Bernoulli Statistics Research Articles

Electron counting statistics and coherent states of electric current

A theory of electron counting statistics in quantum transport is presented. It involves an idealized scheme of current measurement using a spin 1/2 coupled to the current so that it precesses at the rate proportional to the current. Within such an approach, counting charge without breaking the circuit is possible. As an application, we derive the counting statistics in a single channel conductor at finite temperature and bias. For a perfectly transmitting channel the counting distribution is Gaussian, both for zero-point fluctuations and at finite temperature. At constant bias and low temperature the distribution is binomial, i.e., it arises from Bernoulli statistics. Another application considered is the noise due to short current pulses that involve few electrons. We find the time-dependence of the driving potential that produces coherent noise-minimizing current pulses, and display analogies of such current states with quantum-mechanical coherent states.

Synthesis and polymerization of 2‐cyano‐2‐methyltrimethylene carbonate

Abstract2‐Cyano ‐methyltrimethylene carbonate (CMTC, 2) was synthesized starting from 2‐methylacrolein (6), propionaldehyde (8) or 2‐hydroxymethyl‐2‐methyl‐1,3‐propanediol (9). A key position in this synthesis is held by 2,2‐bis(hydroxymethyl)propionitrile (3), which in a first step, is reacted — in conjunction with 2,2‐dimethyl‐1,3‐propanediol (200) — with diphenyl carbonate (17) to result in a statistical copolycarbonate. Ring‐closing depolymerization of this polymer, with a catalyst based on tin, yields in a second step the title monomer, CMTC (2), along with 2,2‐dimethyltrimethylene carbonate (DTC, 1b). Separation of the two monomers can be achieved by selective solubilization of the latter in toluene. Polymerization of CMTC is performed preferentially by using weak nucleophiles as initiators, such as Et2AlOEt, Al(O‐secBu)3 or MgBu2, since strong nucleophiles such as BuLi or ZnEt2 tend to give side reactions with the cyano group. Copolymerization of CMTC with DTC results in statistical copolymers. With Et2AlOEt as initiator copolymers with Bernoulli statistics are obtained, with a ratio of diads DTC/DTC : (DTC/CMTC + CMTC/DTC) : CMTC/CMTC equal to 1:2:1 for equimolar feed composition. With MgBu2 as initiator and the same feed composition, a copolymer with the diad ratio 1:5:1 is obtained. Poly(CMTC) is a crystalline polymer with a melting point of 132.9°C and a glass transition temperature of 68.8 °C.

Studies on the relationship between the statistics of void fraction fluctuations and the parameters of two-phase flows

Based on Bernoulli statistics of bubble dynamics, a model of void fraction fluctuations in two-phase flows is introduced. The model is used to characterize changes of the intensity of void fraction fluctuations at various two-phase flow regimes, i.e. bubbly, slug and annular flows. Depending on the applied frequency band, the behavior of the band-passed variance of void fraction fluctuations changes significantly. At high frequencies, the band-passed variance increases monotonously with increasing void fractions, while the variance over the whole frequency range has a prominent maximum. By performing model calculations, the relationship between microscopic two-phase flow parameters and macroscopic flow quantities has been analyzed.

Supramolecular arrangement of Rhodospirillum rubrum B880 holochrome as studied by radiation inactivation and electron paramagnetic resonance.

Oxidation of the B880 antenna holochrome gives rise to a 3.8-G linewidth electron paramagnetic resonance (EPR) signal that is considerably narrower than the 13-G signal of monomeric bacteriochlorophyll (Bchl) cation. Radiation inactivation was used to verify a model according to which this linewidth narrowing is due to delocalization over several Bchl molecules. Chromatophores of the photoreaction centerless mutant F24 of Rhodospirillum rubrum were subjected to different doses of gamma-radiation. This induced not only a decay of the EPR signal amplitude but also its linewidth broadening. According to target theory, the induced amplitude decay of the EPR signal had a target size of 10.5 kDa. This is attributed to an elementary structure (alpha1beta1Bchl2), whose number in the membrane would limit the rate of encounter with ferricyanide and thus the formation of unpaired spins. We applied Bernoulli statistics to predict, for a given survival probability of the signal, the number of surviving elementary structures in aggregates of (alpha1beta1Bchl2)n where n was varied from 4 to 7. Using an equation that predicted the Bchl special pair in the photo-reaction center, we were able to simulate the observed relationship between the EPR linewidth and the dose of radiation. The best fit was obtained with a hexameric structure alpha1beta1Bchl2)6.

