Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prékopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal—by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.     

Optische Methoden

Analytical and Bioanalytical Chemistry -     

Fluctuation theorems and 1/f noise from a simple matrix

Here we present a model for a small system combined with an explicit entropy bath that iscomparably small. The dynamics of the model is defined by a simple matrix, M. Each row ofM corresponds to a macrostate of the system, e.g. net alignment, while the elements in therow represent microstates. The constant number of elements in each row ensures constantentropy, which allows reversible fluctuations, similar to information theory where aconstant number of bits allows reversible computations. Many elements in M come from themicrostates of the system, but many others come from the bath. Bypassing the bath statesyields fluctuations that exhibit standard white noise; whereas with bath states the powerspectral density varies as S(f) ∝ 1 /f overa wide range of frequencies, f. Thus, the explicit entropy bath is the mechanismof 1/f noisein this model. Both forms of the model match Crooks’ fluctuation theorem exactly,indicating that the theorem applies not only to infinite reservoirs, but also tofinite-sized baths. The model is used to analyze measurements of 1/f-like noise from asub-micron tunnel junction.     

A high-field solid-state 35/37Cl NMR and quantum chemical investigation of the chlorine quadrupolar and chemical shift tensors in amino acid hydrochlorides

A series of six L-amino acid hydrochloride salts has been studied by 35/37Cl solid-state NMR spectroscopy (at 11.75 and 21.1 T) and complementary quantum chemical calculations. Analyses of NMR spectra acquired under static and magic-angle-spinning conditions for the six hydrochloride salts, those of aspartic acid, alanine, cysteine, histidine, methionine and threonine, allowed the extraction of information regarding the chlorine electric field gradient (EFG) and chemical shift tensors, including their relative orientation. Both tensors are found to be highly dependent on the local environment, with chlorine-35 quadrupolar coupling constants (CQ) ranging from -7.1 to 4.41 MHz and chemical shift tensor spans ranging from 60 to 100 ppm; the value of CQ for aspartic acid hydrochloride is the largest in magnitude observed to date for an organic hydrochloride salt. Quantum chemical calculations performed on cluster models of the chloride ion environment demonstrated agreement between experiment and theory, reproducing CQ to within 18%. In addition, the accuracy of the calculated values of the NMR parameters as a function of the quality of the input structure was explored. Selected X-ray structures were determined (L-Asp HCl; L-Thr HCl) or re-determined (L-Cys HCl.H2O) to demonstrate the benefits of having accurate crystal structures for calculations. The self-consistent charge field perturbation model was also employed and was found to improve the accuracy of calculated quadrupolar coupling constants, demonstrating the impact of the neighbouring ions on the EFG tensor of the central chloride ion. Taken together, the present work contributes to an improved understanding of the factors influencing 35/37Cl NMR interaction tensors in organic hydrochlorides.     

The bitwisted Cartesian model for the free loop fibration

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration ΩY→ΛY→Y is obtained. As an application we establish an algebra isomorphism H^∗(ΛY;Z)≈S(U)⊗Λ(s−1U) for the polynomial cohomology algebra H^∗(Y;Z)=S(U).     

Exploring reaction pathways with transition path and umbrella sampling: application to methyl maltoside

The transition path sampling (TPS) method is a powerful approach to study chemical reactions or transitional properties on complex potential energy landscapes. One of the main advantages of the method over potential of mean force methods is that reaction rates can be directly accessed without knowledge of the exact reaction coordinate. We have investigated the complementary nature of these two differing approaches, comparing transition path sampling with the weighted histogram analysis method to study a conformational change in a small model system. In this case study, the transition paths for a transition between two rotational conformers of a model disaccharide molecule, methyl beta-D-maltoside, were compared with a free energy surface constrained by the two commonly used glycosidic (phi,psi) torsional angles. The TPS method revealed a reaction channel that was not apparent from the potential of mean force method, and the suitability of phi and psi as reaction coordinates to describe the isomerization in vacuo was confirmed by examination of the transition path ensemble. Using both transition state theory and transition path sampling methods, the transition rate was estimated. We have estimated a characteristic time between transitions of approximately 160 ns for this rare isomerization event between the two conformations of the carbohydrate. We conclude that transition path sampling can extract subtle information about the dynamics not apparent from the potential of mean force method. However, in calculating the reaction rate, the transition path sampling method required 27.5 times the computational effort than was needed by the potential of mean force method.     

The interplay of inverted redox potentials and aromaticity in the oxidized states of new pi-electron donors: 9-(1,3-dithiol-2-ylidene)fluorene and 9-(1,3-dithiol-2-ylidene)thioxanthene derivatives

Derivatives of 9-(1,3-dithiol-2-ylidene)fluorene (9) and 9-(1,3-dithiol-2-ylidene)thioxanthene (10) have been synthesised using Horner-Wadsworth-Emmons reactions of (1,3-dithiol-2-yl)phosphonate reagents with fluorenone and thioxanthen-9-one. X-ray crystallography, solution electrochemistry, optical spectroscopy, spectroelectrochemistry and simultaneous electrochemistry and electron paramagnetic resonance (SEEPR), combined with theoretical calculations performed at the B3P86/6-31G** level, elucidate the interplay of the electronic and structural properties in these molecules. These compounds are strong two-electron donors, and the oxidation potentials depend on the electronic structure of the oxidised state. Two, single-electron oxidations (E(1)ox < E(1)ox) were observed for 9-(1,3-dithiol-2-ylidene)fluorene systems (9). In contrast, derivatives of 9-(1,3-dithiol-2-ylidene)thioxanthene (10) display the unusual phenomenon of inverted potentials (E(1)ox > E(1)ox) resulting in a single, two-electron oxidation process. The latter is due to the aromatic structure of the thioxanthenium cation (formed on the loss of a second electron), which stabilises the dication state (10(2+)) compared with the radical cation. This contrasts with the nonaromatic structure of the fluorenium cation of system 9. The two-electron oxidation wave in the thioxanthene derivatives is split into two separate one-electron waves in the corresponding sulfoxide and sulfone derivatives 27-29 owing to destabilisation of the dication state.     

Electronic interactions in a new pi-extended tetrathiafulvalene dimer

The first pi-extended tetrathiafulvalene (exTTF) dimer in which the two exTTF units are covalently connected by 1,3-dithiole rings has been obtained in a multistep synthetic procedure involving the Ullmann cross-coupling reaction by using copper(I) thiophene-2-carboxylate (CuTC). The electronic spectrum reveals a significant electronic interaction between the exTTF units. The electrochemical study carried out by cyclic voltammetry in solution and in thin-layer conditions, and the electrochemical simulation and spectroelectrochemical (SEC) measurements confirm the electronic communication and show that the oxidation of dimer 14 occurs as two consecutive 2 e(-) processes D(0)-D(0)-->D(2+)-D(0)-->D(2+)-D(2+). Theoretical calculations, performed at the B3P86/6-31G* level, confirm the experimental findings and predict that 14(2+) exists as a delocalized D(.+)-D(.+) species in the gas phase and as a localized D(2+)-D(0) species in solution (CH(3)CN or CH(2)Cl(2)). Oxidation of 14(2+) forms the tetracation 14(4+) which is constituted by two aromatic anthracene units bearing four aromatic, almost orthogonal 1,3-dithiolium cations.