also affected by the long‐range excluded‐volume interaction.

Applications: The Exponential-Decay Law Applied to Medical Dosages

Many drugs are used up by the human body at rates described in terms of exponential decay. Two simple exponential formulas are given in this article. One describes how a therapeutic level of a drug is achieved in terms of its half-life and other parameters. The other describes how this level is maintained.

Three exponential fluorescence decay of horse liver alcohol dehydrogenase revealed by iodide quenching.

Abstract— The time resolved fluorescence decay of horse liver alcohol dehydrogenase was measured using a frequency doubled picosecond dye laser and time‐correlated single‐photon counting detection. A flow‐cell technique is described which eliminates the photodegradation artifacts which commonly occur with laser excitation. A procedure is introduced which uses fluorescence quenching to reveal minor fluorescence lifetime components. The decay of the unquenched tryptophanyl fluorescence could be described by a double exponential decay law, but experiments conducted in the presence of iodide ion showed that the fluorescence decay must be more complex than this. A model is presented in which the fluorescence decay consists of three exponential components, only two of which are susceptible to quenching by iodide ion. Several possibilities are presented for the origin of this minor decay component, the most reasonable of which is that it arises from conformational heterogeneity in the solvent‐exposed tryptophanyl residue.

Truncated Gamow functions, ?-decay and the exponential law

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an α-decay model to prove general validity of the exponential decay law for periods of several lifetimes.

Adiabatic switching and long-time corrections to the exponential decay law.

Adiabatic switching and long-time corrections to the exponential decay law.

Illustration of resonances and the law of exponential decay in a simple quantum-mechanical problem

The exact quantum-mechanical solution of a particle moving in a one-dimensional well formed by a delta barrier in front of an impenetrable well is used to illustrate the relation between scattering resonances and exponential decay from the well.

Decaying particles do not "heat up" the Universe.

It is usually assumed that a massive relic species, which comes to dominate the mass density of the Universe and later decays, ``heats up'' the Universe when the age of the Universe \ensuremath{\simeq} its lifetime. We show that if its decay follows the usual exponential decay law, then the Universe is never reheated, rather it just cools more slowly. We calculate the evolution of the temperature and entropy, and find that to within numerical factors of order unity, the usual estimates for the entropy increase are correct. Our results have implications for primordial nucleosynthesis in scenarios where a massive relic with lifetime \ensuremath{\simeq} ${10}^{\mathrm{\ensuremath{-}}2}$--${10}^{3}$ sec is present, and for baryogenesis in the new inflationary Universe scenario.

Factors affecting the progressive development of wear-protective oxides on iron-base alloys during sliding at elevated temperatures

The friction behaviour of two commercial Fe-12%Cr alloys during reciprocating sliding in air at 100–400 °C has been studied and a model is proposed to account for the changes observed. After relatively high initial values, associated with metal-metal contact, the friction decreases progressively with sliding time, reaching a minimum value that is very reproducible for a given set of conditions after a short period, the length of which decreases with increasing temperature. Subsequently the friction increases somewhat and attains a steady value which is maintained throughout the remainder of the sliding run. This value can be correlated with oxide-oxide contact only. The decrease in friction in the early stages is associated with the progressive development of adherent compacted oxide regions. The model proposed to account for these changes assumes that these regions are uniform in thickness, that the volume rate of oxide production is proportional to the remaining area of bare metal surface and hence that the area growth rate of compacted oxide follows an exponential decay law. The model relates the changes in coefficient of friction with time to several interfacial metal, oxide and metal-oxide parameters. There is very close correlation between the model and practice at 400 and 300 °C. However, the correlation is less exact at the lower temperatures. These results are considered in the light of two possible mechanisms of oxide generation during sliding wear.

