Rayleigh Process Research Articles

Isotopic and chemical investigations of Fe (III) reduction by Shewanella putrefaciens strain 200R

ABSTRACTExperiments were conducted using the Fe+3‐reducing bacterium Shewanella putrefaciens strain 200R to determine the stable carbon isotope fractionation during dissimilatory Fe (III) reduction and associated lactate oxidation at circum‐neutral pH. Previous studies used equilibrium fractionation factors (~14.3‰) between bacterial biomass and synthesized fatty acids to identify the predominant carbon fixation pathways for some of the most frequently isolated microbes including Shewanella under anaerobic conditions. We investigated the carbon isotope disproportionation among organic carbon substrate (lactate), biomass and respired carbon dioxide at the lag to stationary phase of the growth curve. Ferric citrate and sodium lactate were used as electron acceptor and donor, respectively. Sodium bicarbonate or potassium phosphate was used as buffering agent. Iron (II), iron (III), dissolved inorganic carbon (DIC) and carbon isotope ratios were measured for both bicarbonate‐ and phosphate‐buffered systems. Carbon isotope ratio measurements were made on the respired CO2 (as DIC) and microbial biomass for both buffering conditions. The fraction of lactate consumed was estimated using DIC as a proxy and was verified by direct measurement using HPLC. Our result showed that bicarbonate‐buffered system has an enhancing effect in the reduction process compared to the phosphate system. Both systems resulted in carbon isotope fractionations between the lactate substrate and DIC that could be modelled as a Rayleigh process. The biomass produced under both buffer conditions was depleted on average by ~2‰ relative to the substrate and enriched by ~5‰ relative to the DIC. This translates to an overall isotopic fractionation of 10–12‰ between the biomass and respired CO2 in both buffering systems.

Sr/Ca and Mg/Ca vital effects correlated with skeletal architecture in a scleractinian deep-sea coral and the role of Rayleigh fractionation

Deep-sea corals are a new tool in paleoceanography with the potential to provide century long records of deep ocean change at sub-decadal resolution. Complicating the reconstruction of past deep-sea temperatures, Mg/Ca and Sr/Ca paleothermometers in corals are also influenced by non-environmental factors, termed vital effects. To determine the magnitude, pattern and mechanism of vital effects we measure detailed collocated Sr/Ca and Mg/Ca ratios, using a combination of micromilling and isotope-dilution ICP-MS across skeletal features in recent samples of Desmophyllum dianthus, a scleractinian coral that grows in the near constant environment of the deep-sea. Sr/Ca variability across skeletal features is less than 5% (2σ relative standard deviation) and variability of Sr/Ca within the optically dense central band, composed of small and irregular aragonite crystals, is significantly less than the surrounding skeleton. The mean Sr/Ca of the central band, 10.6 ± 0.1 mmol/mol (2σ standard error), and that of the surrounding skeleton, 10.58±0.09 mmol/mol, are statistically similar, and agree well with the inorganic aragonite Sr/Ca-temperature relationship at the temperature of coral growth. In the central band, Mg/Ca is greater than 3 mmol/mol, more than twice that of the surrounding skeleton, a general result observed in the relative Mg/Ca ratios of D. dianthus collected from separate oceanographic locations. This large vital effect corresponds to a ∼ 10 °C signal, when calibrated via surface coral Mg/Ca-temperature relationships, and has the potential to complicate paleoreconstructions. Outside the central band, Mg/Ca ratios increase with decreasing Sr/Ca. We explain the correlated behavior of Mg/Ca and Sr/Ca outside the central band by Rayleigh fractionation from a closed pool, an explanation that has been proposed elsewhere, but which is tested in this study by a simple and general relationship. We constrain the initial solution and effective partition coefficients for a Rayleigh process consistent with our accurate Metal/Ca measurements. A process other than Rayleigh fractionation influences Mg in the central band and our data constrain a number of possible mechanisms for the precipitation of this aragonite. Understanding the process affecting tracer behavior during coral biomineralization can help us better interpret paleoproxies in biogenic carbonates and lead to an improved deep-sea paleothermometer.

