Pólya number and first return of bursty random walk: Rigorous solutions

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we investigate Pólya number and first return for bursty random walk on a line, in which the walk has different step size and moving probabilities. Using the concept of the Catalan number, we obtain exact results for first return probability, the average first return time and Pólya number for the first time. We show that Pólya number displays two different functional behavior when the walk deviates from the recurrent point. By utilizing the Lagrange inversion formula, we interpret our findings by transferring Pólya number to the closed-form solutions of an inverse function. We also calculate Pólya number using another approach, which corroborates our results and conclusions. Finally, we consider the recurrence properties and Pólya number of two variations of the bursty random walk model.     

Probabilistic derivation of van der Waals' equation

A probabilistic derivation of van der Waals' equation is given where the non-ideal contributions to the entropy coincide with the log-likelihood ratios, at their maxima, of the binomial distribution and the upper or lower bounds on the hypergeometric and Pólya distributions. The hypergeometric distribution accounts for the repulsive interactions of a hard sphere gas while the Pólya distribution accounts for the attractive interactions.1. Work supported in part by contributions from the Ministry of Public Education (M.P.I.) and the National Science Council (C.N.R.).     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

Hidden symmetry of the CKM and neutrino-mapping matrices

We propose that the smallness of the light quark masses is related to the smallness of the T (i.e. CP) violation in hadronic weak interactions. Accordingly, for each of the two quark sectors (“upper” and “lower”) we construct a 3 × 3 mass matrix in a bases of unobserved quark states, such that the “upper” and “lower” basis states correspond exactly via the W^± transitions in the weak interaction. In the zeroth approximation of our formulation, we assume T conservation by making all matrix elements real. In addition, we impose a “hidden symmetry” (invariance under simultaneous translations of all three basis quark states in each sector), which ensures a zero mass eigenstate in each sector.Next, we simultaneously break the hidden symmetry and T invariance by introducing a phase factor e^iχ in the interaction for each sector. The Jarlskog invariant J_CKM, as well as the light quark masses are evaluated in terms of the parameters of the model. Comparing formulas, we find that most unknown factors drop out, resulting in a simple relation with , to leading order in χ and m_s/m_b, with A, λ the Wolfenstein parameters. (Because of the large top quark mass, the contribution from upper quark sector can be neglected.) Setting J_CKM = 3.08 × 10⁻⁵, m_b = 4.7 GeV (1s mass), m_s = 95 MeV, A = 0.818, and λ = 0.227, we find , consistent with the accepted value m_d = 3 − 7 MeV.We make a parallel proposal for the lepton sectors. With the hidden symmetry and in the approximation of T invariance, both the masses of e and ν₁ are zero. The neutrino-mapping matrix V_ν is shown to be of the same Harrison-Perkins-Scott form which is in agreement with experiments. We also examine the correction due to T violation, and evaluate the corresponding Jarlskog invariant J_ν.     

The nano-structural properties of hydrogenated a-Si and Si-C thin films alloys by GISAXS and vibrational spectroscopy

Amorphous hydrogenated silicon (a-Si:H) with high hydrogen content (10-40 at.%), and non-stehiometric silicon-carbon (Si_1−xC_x) thin films with a variation of the carbon to silicon ratio up to 0.3, were deposited by using a magnetron sputtering source. The Si_1−xC_x thin films were partially crystallised after deposition by thermal annealing up to 1050 °C.The GISAXS (Grazing Incidence Small Angle X-ray Scattering) spectra of all of the prepared specimens indicate the presence of “particles” in the “bulk” of the films. For the a-Si:H samples, “particles” are most probably voids agglomerates with a variation in size between 3 and 6 nm. The mean value of the size distribution of the “particles” increases while its width slightly decreases with the hydrogen content in the film. This indicates a better structural ordering which is consistent with the results of Raman spectroscopy that show a decrease of the ratio between intensities of transversal acoustic (TA) and transversal optic (TO) phonon peaks, I_TA/I_TO, and a narrowing of the TO peak with increasing hydrogen content. These results are discussed as a consequence of the beneficial influence of hydrogen bombardment during the film growth.For Si_1−xC_x thin films, the “particles” are assumed to be SiC nano-crystals with a size between 2 and 14 nm and they are larger in films with a higher carbon concentration. Inside each of the films, the crystals are larger closer to surface and they grow faster in the direction parallel to the surface than in that which is perpendicular to it.     