The Law of Exponential Decay for Expanding Transformations of the Unit Interval into Itself

The Law of Exponential Decay for Expanding Transformations of the Unit Interval into Itself

Threshold effects in strong-field photodetachment monitored by spontaneous photo-relaxation

The strong-field photodetachment of an electron from a negative ion, when the laser is tuned near the threshold, displays deviations from the exponential decay law. These deviations manifest themselves in a significant deviation of the photon spectrum from a Lorentzian shape.

Corrections to the exponential decay law for the 2 P → 1S transition in hydrogenlike atoms

We calculate explicit corrections to the Weisskopf-Wigner laws governing the spontaneous radiative transition 2P → 1S in a non-relativistic two level hydrogenlike atom. Deviations from the usual exponential decay and lorentzian emission spectrum prove to be unobservably small, even at the sub-ppm detection levels characterizing experiments searching for atomic parity violations.

The law of exponential decay for expanding transformations of the unit interval into itself

Let T : [ 0 , 1 ] → [ 0 , 1 ] T:[0,1] \to [0,1] be an expanding map of the unit interval and let ξ ε ( x ) {\xi _\varepsilon }(x) be the smallest integer n n for which T n ( x ) ∈ [ 0 , ε ] {T^n}(x) \in [0,\varepsilon ] ; that is, it is the random variable given by the formula \[ ξ ε ( x ) = min { n : T n ( x ) ⩽ ε } . {\xi _\varepsilon }(x) = \min \{ n:{T^n}\;(x) \leqslant \varepsilon \}. \] It is shown that for any z ⩾ 0 z \geqslant 0 and for any integrable function f : [ 0 , 1 ] → R + f:[0,1] \to {R^ + } the measure μ f {\mu _f} (where μ \mu is Lebesgue measure and μ f {\mu _f} is defined by d μ f = f d μ d{\mu _f} = fd\mu ) of the set of points x x for which ξ ε ( x ) ⩽ z / ε {\xi _\varepsilon }(x) \leqslant z/\varepsilon tends to an exponential function of z z as ε \varepsilon tends to zero.

Normalizable Resonance Wave Function, Analyticity, and Decay Law

The authors show how to construct a normalizable resonance wave function produced by a sharp-cutoff two-body potential using analyticity and basic quantum mechanical requirements. To this end they propose a saturation scheme, and estimate the duration of validity of the exponential decay law. The viability of the model is tested in a deuteron stripping to unbound-state calculation.

The law of exponential decay for r-adic transformations

The law of exponential decay for r-adic transformations

Open-system approach to Jahn-Teller systems

We study the dynamical Jahn-Teller effect in crystals for weak coupling by means of an open-system perturbative approach. This gives the evaluation of the relaxation to equilibrium of populated electronic excited states and of the optical response. The different roles of the evolution with and without memory are explicitly shown. As far as electronic relaxation is concerned, the presence of memory gives corrections to the exponential decay law due to dynamics without memory. In optical absorption the terms without memory are responsible for the energy shift and the Lorentzian shape of the zero-phonon line. The terms with memory yield the many-phonon structure and the Huang-Rhys factor. Also, a change is found in the coupling constants and in the phonon state densities. The latter yields vibronic states in a nonperturbative approach.

Fermi's golden rule and exponential decay in two-level systems

The decay of isolated quantum systems is studied. In particular, a specific model for the radiative decay of an excited atomic state is considered. The atom is taken to have two levels and is described by a current distribution $\stackrel{\ensuremath{\rightarrow}}{\mathrm{j}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};1,2)$. Both the case in which the modes of the electromagnetic field are continuous and the case in which they are discrete are considered. Conditions on $\stackrel{\ensuremath{\rightarrow}}{\mathrm{j}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};1,2)$ are found under which the decay law for the excited state is approximately exponential with a lifetime given by Fermi's golden rule. The current should be well localized, not too strong, and not too singular. The $2p\ensuremath{\rightarrow}1s$ decay in hydrogen is examined in detail. It is also found that the existence of an approximately exponential decay law does not depend upon the ability to extend the resolvent operator of the system to a second Riemann sheet. The case in which the electromagnetic field is quantized in a cavity illustrates this point.