Molecular interaction energies and optimal configuration of a cubane dimer

We have studied the dependence of the binding energy of a cubane dimer on the mutual orientation of and the distance between the composing monomers employing the second‐order Møller–Plesset perturbation scheme (MP2) with the cc‐pVDZ molecular basis set. We have found that the MP2 contribution from the molecular correlations is responsible for the bound state of the cubane dimer, whereas the Hartree–Fock contribution remains anti‐bonding at all intermolecular distances. Starting with two molecules in the standard orientation and centers of mass at (0,0,0) and (0,0,d), respectively, the maximal binding energy is found at d = 5.125 Å and one of the monomers rotated by 45° about the z‐axis. This configuration implies that the hydrogen atoms belonging to different monomers tend to repel each other. The results are in agreement with experimental data on the optimal packing of cubane molecules in the solid state. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Oriented tin on the metal surface obtained by crystallization in magnetic field

¹¹⁹Sn NMR showed that β-tin areas that formed upon freezing at magnetic field 9.4 T on the surface of tin polycrystalline samples have their symmetry axes strongly aligned along a particular direction. The fraction of the oriented frozen tin was found to approach the total amount of the Sn melt on the sample surface.     

A model of the holographic principle: Randomness and additional dimension

In recent years an idea has emerged that a system in a 3-dimensional space can be described from an information point of view by a system on its 2-dimensional boundary. This mysterious correspondence is called the Holographic Principle and has had profound effects in string theory and our perception of space-time. In this note we describe a purely mathematical model of the Holographic Principle using ideas from nonlinear dynamical systems theory. We show that a random map on the surface S² of a 3-dimensional open ball B has a natural counterpart in B, and the two maps acting in different dimensional spaces have the same entropy. We can reverse this construction if we start with a special 3-dimensional map in B called a skew product. The key idea is to use the randomness, as imbedded in the parameter of the 2-dimensional random map, to define a third dimension. The main result shows that if we start with an arbitrary dynamical system in B with entropy E we can construct a random map on S² whose entropy is arbitrarily close to E.     

Size-dependent word frequencies and translational invariance of books

It is shown that a real novel shares many characteristic features with a null model in which the words are randomly distributed throughout the text. Such a common feature is a certain translational invariance of the text. Another is that the functional form of the word-frequency distribution of a novel depends on the length of the text in the same way as the null model. This means that an approximate power-law tail ascribed to the data will have an exponent which changes with the size of the text-section which is analyzed. A further consequence is that a novel cannot be described by text-evolution models such as the Simon model. The size-transformation of a novel is found to be well described by a specific Random Book Transformation. This size transformation in addition enables a more precise determination of the functional form of the word-frequency distribution. The implications of the results are discussed.     

Multifractality and freezing phenomena in random energy landscapes: An introduction

We start our lectures with introducing and discussing the general notion of multifractality spectrum for random measures on lattices, and how it can be probed using moments of that measure. Then we show that the Boltzmann-Gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet non-trivial example of disorder-induced multifractal measures. The typical values of the multifractality exponents can be extracted from calculating the free energy of the associated Statistical Mechanics problem. To succeed in such a calculation we introduce and discuss in some detail two analytically tractable models for logarithmically correlated potentials. The first model uses a special definition of distances between points in space and is based on the idea of multiplicative cascades which originated in theory of turbulent motion. It is essentially equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by Derrida and Spohn (1988) in Ref. [12]. In this way we introduce the notion of the freezing transition which is identified with an abrupt change in the multifractality spectrum. Second model which allows for explicit analytical evaluation of the free energy is the infinite-dimensional version of the problem which can be solved by employing the replica trick. In particular, the latter version allows one to identify the freezing phenomenon with a mechanism of the replica symmetry breaking (RSB) and to elucidate its physical meaning. The corresponding one-step RSB solution turns out to be marginally stable everywhere in the low-temperature phase. We finish with a short discussion of recent developments and extensions of models with logarithmic correlations, in particular in the context of extreme value statistics. The first appendix summarizes the standard elementary information about Gaussian integrals and related subjects, and introduces the notion of the Gaussian free field characterized by logarithmic correlations. Three other appendices provide the detailed exposition of a few technical details underlying the replica analysis of the model discussed in the lectures.     

Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x₀=0,x₁,x₂,…,x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek-Spitzer formula and the associated Sparre Andersen theorem.     

Universal correlations and power-law tails in financial covariance matrices

We investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power-law tails. In order to generate individual eigenvalue distributions a chopping procedure is devised, which produces a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively. This suggests a strong degree of robustness especially in the low-lying sector of the spectra, most relevant for portfolio selections. Conversely, the global spectral density of a single covariance matrix as well as the average over all unfolded nearest-neighbour spacing distributions deviate from standard Gaussian random matrix predictions. The data are in fair agreement with a recently introduced generalised random matrix model, with correlations showing a power-law decay.     

The effective dielectric constant of plasmas — A mean field theory built from the electromagnetic ionic T-matrix

This work aims to obtain the effective dielectric constant tensor of a warm plasma in the spirit of the derivation of a mixing law. The medium is made of non point-like ions immersed in an electron gas with usual conditions relating the various lengths which define the problem. In this paper the ion dielectric constants are taken from their RPA responses as developed in a previous paper [1]. Furthermore the treatment of the screening effects is made through a mathematical redefinition of the initial problem as proposed in Ref. [1]. Here the complete calculation of the T-matrix describing the scattering of an electromagnetic wave on an isolated ion immersed in an “effective medium” is given. It is used for building , in the spirit of a mixing law, a self-consistent effective medium theory for the plasma dielectric tensor. We then extend the results obtained in Ref. [1] to higher orders in ion or dielectric inclusion densities. The techniques presented are generic and can be used in areas such as elasticity, thermoelasticity, and piezoelectricity.     

Modulational instability in optical fiber with stochastic parameters and noninstantaneous response

We investigate analytically and numerically the modulational instability (MI) in optical fiber, where the effect of noninstantaneous nonlinear response as well as stochastic coefficients are taken into account. Applying the linear stability analysis and stochastic calculus, we show that the MI gain spectrum reads as the maximal eigenvalue of a constant matrix. In the limiting cases of small fluctuations, we give explicit expressions for the MI gain spectra. In the general configurations, we derive an explicit form of the effective matrix and compute numerically the maximal eigenvalue. The moment MI peak is enhanced and the delayed Raman response reduces the maximum MI gain caused by stochasticity both in anomalous and normal dispersion regimes. Numerical simulations of the full stochastic nonlinear Schródinger equation show that, the phenomenon of MI gives rise to periodic pulse arrays of waves train, as well as to a chain of peaks with continuously growing amplitudes.     

Effects of pulse electrodeposition parameters on the properties of Ni-TiO2 nanocomposite coatings

The aim of this study is to investigate the effects of pulse electrodeposition parameters on the properties of nickel-titania composite coatings electrodeposited from a nickel Watts type bath. The effects of average current density, frequency and duty cycle on the surface morphology, crystal size, preferred orientation of the deposits and the amount of embedded nano-TiO₂ particles in the composite coatings were investigated. The results represented the optimum amount of average current density (e.g., 4 A dm⁻²) for obtaining the highest volume percentage of the incorporated titania particles and subsequently the maximum microhardness. Moreover, by increasing the frequency up to 10 Hz while reducing the duty cycle to 10% at constant peak current density, the volume percentage of particles increased to about 7% which is almost twice as much the volume percentage as deposited particles in direct current method. According to the results the composite coating exhibited obviously [1 0 0] + [2 1 1] as preferred orientation at low pulse frequency and the diffraction intensity of the [2 1 1] fiber orientation is increased, probably due to the pH increase adjacent to the electrolyte/cathode interface at higher frequencies. Also, it has been found that by reduction in the duty cycle, more titania particles were incorporated in the composite coatings and this promoted the nickel crystals growth on [1 0 0] planes and consequently the coatings preferred orientation changed from the [2 1 1] to [1 0 0] + [2 1 1] planes.