The Stochastic Rayleigh diffusion model: Statistical inference and computational aspects. Applications to modelling of real cases

In this paper, we consider a Stochastic System modelling by the Stochastic Rayleigh Diffusion Process and we discuss theoretical aspects of the latter and establish a statistical methodology to adjust it to real cases, particulary, in the field of biometry and related areas. The Rayleigh process, according to the definition of [C. Giorno, A. Nobile, L. Ricciardi, L. Sacerdote, Some remarks on the Rayleigh process, Journal of Applied Probability 23 (1986) 398–408], is examined from the perspective of the corresponding nonlinear stochastic differential equation, and from its associated probability density function we obtain the corresponding mean functions (trend function and conditional trend function), which depend of Kummer functions. The drift parameters are estimated by maximum likelihood on the basis of continuous sampling of the process and they are calculated by computational methods. We propose numerical approximations for the diffusion coefficient, from an extension of the [M. Chesney, J. Elliot, Estimating the instantaneous volatility and covariance of risky assets, Applied Stochastic Models and Data Analysis 11 (1995) 51–58] procedure to the case of nonlinear stochastic differential equations and we establish also computational procedures and simulation algorithm, that are applied to obtened simulated paths of the fitted process. The proposed methodology is applied to two studies carried out in Andalusia (Spain) on females and males life expectancy at birth, between 1944 and 2001.

Silicon isotope study on rice plants from the Zhejiang province, China

Silicon isotope compositions of 16 rice plants from the Zhejiang province were studied, and potential applications of silicon isotope study to agriculture and biogeochemistry were discussed. It is found that the SiO 2 contents in ignited rice plant samples increased generally from roots, through SL (stem and leaves), to husks. Large and systematic silicon isotope fractionation was observed among different organs of individual rice plants. Their δ 30Si values show a generally increasing trend from roots and SL, through husks, to grains, which can be explained by kinetic isotope fractionation in a Rayleigh process. A δ 30Si value of 6.1‰ was obtained from two samples of rice grains, which is much greater than the highest value (3.4‰) observed in terrestrial samples before. It is inferred that monosilicic acid is the major silicon-bearing component taken up by rice roots. Passive uptake of silicon is important for rice and evapotranspiration may be the major mechanism for the transportation and precipitation of silicon in rice plants. It is suggested that silicon isotope study has a potential application to agriculture and biogeochemistry.

Estimation of denitrification potential in a karst aquifer using the 15N and 18O isotopes of NO 3 −

A confined aquifer in the Malm Karst of the Franconian Alb, South Germany was investigated in order to understand the role of the vadose zone in denitrifiaction processes. The concentrations of chemical tracers Sr2+ and Cl− and concentrations of stable isotope 18O were measured in spring water and precipitation during storm events. Based on these measurements a conceptual model for runoff was constructed. The results indicate that pre-event water, already stored in the system at the beginning of the event, flows downslope on vertical and lateral preferential flow paths. Chemical tracers used in a mixing model for hydrograph separation have shown that the pre-event water contribution is up to 30%. Applying this information to a conceptual runoff generation model, the values of δ15N and δ18O in nitrate could be calculated. Field observations showed the occurence of significant microbial denitrification processes above the soil/bedrock interface before nitrate percolates through to the deeper horizon of the vadose zone. The source of nitrate could be determined and denitrification processes were calculated. Assuming that the nitrate reduction follows a Rayleigh process one could approximate a nitrate input concentration of about 170 mg/l and a residual nitrate concentration of only about 15%. The results of the chemical and isotopic tracers postulate fertilizers as nitrate source with some influence of atmospheric nitrate. The combined application of hydrograph separation and determination of isotope values in δ15N and δ18O of nitrate lead to an improved understanding of microbial processes (nitrification, denitrification) in dynamic systems.

Rayleigh type behavior of the Young's modulus of unpoled ferroelectric ceramics and its dependence on temperature

The dependence on stress of the low frequency Young's modulus and mechanical losses of unpoled ferroelectric ceramics has been studied as a function of temperature. The Young's modulus of unpoled Pb(Zr,Ti)O3 (PZT) showed a Rayleigh type dependence, analogous to the one already described for the longitudinal piezoelectric coefficient. This has been associated to ferroelectric/ferroelastic domain wall movements across, and their pinning/depinning on, randomly distributed defects. The Rayleigh coefficient was found to increase with temperature. The activation energy of the Rayleigh process was obtained, which must be related to the pinning energy. The Young's modulus of Mn doped 0.65Pb(Mg1/3Nb2/3)O3–0.35PbTiO3 showed no stress dependence.

