Using bimodal probability distributions in the problems of Brownian diffusion

When analyzing nonlinear stochastic systems, we deal with the chains of differential equations for the moments or cumulants of dynamic variables. To disconnect such chains, the well-known cumulant approach, which is adequate to the quasi-Gaussian expansion of the higher-order moments is used. However, this method is inefficient in the problems of Brownian diffusion in bimodal potential profiles, and the disconnection problem should be solved on the basis of bimodal probability distributions. To this end, we propose to construct bimodal model distributions, in particular, the bi-Gaussian distribution. Cumulants and the expansions of the higher-order moments for symmetric and nonsymmetric bi-Gaussian models. On this basis, we consider relaxation of probability characteristics of one-dimensional Brownian motion in the bimodal potential profile. The dependences of relaxation of the mean value and variance of particle coordinate on the potential barrier “power,” the noise intensity, and the initial distribution of particles are analyzed numerically. In particular, it is shown that relaxation proceeds by stages with different temporal scales in the case of a powerful barrier. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 8, pp. 718–729, August 2006.     

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv ²), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA ₁ exp[−B ₁(t)v ²] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Water transbilayer diffusion in macroscopically oriented lipid bilayers as studied by pulsed field gradient NMR

Water self-diffusion in lipid bilayers macroscopically oriented on glass plates was studied by pulsed field gradient¹H nuclear magnetic resonance technique. Diffusion decays were multicomponent with a distribution of diffusion coefficients ranging from about 10⁻¹⁰ to about 10⁻¹³ m²/s. A number of measurements with variations of the sample orientation, diffusion time and the distance between the glass plates showed that the “fast” component of diffusion corresponds to water in the bilayer “cracks”. The “slow” component of diffusion corresponds to transbilayer water diffusion in the long-diffusion-time regime. For a more reliable separation of parts corresponding to fast and slow diffusion of water, a “component-resolved spectroscopy” method for the global analysis of correlated spectral data (P. Stilbs, K. Paulsen, P.C. Griffiths: J. Phys. Chem. 100, 8180, 1996) was applied.     

Nonlinear filtration of MS-flux intensity in the presence of dead time

The filtration of MS-flux intensity in the presence of dead time is considered in the case when the controlling process is a diffusional Markov process with known drift and diffusion coefficients. Equations determining the evolution of thea posteriori probability density of the control-process values in the time intervals between the moments of onset of events of the observable flux are obtained, as well as a formula for thea posteriori probability density at those moments. Under the assumption of a Gaussiana posteriori probability density, a quasi-optimal filtration algorithm for the control process is synthesized. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 38–44, April, 1997.     

Orthotropic ambipolar diffusion model for γ-manganese dioxide. The role of the reactive sub-surface

γ-manganese dioxide is considered as a continuing solid medium, when it is submitted to diffusion laws. These are derived from an appropriate adoptation of “the analytical theory of heat diffusion in the anisotropic medium”. We have taken into account the ambipolar nature of the diffusion and the internal electrical field which accompanies it. The “instantaneous or continuous point source” theory, which gives a physical interpretation to the analytical solution of the heat conduction differential equation, was applied. With this we explain how particles diffuse from the discontinuous reactive area into the γ-MnO₂ bulk.     

Estimating volume changes in the diffusion zone. 2. Interaction of two finite media

We solve the problem of full diffusion homogenization of bimetallic samples of finite thickness. In the special case when the thickness of one of the materials is small, we obtain an approximate analytic solution. Numerical analysis enables us to construct the concentration distribution, the relative volume change of the sample, and also the stress and deformation distribution at various moments of time. We determine the full homogenization time and the relative volume change as a function of the relative thickness of the initial samples. Institute of Strength Physics and Materials Production. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 48–56, June, 1997.     

Symmetry Transitions in Random Matrix Theory &; <Emphasis Type="Italic">L</Emphasis>-functions

We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum . Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = π y/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures. JPK is supported by an EPSRC Senior Research Fellowship. BEO was supported by an Overseas Research Scholarship and a University of Bristol Research Scholarship.     