Phantom Friedmann cosmologies and higher-order characteristics of expansion

We discuss a more general class of phantom (p < −?) cosmologies with various forms of both phantom (w < −1), and standard (w > −1) matter. We show that many types of evolution which include both Big-Bang and Big-Rip singularities are admitted and give explicit examples. Among some interesting models, there exist non-singular oscillating (or “bounce”) cosmologies, which appear due to a competition between positive and negative pressure of variety of matter content. From the point of view of the current observations the most interesting cosmologies are the ones which start with a Big-Bang and terminate at a Big-Rip. A related consequence of having a possibility of two types of singularities is that there exists an unstable static universe approached by the two asymptotic models—one of them reaches Big-Bang, and another reaches Big-Rip. We also give explicit relations between density parameters Ω and the dynamical characteristics for these generalized phantom models, including higher-order observational characteristics such as jerk and “kerk.” Finally, we discuss the observational quantities such as luminosity distance, angular diameter, and source counts, both in series expansion and explicitly, for phantom models. Our series expansion formulas for the luminosity distance and the apparent magnitude go as far as to the fourth-order in redshift z term, which includes explicitly not only the jerk, but also the “kerk” (or “snap”) which may serve as an indicator of the curvature of the universe.     

Magnetic response of chiral carbon nanotori: The dependence of torus radius

Based on tight-binding approximation, the magnetic moments of chiral “metallic” carbon nanotori (i.e., nanotori formed from metallic carbon nanotubes) are investigated by application of London theory. It is shown that in the vicinity of zero magnetic field, a chiral “metallic” carbon nanotorus (n,m,p=3I) would exhibit diamagnetic response, while paramagnetic response for these “metallic” carbon nanotorus (n,m,p≠3I), with I an integer. This prediction differs from previous conclusions, which can be ascribed to curvature effect considered in our calculation.     

Morphological and dynamical aspects of the room evacuation process

We study the evacuation of a set of 200 pedestrians from a room under a state of panic. The dynamics of the pedestrians is given by the Social Force Model. The degree of panic is controlled by a parameter v_d which represents the velocity at which pedestrians wish to move. We show that the “faster is slower effect” can be understood in terms of the works performed by the different forces present in the system and the role played by dissipative terms in the model. Beyond the maximum flow rate the “granular cluster” mass distribution displays a transition from exponentially decaying to “U-shaped” as this value of v_d evacuation efficiency begins to decrease rapidly.     

Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique

We demonstrate that in addition to their role in tuning the wavelength of an N-stage hybrid liquid-crystal Šolc filter, liquid-crystal cells can also be used to vary the transmission bandwidth of such filter around any of the tuned wavelength. This bandwidth tuning is based on the variation of the number of stages by what we call here an “optical cancelling technique”. This is achieved by varying the birefringence of the liquid-crystal cells whose optical path difference switches between two particular values. We show that for a 10-stage filter and at λ_i = 1.532 μm, the calculated 3-dB bandwidth varies from 2.6 to 11.8 nm when the number of “optically-cancelled” hybrid plates increases from 0 to 8. During the tuning process, the contrast ratio remains equal to that of the equivalent classical Šolc filter.     