Relationship between the variation of isotopic ratios and the source of summer precipitation in eastern Siberia

The response of the isotopic ratio of summer precipitation in eastern Siberia to precipitation sources is estimated by examination of stable isotope ratios in the summer precipitation and by atmospheric data analysis, based on water balance considerations. The isotopic ratio (δ18O) was analyzed in samples of precipitation that were collected daily or weekly, from May 2000 to August 2000, in the Yakutsk region of eastern Siberia. The results of the isotopic data show that during spring, the variation of isotopic ratio is controlled mainly by the Rayleigh distillation process. During summer, on the other hand, the variation of isotopic ratio bears no relation to the Rayleigh process, and isotopic variation at this time is mainly caused by changes in the sources of precipitation. This is reinforced by comparison with the analysis of atmospheric data and observed isotopic data. The comparison shows that only in the summer samples, there was a good correlation (R2 = 0.88) between heavier isotopic ratios and increases in the proportion of the moisture supply that originates from the west, which is the dominant source of precipitation (about one half of the moisture). This linear relationship indicates the summer precipitation in eastern Siberia is a composite of source water from the west with heavy isotopic ratio and other sources with light isotopic ratio. The origin of this light isotopic ratio in summer precipitation might be recycled water from the land surface. Recycled water with light isotopic ratio results from the contribution of thaw water.

Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion

Symmetry properties of the one-dimensional Fokker-Planck equations with arbitrary coefficients of drift and diffusion are investigated. It is proved that the group symmetry of these equations can be one, two, four or six parametric and corresponding criteria are obtained. The changes of the variables reducing Fokker-Planck equations to the heat equation and Schrodinger equation with certain potentials are determined. Using the substructure of the invariance algebra of the Rayleigh process equation we obtain the differential invariants and construct some classes of the exact solutions of this equation.

New derivation of Craig's formula for the Gaussian probability function

The product of a sinusoidal random process and an independent Rayleigh process is known to produce a Gaussian process. The corresponding relationship between the respective probability density functions can be expressed via the Mellin convolution, and this yields a new derivation, together with a physical interpretation, of an alternative form of the Gaussian probability function Q(x).

A Study on Estimation of Extreme Value of Moored Vessels by Means of Path Integral Solution Method

The paper investigates the path integral solution method combined with the gamma-distribution approximation for exciting forces, for calculating the response statistics of nonlinear dynamic systems whose equations of motion can be modelled by the use of Ito stochastic differential equations. The state vector process of motions is generally a diffusion process, and the transition probability density function (TPDF) of the process satisfies a Fokker-Planck-Kolmogorov (FPK) equation. The method solve FPK equation for short time TPDF. Naess assumed that second order wave exciting force is described by one term of square of Rayleigh process, but this assumption is not actual. Authors propose the new path integral solution method which express the wave exciting force by gamma process. In this paper the new method is applied to the cases whose exact solution exist, and is verified its effectiveness.

Sulphur isotope geochemistry of the ores and country rocks at the Almadén mercury deposit, Ciudad Real, Spain

Seventy-four new S isotope analyses of ore minerals and country rocks are given for the Hg deposit of Almadén. The spread of the cinnabar δ 34 S is narrow within each of the three orebodies, but the δ 34 S average values differ sufficiently between them (mean δ 34 S: San Nicolas = 0.2 ± 1.1 %., San Francisco = 8.1 ± 0.7%., San Pedro = 5.9 ± 1.0%.) to indicate three different mineralization episodes and possibly processes. The unweighted mean for all cinnabar samples is 5.6%. and the S source is considered to be the host-rocks, either the Footwall Shales ( δ 34 S = 5.5%.) or the spilites ( δ 34 S = 5.1 ± 1.3%.). For geometric and chronologic reasons, the former seem the best potential source. However, the high δ 34 S values of the San Francisco cinnabar cannot be explained without addition of heavy S from reduction of seawater sulphate. Orderly distributions of the δ 34 S values are observed in all three orebodies: 1. (1) their increase from the stratigraphic bottom to the top in the San Pedro orebody is explained by a Rayleigh process, and 2. (2) the maxima in the centres of the San Francisco and San Nicolas orebodies are explained by mixing of the S transporting hydrothermal fluids with seawater within the sediments. Associated pyrite and cinnabar were deposited under isotopic disequilibrium, probably because the low solubility of cinnabar caused rapid precipitation of cinnabar. The different morphological pyrite types have their own isotopic δ 34 S signatures. The spilites are notably enriched in S ( n = 3; average S content = 0.56%) compared to normal basalts (1000 ppm) and have an average δ 34 S = 5.1 ± 1.3%.. The linear relationship between the δ 34 S and the S content of the spilites is interpreted as a mixing line between mantle S and a constant S source, probably an infinite open reservoir. An incomplete basalt-seawater reaction at nearly constant temperature is the best explanation for this relation. The S (predominantly pyrite) of the black shales ( n = 3; δ 34 S from −6.7%. to 14.4%.) is interpreted as resulting from bacterial seawater sulphate reduction under variable euxinic conditions. The stratabound pyrite ( n = 16) found in the sediments near, but independent of the Hg ores, also has a wide range of δ 34 S between 15.0%. and 16.4%., with a mode of about 8%.; the pyrite rims of the diagenetic dolomite nodules ( n = 2; δ 34 S = −14.7 ± 0.35%.) in the Hanging Wall Shales display the most negative values found at Almadén.

