Prediction of precision from signal and noise measurement in liquid chromatography: Limit of detection

Summary A method to determine the limit of detection (LOD) in high performance liquid chromatography (HPLC) is described. The power spectral density of instrumental baseline variation is fitted by the simplex least squares methods with a mixed random process of white noise and Markov process as a model. The white noise is characterized by standard deviation (SD),; the Markov process by the SD, , and auto-correlation degree, ρ. All required parameters for calculating the LOD signal are obtained by experiment without repeat measurements. No arbitrary constants are needed. The LOD signal is uniquely determined and is characterized by 33.3 % relative standard deviation (RSD) of analyte measurements and 0.13 % of the error of the first type. This signal also specifies that the signal-to-noise ratio =3, using the definition of noise originating from the white noise and Markov process. The theoretical conclusion is verified by the Monte Carlo simulation using real baseline and peaks. The LOD concentrations for naphthalene, acenaphthene, pyrene and perylene are given. First part of series cited as Ref. [1].     

Prediction of precision from signal and noise measurement in liquid chromatography: Limit of detection

Summary A method to determine the limit of detection (LOD) in high performance liquid chromatography (HPLC) is described. The power spectral density of instrumental baseline variation is fitted by the simplex least squares methods with a mixed random process of white noise and Markov process as a model. The white noise is characterized by standard deviation (SD), w; the Markov process by the SD, m, and auto-correlation degree, . All required parameters for calculating the LOD signal are obtained by experiment without repeat measurements. No arbitrary constants are needed. The LOD signal is uniquely determined and is characterized by 33.3% relative standard deviation (RSD) of analyte measurements and 0.13% of the error of the first type. This signal also specifies that the signal-to-noise ratio=3, using the definition of noise originating from the white noise and Markov process. The theoretical conclusion is verified by the Monte Carlo simulation using real baseline and peaks. The LOD concentrations for naphthalene, acenaphthene, pyrene and perylene are given.First part of series cited as Ref. [1].     

Prediction of measurement precision of apparatus using a chemometric tool in electrochemical detection of high-performance liquid chromatography

The relative standard deviation (RSD) of measurements in high-performance liquid chromatography with electrochemical detection (HPLC-ECD) was predicted by a chemometric tool based on the 1/f fluctuation model which is made up of white noise and a Markov process, called the Function of Mutual Information (FUMI) theory. FUMI theory can provide aprecise and reliable detection limit from a single measurement of noise and signal in HPLC-ECD. To obtain RSD (n = 5) for determination of (-)-epicatechin at five concentrations required 12.5 h, while the predicted RSD by FUMI theory required only 0.5 h (one measurement). Moreover, to trace the source of instrumental noise, power spectra of chromatographic baseline were used. Selection of a suitable apparatus in HPLC-ECD system, acquisition of RSD, and detection limits for determination of catechins by HPLC-ECD were simply and easily made by this chemometric tool within a very short time. The use of the FUMI theory for the prediction of measuring precision was more efficient and the optimization was less time-consuming to be suited for determination.     

A Markov model for relaxation and exchange in NMR spectroscopy

A two-state Markov noise process for lattice fluctuations and chemical exchange dynamics is used to derive a stochastic Liouville equation describing the evolution of the spin-density operator in nuclear magnetic resonance spectroscopy. Relaxation through lattice fluctuations and chemical exchange processes is incorporated into the theory at the same fundamental level, and the results are valid for all time scales provided that lattice fluctuations are much faster than chemical exchange kinetics. Time-scale separation emerges as an essential feature from the lowest-order perturbation expansion of the average resolvent in the Laplace domain.     

Discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise

A discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise is developed. Based on the discrete version of Wiener-Khinchin theorem, the theoretical discrete Chebyshev spectrum for the Markov random process is calculated. It is characterized by two parameters: the dispersion and the relaxation frequency (or relaxation time). The noise of corrosion process and the noise of recording equipment are measured. Using the theoretical Chebyshev spectrum, the Markov parameters were found both for the noise of the corrosion process and for the noise of the measuring equipment.     

Prediction of precision from signal and noise measurement in liquid chromatography: Mathematical relationship between integration domain and precision

Summary The precision of integration over noisy instrumental output for quantitative analysis is studied. A probability theory is developed to predict the relative standard deviation (RSD) of integration results over an integration domain from one-point integration (peak height measurement) to entire area integration in HPLC. Common integration modes of horizontal zero line and oblique zero line are taken into account, but no peak overlap is assumed. The question of the analytical superiority of peak height measurement or integration for quantitation is answered. In the HPLC apparatus used, the minimum RSD of measurements is found in the integration domain of ca. ±0.5 σ for analytes [peaks are approximated by the Gaussian signal of width, σ (standard deviation)]. The RSD of integration measurements is also shown to depend on the stochastic properties of background noise (uncorrelated noise and correlated 1/f type noise). The theoretical conclusion is verified by Monte Carlo simulation and HPLC experiments for some aromatic compounds. Second Part of series cited as Ref. [1].     