Separation and structural analysis by 1H‐NMR of the telomers of methyl methacrylate with thiophenol

AbstractThe stereoselectivity of the addition step and the transfer step of the radical telomerization of methyl methacrylate with thiophenol has been studied by determining the structure and the amount of each isomer by 1H‐NMR. We found, in this case, that the addition step gave preferentially a diad r and that the transfer step gave preferentially a diad m. On the other hand, the tacticity observed for higher telomers is preferentially syndiotactic, similar to that of the poly(methyl methacrylate) prepared by radical initiation. The propagation process for radical telomerization obeys, as for the radical polymerization, Bernoulli's statistics.

NMR studies of ionomers. 3. Synthesis and microstructure of poly(ethyl acrylate‐co‐lithium acrylate)

AbstractCopolymers of ethyl acrylate (EA) with lithium acrylate (LiA) with relatively low contents of LiA units (up to 30%) were synthesized by alkaline hydrolysis of poly(EA). 13C‐NMR spectra of these copolymers registered in a mixture of deuterated benzene with deuterated methanol (85:15 v/v) presented configurational splittings of the ester methylene, methylene main chain, and ester carbonyl group signals. Assignments of these splittings were made and Bernoulli statistics with a meso probability of 0.48 could fit both poly(EA) and poly(EA‐co‐LiA). Only the CO signal of the copolymer was sensitive to sequence distribution and indicated that the LiA units have a tendency to be isolated on the chain. Two‐dimensional HETCOR pulse sequence helped to assign the main chain proton spectrum.

Etude structurale des télomères du méthacrylate d'époxy‐2,3 propyle avec les thiols

AbstractThe structure of telomers 5a, b, prepared from 2,3‐epoxypropyl methacrylate (2) and benzenethiol (1a) or 1‐dodecanethiol (1b) by UV irradiation, was studied by MS, IR, 1H and 13C NMR spectroscopy. The IR spectra of the products show that thiol addition occurs only at the double bond of monomer 2. Mono‐ and diadducts could be isolated by fractionation and were characterized spectroscopically together with higher telomers (10 < DPn < 200). Coupling constants and chemical shifts were assigned for the mono‐ and diadducts unambiguously. The tacticity of higher telomers was determined and found to follow the Bernoulli statistics.

13C and 1H NMR study of polymers obtained by polymerization of styrene oxide using diethylzinc/water system as initiator

AbstractStyrene oxide was polymerized by the initiator ZnEt2/H2O and the structure of the resulting polymers was determined by means of 1H and 13C NMR spectroscopy. Both the crystalline and the amorphous fraction have regular head‐to‐tail structure indicating that the oxirane ring opens exclusively at the β position. The stereochemistry of the growth of the crystalline, highly isotactic fraction follows first order Markov statistics, while that of the amorphous fraction follows Bernoulli statistics.

Molecular design of functional polymers having amino groups

Three topics on molecular design of functional polymers having amino groups will be discussed. The first topic is on the synthesis of functional isoprene oligomers with one amino end group. The best condition for the synthesis is to seed a small amount of the amino-ended oligomer into a reaction system where isoprene was polymerized with lithium diisopropylamide as initiator. The highly active behavior of the amino-ended oligomer as chain transfer agent was ascribed to a “macromolecular effect”. The second topic is concerned with a study on the soluble poly(1,4-divinylbenzene) which was formed by polymerizing 1,4-divinylbenzene with lithium diisopropylamide in the presence of excess diisopropylamine in THF. Structural analyses with 400MHz1H NMR and 25 MHz13C NMR showed that there were 2 or 3 branchings per 100 monomeric units along the main chain. Population of the triad tacticities suggested the steric course of polymerization to proceed according to the Bernoulli statistics with respect to diad placements, m and r. Third topic is on a polyamine macromer which was prepared by allowing an ethylenediamine homologue, N,N′-diethylethylene-diamine (DEDA), to react with 1,4-divinylbenzene in the presence of lithium diisopropylamide as catalyst. Results on syntheses and properties of polystyrene-polyamine (SA) and poly(2-hydroxyethyl methacrylate)-polyamine (HA) graft copolymers will be presented with the emphasis on their biomedical behavior.