Unifying stochastic Markov process and its transition probability density function.

A stochastic Markov process is defined which generalizes a class of processes previously considered by the author [Phys. Rev. A 42, 4485 (1990)]. The enlarged class unifies such apparently unrelated processes as the Ornstein-Uhlenbeck process, the generalized Verhulst-Landau processes, the generalized Rayleigh process, the hyperbolic tangent processes, and many other cases of physico-mathematical interest. The Fokker-Planck equation associated with this unifying stochastic process is solved analytically for the transition probability density function, using a similar constructive solution method as in the author's previous work. The result is obtained as an eigenfunction expansion over a generally mixed spectrum. The discrete eigenfunctions are related (but not identical) to Jacobi polynomials. It is shown that the results for the above-mentioned processes are obtainable as limiting cases, and that some additional solvable equivalent Schr\"odinger problems can be defined.

Effect of maturity on carbon isotope ratios of oils and condensates

For modelling isotopic variations in oils it is convenient to differentiate the effects of oil generation (∼ 100–150°C) from the effects of oil to gas cracking (∼ 150–180°C). During generation, δ 13 C of kerogen may increase by up to 1% due to release of isotopically light oil and gas, although most kerogens show little or no chan δ 13 C of the generated oil increases by between 0 and 1% (av. 0.5%) due to mixing of isotopically heavy oil with an initial isotopically light unbound fraction, possibly of bacterial origin. The change occurs mostly over the first 20% of generation. During oil to gas cracking, kinetic isotope effects become important and the effect on δ 13 C of the remaining oil can be modelled as a Rayleigh process. δ 13 C increases by 1.5% by 50% cracking. Insufficient data are available to calibrate the effects at higher levels of cracking, and modelling these variations is hindered by a lack of understanding of the mechanism of pyrobitumen formation. However, increases greater than about 4% are unlikely to be observed. With increasing maturity, the low molecular weight fractions become isotopically heavy faster than the high molecular weight fractions. As a result, any separation of the low molecular weight fraction into a gas phase (“condensate formation”) will produce an isotopic difference between oil and condensate that depends on maturity. In the early stages of generation the condensate may be up to 1% lighter than the remaining oil. With increasing maturity, this difference at first decreases and then increases in the opposite sense. By half way through oil to gas cracking the condensate may be 1.5% heavier than the residual liquid. More subtle rearrangement reactions may result in small, but significant, changes to the shape of the isotope “type-curves” when different oil fractions are compared.

Dephasing of excitons in quantum well structures studied by picosecond time-resolved resonant Rayleigh scattering

Electronic states in solids with disorder give rise to an elastic (Rayleigh) contribution to the scattering spectrum which becomes resonantly enhanced for excitation in the electronic transition. It is shown theoretically that from this resonant Rayleigh process, if temporally resolved, the coherence time of the electronic states may be deduced. Experimentally this is demonstrated for the first time by studying the n = 1 heavy-hole exciton in GaAs/AlGaAs quantum well structures. Employing picosecond time-resolved spectroscopy and analyzing the data within the developed theory, coherence times are found between 5 and 30 ps in agreement with earlier results obtained by non-linear optical techniques.

Use of isotope fractionation of sulfate-sulfur and sulfate-oxygen to assess bacterial desulfurication in a sandy aquifer