CO oxidation on Ir(111) surfaces under large non-Gaussian noise

In this article we consider the CO oxidation on Ir(111) surfaces under large external noise with large autocorrelation imposed on the composition of the feed gas, both in experiments and in theory. We report new experimental results that show how the fluctuations force the reaction rate to jump between two well defined states. The statistics of the reaction rate depend on those of the external noise, and neither of them have a Gaussian distribution, and thus they cannot be modeled by white or colored noise. A continuous-time discrete-state Markov process is proposed as a suitable model for the observed phenomena. The model captures the main features of the observed fluctuations and can be modified to accommodate other surface reactions and other systems under non-Gaussian external noise.     

Theoretical study on the effects of internal noise for rate oscillations during CO oxidation on platinum(110) surfaces

Very recently, the effects of internal molecular noise in mesoscopic chemical reaction systems have gained growing attention. Using a mesoscopic stochastic model, the effect of internal noise for rate oscillation during CO oxidation on Pt(110) surface is studied analytically. In a parameter region outside but close to the supercritical Hopf bifurcation, a stochastic normal form is obtained from the chemical Langevin equation. By stochastic averaging procedure, the system is simplified and solvable. Noise-induced oscillation and internal noise coherent resonance (which is related to an optimal system size), observed from simulations, are well reproduced by the theory. The theoretical analysis helps to clearly figure out when and how the internal noise affects the system's oscillating dynamics.     

Long time behaviour of a stochastic model for continuous flow bioreactor

The global dynamics of a deterministic model in wastewater treatment has been investigated in Zhang (J Math Chem 50:2239–2247, 2012). The stochastic version, which can be used for continuous flow bioreactor and membrane reactor is presented in this study. Precisely, we assume there is some uncertainty in the part describing the recycle, which results in a set of stochastic differential equations with white noise. We first show that the stochastic model has always a unique positive solution. Then long time behavior of the model is studied. Our study shows that both the washout equilibrium and non-washout equilibrium are stochastically stable. At the end, we carry out some numerical simulations, which support our theoretical conclusions well.     

A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.     

Coherent Resonance for Rate Oscillations During CO Oxidation on Pt(110) Surfaces: Interplay Between Internal and External Noise

Effects of noise on rate oscillations during CO oxidation on Pt(110) surface were investigated, both theo-retically and numerically, by focusing on the interplay of internal noise (IN) due to stochasticity in reaction events, and external noise (EN) resulting from parameter perturbation. The surface is divided into cells of variable size which are assumed to be well mixed, and we consider the behavior inside a single cell. At-tention is paid to parameter regions subthreshold of the deterministic Hopf bifurcation, where noise can induce stochastic oscillations, the signal-to-noise ratio (SNR) of which shows a maximum with the variation of noise intensity, known as coherent resonance (CR). By stochastic normal theory, we show that IN and EN contribute in a weighted additive way to an effective noise that lead to CR, such that SNR shows a ridge shape in the D-1/N plane, where D and 1/N measures the strength of EN and IN, respectively. It is shown that for too large IN (EN), CR behavior with EN (IN) no longer exists. Numerical simulations show good agreements with the theoretical results     

Using Markov models to simulate electron spin resonance spectra from molecular dynamics trajectories

Simulating electron spin resonance (ESR) spectra directly from molecular dynamics simulations of a spin-labeled protein necessitates a large number (hundreds or thousands) of relatively long (hundreds of nanoseconds) trajectories. To meet this challenge, we explore the possibility of constructing accurate stochastic models of the spin label dynamics from atomistic trajectories. A systematic, two-step procedure, based on the probabilistic framework of hidden Markov models, is developed to build a discrete-time Markov chain process that faithfully captures the internal spin label dynamics on time scales longer than about 150 ps. The constructed Markov model is used both to gain insight into the long-lived conformations of the spin label and to generate the stochastic trajectories required for the simulation of ESR spectra. The methodology is illustrated with an application to the case of a spin-labeled poly alanine alpha helix in explicit solvent.     