The analytical treatment of magnetic properties of three-layer ferrimagnetic superlattice

The Hamiltonian of the magnetic superlattice with three-layer unit cell was treated within the Boson formalism. The Boson Green’s functions (BGFs) were derived and it was shown that the system for BGFs splits into two sets which lead to the energies with opposite signs, although the energies of elementary excitations are strictly only the positive ones. However, when corresponding energies are used, the correlation functions calculated from both sets are the same. All the physically relevant quantities: total energy of the system, ground state energy, layer magnetization and zero-point (quantum) fluctuations are derived analytically by using both sets, showing that they lead to the same expressions. The Hamiltonian was also diagonalized by the so-called “u-v” transformation of the operators. It is shown that in spite of formal independence of the approaches, there exists a close relationship between BGF and “u-v” transformations.     

Anomalous B–H behaviour of electrical steels at very low flux density

The behaviour of ferromagnetic materials under very low magnetic field was investigated more than a century ago by Lord Rayleigh. However, it has been shown since that the so-called Rayleigh law fails for very low magnetic fields, although the explanation for this phenomenon was not given. An anomalous B–H behaviour at very low alternating peak flux density in conventional grain-oriented (GO) and non-oriented (NO) electrical steels is reported. It has been found that the initial permeability is constant for all the measured frequencies (from 20 to 400 Hz) at peak flux density below 0.1 mT, and in this region the magnetisation is almost reversible (for both GO and NO). At higher flux density the B–H loops become visibly irreversible, with a relatively narrow (for GO) or very wide (for NO) transition region. For GO the B–H loop becomes visibly “distorted” for all frequencies at around 2 mT. The eddy current loss calculated from the so-called “classical” equation gives values higher than the measured total losses at lower frequencies. Both these measured results are difficult to explain.     

Quantum phase properties associated to solvable quantum systems using the nonlinear coherent states approach

In this paper we study the quantum phase properties of “nonlinear coherent states” and “solvable quantum systems with discrete spectra” using the Pegg-Barnett formalism in a unified approach. The presented procedure will then be applied to few special solvable quantum systems with known discrete spectrum as well as to some new classes of nonlinear oscillators with particular nonlinearity functions. Finally the associated phase distributions and their nonclasscial properties such as the squeezing in number and phase operators have been investigated, numerically.     

A probabilistic walk up power laws

We establish a path leading from Pareto’s law to anomalous diffusion, and present along the way a panoramic overview of power-law statistics. Pareto’s law is shown to universally emerge from “Central Limit Theorems” for rank distributions and exceedances, and is further shown to be a finite-dimensional projection of an infinite-dimensional underlying object — Pareto’s Poisson process  . The fundamental importance and centrality of Pareto’s Poisson process is described, and we demonstrate how this process universally generates an array of anomalous diffusion statistics characterized by intrinsic power-law structures: sub-diffusion and super-diffusion, Lévy laws and the “Noah effect”, long-range dependence and the “Joseph effect”, 1/f$1/f$ noises, and anomalous relaxation.     

Relationship between loudness perception and noise indices in Valdivia, Chile

Residents perception of road traffic noise loudness in relation to the measured noise indices close to their dwellings was studied. Percentile distributions of five loudness categories as a function of the Day-Night noise index L_DN were obtained. Hearing sensitivity was considered as a factor in loudness perception. In addition, the prevalence of people’s perception of traffic noise in the “Extremely Loud” loudness category was compared with percentage of people stating that they were “Highly Annoyed” by noise. It is concluded that hearing sensitivity for noise is one of the variables that explains the loudness classification difference in different L_DN index ranges and that the percentages of people “Highly Annoyed” by noise are slightly higher than the percentages obtained in the “Extremely Loud” category of loudness perception.     

Multi-fractal formulation of compressible fully developed turbulence: Parabolic-profile approximation for the singularity spectrum

A parabolic-profile approximation (PPA) for the singularity spectrum D (h) in the multi-fractal model for compressible fully developed turbulence (FDT) is considered and is then extrapolated to the Kolmogorov microscale regime. The generalization of Kolmogorov’s “4/5th law” relating the third-order velocity structure function to the mean energy dissipation rate ε to compressible FDT is considered. The PPA is also shown to afford, unlike the generic multi-fractal model, an analytical calculation of probability distribution function (PDF) of velocity gradients and to describe intermittency corrections for this PDF that complement those provided by homogeneous-fractal model.     