The unconfined aquifer of the “Fuhrberger Feld” consists of 20–30m sands and gravelly sands with unevenly distributed reduced sulfur compounds and small lignitic pebbles. In vertical direction two hydrochemically different zones with specific solute transformations are present, an upper portion with bacterial denitrification (with reduced sulfur compounds as electron donors) and a deeper portion with hydrochemical indications of reduction of the aqueous sulfate. In the study area ten multilevel groundwater sampling wells were installed. Sampling and interpretation was based on a two-dimensional vertical-plane groundwater flownet and on the knowledge of the landuse upgradient of the multilevel wells. The concentration and isotopic composition of the sulfate at the entrance into the zone of sulfate reduction is controlled by sulfate of groundwater recharge as well as sulfate formed during denitrification. The recharge-sulfate shows clear landuse specific differences in concentration and isotopic composition: under arable land an average concentration of ≈ 60 mg SO 4l −1 with values of δ 34S close to +3‰ CDT Diablo Meteorite Standard (Can̄on) and of δ 18O about + 8‰ standard mean ocean water (SMOW) was measured, whereas under coniferous forest an average of ∼ 100 mg SO 4l −1 with δ 34 S ≈ + 2‰ and δ 18 O ≈ 4.5‰ was found. In the denitrification zone no significant sulfur isotope fractionation occurs during oxidation of reduced sulfur. The oxygenisotopic composition of the newly formed sulfate (“denitrification-sulfate”) depends on the 18O content of nitrate acting as oxygen donor and on the 18O content of groundwater in which the oxidation occurs; under certain assumptions a δ 18 O = +0.54‰ was calculated for the oxygen of the newly formed sulfate. Consequently, with increasing sulfate concentrations, i.e. with the admixture of “denitrification-sulfate”, a decrease of the δ 18O values from about +8‰ to +2‰ was observed. In the zone of sulfate reduction, both the δ 34S and δ 18O-values of the residual sulfate increase with decreasing sulfate concentrations thus proving the existence of bacterial reduction of the aqueous sulfate. This sulfate reduction follows a Rayleigh process with calculated enrichment factors of ϵ 34 = −9.7‰ for 34S and ϵ 18 ≈ −6‰ for 18O, respectively. A plot of δ 34S vs. δ 18O gives a good linear correlation with a slope close to 1: 1.4. Data on concentration and isotopic composition of sulfate and on the hydraulically derived groundwater age are used for kinetic considerations. The results of a multilevel well for depths between 10 and 17m are given as an example. The bacterial reduction can be considered as first-order reaction with a half-life between 75 and 100 years.

Green's function for one-dimensional Fokker-Planck equations with exponentially dependent drift and diffusion coefficients.

The class of Fokker-Planck equations considered in this paper bridges the gap between the generalized Verhulst-Landau process and the Rayleigh process, which were apparently unrelated up to now. The equations are solved analytically for the transition probability density function of the underlying stochastic processes. The solution method uses operational calculus, and simultaneously produces all components of the spectral problem: spectrum, eigenfunctions, and their normalization. The general results obtained are benchmarked with the known results for the above-mentioned processes by inserting suitable parameters.

Using isotope fractionation of nitrate-nitrogen and nitrate-oxygen for evaluation of microbial denitrification in a sandy aquifer

The nitrate input into the groundwater from forest and agricultural land in the ‘Fuhrberger Feld’ catchment area is virtually completely degraded by microbial denitrification in the aquifer. Five multilevel groundwater sampling wells show characteristic depth functions of nitrate, proving the denitrification process. A series of 10 samples of ‘arable land groundwater’ was taken from different depths at these multilevel wells and isotope ratios of 15 N 14 N and 18 O 16 O in the nitrate were measured. The sampling was based on the two-dimensional vertical-plane groundwater flownet and on the knowledge of the landuse upgradient of the multilevel wells. High nitrate concentrations are associated with low δ 15N and low δ 18O values, and vice versa. The δ 15 N δ 18 O ratio indicates a linear increase with a slope of 1:2.1. The microbial dentrification accords with a Rayleigh process with calculated enrichment factors of ε = −15.9‰ for 15N and ε = −8.0 ‰ for 18O. Three samples taken from groundwater recharged under coniferous forest show deviating characteristics: low nitrate concentrations with low δ 15N values (1–4‰) but very high δ 18O values (40–50‰). Using the previously quantified enrichment factors for 15N and 18O, an attempt was made to ascertain the significance of denitrification in specific depth sections at two other multilevel wells (7 samples). In two cases quantitative assessments on the degree of nitrate concentration decrease owing to denitrification were possible. However, in other cases the specific nitrate isotope fractionation was masked by the variability of nitrate concentration and of nitrate isotope ratios in the corresponding groundwater recharge preventing definite interpretations.

First and second passage times of Rayleigh processes (Corresp.)

The first and second passage times of a stationary Rayleigh process R(t,a) are discussed. R(t,a) represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequency f_{0} plus stationary Gaussian noise of unit variance having a narrow-band power spectral density which is symmetrical about f_{0} . Approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times of R(t,a) . The resulting probability functions are presented in graphs for the case when the power spectral density of the noise is Gaussian. Related results concerning the approximate distribution function of the absolute minimum or absolute maximum of R(t,a) in the closed interval [0,\tau] are also presented. The exact probability densities are expressed in the form of an infinite series of multiple integrals.

Some remarks on the Rayleigh process

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

Some remarks on the Rayleigh process

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.