The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

Two models for the Freedericksz transition in a fluctuating magnetic field are considered: one is based on a dichotomic and the other on a nonlinear Gaussian noise. Both noises are characterized by a finite correlation time. It is shown that the linear response assumption leading to the best Fokker-Planck approximation in the dichotomic and nonlinear Gaussian cases can be trusted only up to the order ¹ and ⁰, respectively. The role of the corrections to the linear response approximation is discussed and it is shown how to replace the non-Fokker-Planck terms stemming from these corrections with equivalent terms of standard type. This technique is shown to produce perfect agreement with the exact analytical results (dichotomic noise) and to satisfactorily fit the results of analog simulation (nonlinear Gaussian noise).     

The projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator into a perturbation ₁ and an unperturbed part ₀. The standard Fokker-Planck structure is recovered at the second order in ₁, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in ₁ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time, a resummation up to infinite order in must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in ₁, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order in ₁ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady-state distributions. Therefore, over the whole range 0 the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in ₁ vanish. In the short- region the LL leads to results virtually coincident with those of the BFPA. In the large- region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.     

Rational synthesis of covalently bonded organic-inorganic hybrids

Hybrid materials based on covalently linked inorganic polyoxometalates (POMs) and organic species containing a delocalized pi system have drawn increasing attention. These hybrids, traditionally prepared by cluster assembly approaches that lack predictability and controllability, can now be synthesized through common organic reactions by using organically functionalized POM clusters as building blocks. This minireview highlights some of the most recent advances on a particular type of hybrids where the organic and inorganic components are connected by an imido linkage.     

Mesoporous micelle templated silica with incorporated C<Subscript>8</Subscript> and C<Subscript>18</Subscript> phase

Mesoporous silica material of MCM-41 type was synthesized by co-condensation of highly concentrated octyltriethoxysilane (OTEOS), octadecyltriethoxysilane (ODTEOS) and tetraethoxysilane (TEOS). The obtained hybrid materials were characterized using XRD, TG-DSC and low temperature adsorption/desorption of nitrogen. It was shown that the applied method of synthesis allows to obtain silica of MCM-41 type with a high degree of hydrocarbon saturation.     

一种新颖一维铅碘聚合物[(Pb_2I_5)(L-H~ )·2DMF]_n(L=Piperazine)的自组装合成和晶体结构(英文)

0IntroductionIncreasinginterestisfocusedonthestudyofmagnetic,electronicandoptoelectronicpropertiesoflow鄄dimensionalorganic鄄inorganichybridcompoun鄄ds[1~3].Studiesshowthatcomplexconsistingoforganicandinorganiccomponentshavegreatpotentialforthecreationoff…     

Silica hybrid nanocomposites

In this work we present experimental results about the formation, properties and structure of sol — gel silica based biocomposite containing Calcium alginate as an organic compound. Two different types of silicon precursors have been used in the synthesis: tetramethylortosilicate (TMOS) and ethyltrimethoxysilane (ETMS). The samples have been prepared at room temperature. The hybrids have been synthesized by replacing different quantitis of the inorganic precursor with alginate. The structure of the obtained hybrid materials has been studied by XRD, IR Spectroscopy, EDS, BET and AFM. The results proved that all samples are amorphous possessing a surface area from 70 to 290 m²/g. It has also been established by FT IR spectra that the hybrids containing TMOS display Van der Walls and Hydrogen bonding or electrostatic interactions between the organic and inorganic components. Strong chemical bonds between the inorganic and organic components in the samples with ETMS are present. A self-organized nanostructure has been observed by AFM. In the obtained hybrids the nanobuilding blocks average in size at about 8–14 nm for the particles.     

{[Gd(DMF)_3(DMSO)(H_2O)_3](μ-CN)[Fe(CN)_5]}·2H_2O的合成,结构表征和磁性(英文)

0IntroductionThefirstmolecule鄄basedmagnetswerereportedin1986[1].Duringthelasttwodecades,moreandmoreattentionhasbeenpaidtothesynthesisofmolecule鄄basedmagnets,especiallytoPrussianBlueanalogu鄄es[2].TheoldPrussianBlueanalogueshavebeenbroughtuptodateduetoth…     

Product or sum: comparative tests of Voigt,and product or sum of Gaussian and Lorentzian functions in the fitting of synthetic Voigt‐based X‐ray photoelectron spectra

A comparative study for the fitting of X‐ray photoelectron spectra (XPS) using different model functions is presented. Synthetically generated test spectra using Gaussian/Lorentzian convolution and a real measured spectrum are fitted with the three commonly used models: product, sum and Gaussian/Lorentzian convolution functions. In these limited tests, it was found that the sum function is superior to the product function, particularly for low‐noise spectra. Copyright © 2007 John Wiley & Sons, Ltd.     

Siloxane-Based Nanobuilding Blocks by Reaction Between Silanediol and Trifunctional Silicon Alkoxides

The preparation of nanostructured organic-inorganic materials by assembling of nanobuilding blocks allows controlling the extent of phase interaction, which in its turn governs structure-properties relationships. We present here the synthesis of siloxane-based nanobuilding blocks prepared by reacting diphenylsilanediol with vinyltriethoxysilane and triethoxysilane. The reaction products were obtained by non-hydrolytic condensation between silanediol and ethoxide groups in inert atmosphere, in the presence of pyridine, triethylamine or butyl lithium. Different synthetic conditions were examined by means of ATR-FTIR and NMR spectroscopies, showing the formation of siloxane bonds. In the case of triethoxysilane the reaction carried out in the presence of pyridine leads to Si–H bond preservation in the final product. Air stable products with improved Si–O–Si hydrolytic stability can be obtained by removal of the base after the reaction completion. The condensation products can be described as a mixture of siloxane rings involving difunctional and trifunctional silicon units.     

Lamellar Organo-Bridged Silicones

The acid-hydrolysis of an organo-bridged bisdiethoxysilylated molecular precursor bearing urea groups, (EtO)₂MeSi(CH₂)₃NHCONH(CH₂)₁₂NHCONH(CH₂)₃SiMe(OEt)₂, has been performed in pure aqueous medium. Scanning electron microscopy (SEM) analysis of the resulting insoluble solid revealed plate-like forms with a lamellar structure as determined by powder X-ray diffraction (PXRD) studies with a sharp peak at 28.5 Å. The solid state ²⁹Si MAS-NMR spectrum of this bridged siloxane hybrid is consistent with a moderately condensed material with complete preservation of the Si–C bonds throughout the hybrid network. In comparison, the classical sol–gel hydrolysis-condensation of the molecular precursor in ethanol with stoichiometric amount of water and fluoride anion as catalyst produced an amorphous featureless solid.