Crystallization of bisphenol‐A polycarbonate induced by organic salts: Chemical modification of the polymer. II. Model reactions

AbstractIn the presence of sodium o‐chlorobenzoate, bisphenol‐A diphenylcarbonate (BADPC) undergoes complex chemical modifications from 170°C. Reaction between BADPC and minute amounts of salt produces sodium phenoxide end groups which undergo a fast exchange reaction with the carbonate groups of BADPC. In a closed system, an equilibrium molecular weight distribution is progressively attained which agrees fairly well with the one predicted from a simple model based on Bernoulli statistics. In an open system, the equilibrium is continuously displaced by the evaporation of volatile diphenyl carbonate. This leads to a steady increase of the molecular weight.

Use of lanthanide shift reagents for analysing the microstructure of polydienes and polyalkenamers by 1H-NMR spectroscopy

A method is proposed for determining the content of the cis- and trans-units and their distribution in the chain of polybutadiene and the trans-polypentanamer by 1H-NMR spectroscopy. It is shown that tris-1,1,1,2,2,3,3-heptafluoro-7,7-dimethyl-4,6-europium octanedionate combined with AgNO 3 will add at the double bonds of the polymer chain. With the method it is possible to differentiate the diad configurations of cis- ( c) and trans- ( t) units in the chain of the polydienes and polyalkenamers cc, tc(ct) and tt. Statistical analysis of the data on the diad composition of the compounds studied showed that the distribution of the cis- and trans-units in the trans-polypentenamer chain is governed by Bernoulli statistics. Polybutadiene obtained with a nickel catalytic system has a block distribution of the cis- and trans-units along the chain fundamentally differing from the Bernoulli statistics.

Study of the structure of atactic poly-4-vinylpyridine using paramagnetic shearing reagents

A study was made of the addition of paramagnetic shearing reagents of Eu(DPM) 3 and Pr(DPM) 3 on 1H and 13C NMR spectra of atactic poly-4-vinylpyridine in a deuterochloroform solution in the temperature range of 28–100°. Form an analysis of 1H (360 Mc/s) and 13C (22·63 Mc/s) spectra microtactivity of radical poly-4-vinylpyridine was determined and described by Bernoulli's statistics with P m = 0·52. It was shown that 1H shear of the polymer induced using Eu(DPM) 3 is pseudo-contact. Geometrical factors (proportional to the value of pseudo-contact shear) were calculated for fragments of regular conformations of macromolecular iso- and syndiochains. A comparative analysis was made (using the principle of maximum probability) of experimental and calculated shear values induced using Eu(DPM) 3 to determine the conformation composition of the polymer, which is in agreement with the proposed model of microtactivity of poly-4-vinylpyridine. From the temperature dependence of 1H of the shear of the polymer it was concluded that the conformation of poly-4-vinylpyridine changes in the direction of higher content of the syndio- trans-form.

Radical polymerization of methyl methacrylate in the presence of stereoregular poly(methyl methacrylate). III. Influence of temperature

AbstractMethyl methacrylate (MMA) was polymerized by radical initiation at 0, 25, 50, 75 and 100°C in DMF in the presence of preformed isotactic PMMA (iMA) or preformed syndiotactic PMMA (sMA) with different M̄v and also without preformed PMMA (“blank” polymerizations). From the tacticities of the formed polymers it is concluded that blank polymerization does not conform to simple Bernoulli statistics, but follows at least first‐order Markov statistics. The formation of long syndiotactic sequences in the presence of iMA and long isotactic sequences in the presence of sMA denotes still higher‐order Markov statistics. The stereoregulating action is improved by higher M̄v of the preformed polymer (matrix) and lower reaction temperature. These influences can be explained by assuming an equilibrium between polymer growth on the matrix and in the “free” solution. For polymerizations in the presence of iMA or sMA below 300°K, the differences in activation enthalpies (ΔHs/i‡ – ΔHi/s‡) are practically equal to that for the blank polymerization, ca. 900 cal/mole, whereas the differences in activation entropies (ΔSs/i‡ – ΔSi/s‡) differ considerably. (ΔSs/i‡ – ΔSi/s‡) values are highly negative in the presence of iMA and highly positive in the presence of sMA. From these results it is concluded that the isotactic and syndiotactic polymer matrices exert a steric influence on the monomer addition process, thus promoting so‐called stereospecific replica polymerization.