Prediction of precision from signal and noise measurement in liquid chromatography: Mathematical relationship between integration domain and precision

Summary The precision of integration over noisy instrumental output for quantitative analysis is studied. A probability theory is developed to predict the relative standard deviation (RSD) of integration results over an integration domain from one-point integation (peak height measurement) to entire area integration in HPLC. Common integration modes of horizontal zero line and oblique zero line are taken into account, but no peak overlap is assumed. The question of the analytical superiority of peak height measurement or integration for quantitation is answered. In the HPLC apparatus used, the minimum RSD of measurements is found in the integration domain of ca. ±0.5 for analytes [peaks are approximated by the Gaussian signal of width, (standard deviation)]. The RSD of integration measurements is also shown to depend on the stochastic properties of back-ground noise (uncorrelated noise and correlated 1/f type noise). The theoretical conclusion is verified by Monte Carlo simulation and HPLC experiments for some aromatic compounds.Second Part of series cited as Ref. [1].     

Investigating aluminum thin films properties by stochastic analysis

In this article, we investigate about stochastic properties of a thin film such as aluminum thin film that is formed by thermal evaporation in high vacuum. So we prepare thin films with different thicknesses on glass surfaces in high vacuum chamber with 10^?6 torr. Then, we use Atomic Force Microscopy (AFM), for determining the height fluctuations of aluminum thin films in different growth stages and use an inverse method that utilizes data from AFM to construct a simple equation that governs the stochastic process and a relation between stochastic parameters such as roughness, correlation and Markov length, drift, and diffusion coefficients during the growth process. Also we show that aluminum thin films, during growth by evaporating in high vacuum chamber, have a mono fractal behavior in small distances(little than 100 nm) on the surfaces. Copyright © 2008 John Wiley & Sons, Ltd.     

IO3--AsO33-化学反应中的随机共振

通过计算机模拟,在IO3--AsO33-化学双稳体系中,对反应控制参量加入周期信号和高斯噪声,体系由于信号和噪声的作用发生上下态之间相互跃迁.计算输出时间序列的功率谱,发现信号在噪声的作用下被放大,在某一最佳噪声强度下,存在一个最大的信噪比输出,表现出随机共振的特征.     

Dynamics of a stochastic model for continuous flow bioreactor with Contois growth rate

In the modelling of the continuous flow bioreactor, due to uncertainties in the environment the growth rate parameter is under perturbation of white noise, which results in a mathematical model governed by a set of stochastic differential equations. In this paper, assume the Contois growth rate is used and then we first show that the stochastic model has always a unique positive solution. Then long time behavior of the model is studied. Our study shows that both the washout and non-washout equilibria are stochastically stable. At the end, we carry out some numerical simulation, which supports our theoretical conclusion well. Also, by the quantities introduced in the last section, both residence time and intensity of the noise have significant effect on the performance of the reactor.     

Simulation of noise characteristic effects on stochastic resonance in Belousov-Zhabotinsky flow reaction

A three-variable model, which was proposed to account for the stochastic resonance (SR) in Belousov-Zhabotinsky (B-Z) reaction in a continuous-flow stirred-tank reactor, is investigated when the control parameter kr, the flow rate, is modulated by noise near supercritical Hopf bifurcation point. Using the computer simulation, noise-induced oscillations are observed, and the signal-to-noise ratio (SNR) goes through a maximum with the increment of noise intensity, which means occurrence of stochastic resonance. in addition, we have also investigated the effects of correlation time of colored noise and the duration time of white noise on the system's dynamics.     

Escape over fluctuating barriers driven by colored noise

Escape over fluctuating barriers in the presence of thermal white noise is addressed. Several general results are established, for stochastic barrier fluctuations being controlled by colored Gaussian noise. Our findings are exact in the limit of white noise sources and (partially) in the limit of extreme large noise color, and are approximate for intermediate noise color. As one main result we find that the escape time can generically exhibit a minimum resonant activation, whenever the colored noise intensity is an increasing function of the noise correlation time. The effects induced by correlated noise sources are addressed as well.     

化学体系随机共振现象的理论研究

化学体系中随机共振现象的特性、机理、本质和应用的研究是非常重要的前沿课题。综述中国科学技术大学非线性化学实验室,在化学体系随机共振现象的理论研究中,所取得的主要成果,并阐述化学体系随机共振现象研究中所提出来的若干重要问题：非均相化学体系中随机共振;化学体系中内信号随机共振;化学体系中多重随机共振;化学体系中随机共振的调制;化学体系中色噪声作用下的随机共振;化学体系中时空随机共振;耦合化学体系中随机共振;化学体系中双参量随机共振。