Structure refinements of cadmium substituted lead fluoroapatites by powder X-ray pattern fitting

Solid solutions of cadmium and lead fluoroapatite [Pb_(10−x)Cd_x(PO₄)₆F₂ (0?x?5)] were synthesized by a wet process in a basic medium. Replacement of lead by cadmium induces a linear variation of the crystallographic parameters “a” and “c” according to Végard's law. The cadmium content, as obtained from the refinement, is in agreement with the chemical analysis. The distribution of the lead and cadmium ions between two non-equivalent crystallographic sites, M(1) and M(2), was determined by the Rietveld method. The site-occupancy factors of atoms clearly indicate a preference of cadmium for site M(1) in the apatite structure in agreement with its smaller ionic radii. A progressive shift of the F⁻ ion toward the center of the triangles formed by the site M(2) metals has been observed with increasing cadmium content.     

A novel high-k ‘Y5V’ barium titanate ceramics co-doped with lanthanum and cerium

Structural, dielectric, and ferroelectric properties of a novel high-k ‘Y5V’ (Ba_1−xLa_x)(Ti_1−x/4−yCe_y)O₃ ceramics (where x=0.03 and y=0.05, denoted by BL3TC5) with the highest ‘Y5V’ dielectric response (ε′>10 000) among rare-earth-doped BaTiO₃ ceramics to date are investigated in detail using SEM, TEM, XRD, DSC, EPR, Raman spectroscopy (RS), temperature and frequency, electric field dependences of dielectric permittivity (ε′), and temperature and electric field dependences of ferroelectric hysteresis loops. The BL3TC5 diffusion of ferroelectric phase transition occurs around dielectric peak temperatures (T_m) near a room temperature characteristic of dielectric thermal relaxation. Powder XRD data and defect complex model were given. “Relaxor” behavior associated with an order/disorder model and formation of a solid solution were discussed. The EPR results provided the evidence of Ti vacancies as compensating for lattice defects. High-k relaxor nature of BL3TC5 is characterized by an average cubic structure with long-range lattice disordering and local polar ordering; a slow change of the ε′ (T) and P_r(T) curves around T_m; no phase transition observed by DSC; and a broad, red-shifted A₁ (TO₂) Raman phonon mode at 251 cm⁻¹ accompanying the disappearance of the “silent” mode at 305 cm⁻¹ and a clear anti-resonance effect at 126 cm⁻¹ at room temperature.     

The effect of hydrogen stoichiometry on palladium strain and resistivity

The strain and the electrical resistivity of a Pd sample stressed by a constant tension have been investigated through a series of hydrogenation cycles in a continuous H stoichiometry [0?x?0.8] range. The isotropic lattice expansion for both “as drawn” and “annealed” Pd sample reveals a strain of only 1% from pure Pd to PdH_0.8 in disagreement with literature data available; the measured effect is minimum at x=0.13 (α+β phase) and then from x=0.6 (β phase) it has an exponential increase. The contribution of the mechanical tensile stress on the total relative elongation of the wire is also investigated. An increase of the Pd sample tensile strain after each hydrogenation cycle is reported for “as drawn” samples, while for “annealed” samples the reverse behaviour is observed. Moreover, annealed samples show considerably higher value of tensile strain compared to “as drawn”. The variation of mechanical strain versus H content, for both “annealed” and “as drawn”, has a maximum at x=0.52. Strain variation and resistivity variation versus H content exhibit similar behaviour.     

On the “expanded local mode” approach applied to the methane molecule

On the base of the “expanded local mode approach” (see, Ref. [12]) a simple expression of the ambiguity parameter sinγ of the CH₄ molecule is estimated and then, using empirical relations between F_i…j force coefficients, simple relations between different spectroscopic parameters of the methane molecule are derived. Comparison with corresponding “experimental” values is made, that shows more than satisfactory correlations between both (predicted and obtained from experimental data) sets of parameters.