Searching for fractal phenomena in multidimensional phase-spaces

A unified point of view on the fractal analysis ind-dimensional phase-spaces is presented. It is applicable to the data coming from the counting experiments. Explicit expressions are formulated for the fundamental types of factorial moments characterizing the presence of the fractal phenomena, their number being given by (2 ^d+1 − 1), as well as for a variety of associated statistical moments; special attention is paid to two and three dimensions. In particular, it is found that scaling properties of the modified dispersion moments are directly related with the presence of empty bins in the corresponding distributions. As to the high-energy experiments, those expressions can be applied to the data presently available, e.g. from LEP, as well as to the data arising in the near future from heavy-ion collisions performed at the CERN collider and from the pp collisions observed at the Tevatron, Fermilab.     

Magnetic moments of decuplet baryons in an independent particle potential model

Magnetic moments of decuplet baryons have been calculated in a relativistic independent quark model with a phenomenological potential in equally mixed scalar-vector harmonic form. Such a model has been successful in describing wide ranging hadronic phenomena in mesonic and baryonic sectors. Using the solutions of the constituent quark orbitals with the model parameters taken from its earlier applications, the magnetic moments of decuplet baryons Δ⁺⁺ and Ω⁻ have been obtained which are in good agreement with the available experimental data. However, the agreement is found to be much better when the magnetic moment ratios such as μ_δ++/μ _p and μ_Ω-/μ_Λ are considered. Model predictions for the magnetic moments of other decuplet baryons together with the charge radii have also been calculated which may be verified in future experiments.     

Absorption and scattering of light in quasi-zero-dimensional structures. I. Transition dipole moments of the charge carriers

A theory is developed for of the interaction of an electromagnetic field with local charge-carrier states of arising near a small dielectric particle. It is shown that the transition dipole moments between local states are large, higher than the typical values of the transition dipole moments for semiconductors. Fiz. Tverd. Tela (St. Petersburg) 39, 720–722 (April 1997)     

Magnetic moments of mirror nuclei with tilted-foil polarization

We report here on an ongoing experimental program initiated at the ISOLDE facility at CERN for the measurement of magnetic moments of short-lived radionuclides, with the emphasis on magnetic moments of mirror nuclei in far-from-stability regions. The nuclei are polarized by the tilted foil technique and the resulting 0–180^○ βasymmetry is monitored as a function of rf frequency applied in an NMR setup. In order to achieve sufficiently high energy for transmission through the foils, the experimental setup is mounted on a high voltage platform. The first experiment in this program was the measurement of the βasymmetry and the NMR resonance for the ground state of ²³Mg (I=3/2, T_1/2=11.3 s), yielding μ=−0.533(6) nm. Improvements to the experimental setup are presently being designed, to be used in conjunction with the new developments at ISOLDE for obtaining high charge-state ions from the EBIS (REX-ISOLDE) ion source. This will help pave the way for measurements of magnetic moments of T=3/2 nuclei in the s–d shell and of T=1/2 f-shell nuclei. The study of relaxation times and other solid-state phenomena in semiconductors and other materials of interest using this technique is also contemplated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solvatochromic Effect on the Photophysical Properties of Two Coumarins

The absorption and emission spectra of two coumarins namely 7, 8 benzo-4-azidomethyl coumarin (C₁) and 6-methoxy-4-azidomethyl coumarin (C₂) have been recorded at room temperature in solvents of different polarities. The ground state dipole moments (μ _g ) of two coumarins were determined experimentally by Guggenheim method. The exited state (μ _e ) dipole moments were estimated from Lippert’s, Bakhshievs and Chamma-Viallet’s equations by using the variation of Stoke’s shift with the solvent dielectric constant and refractive index. The ground and excited state dipole moments were calculated by means of solvatochromic shift method and also the excited state dipole moments are determined in combination with ground state dipole moments. It was observed that dipole moments of excited state were higher than those of the ground state, indicating a substantial redistribution of the π-electron densities in a more polar excited state for two coumarins.     

Surface self-diffusion of a Pt adatom on cuboctahedral and truncated decahedral clusters,size dependence

The self-diffusion of single Pt adatom on the surface of cuboctahedral and truncated decahedral clusters with 561–10 179 atoms are studied within the context of the many-body potentials obtained via the embedded atom method. The minimum energy diffusion path and the corresponding energy barrier for adatom diffusion on the cuboctahedral and truncated decahedral clusters surfaces are determined through a combination of the quenched molecular dynamics and the nudged elastic band method. The calculated energy barriers are consistent with the available experimental data. The dependence of energy barrier for adatom diffusion across the step edge on the cluster size is found. For the larger cuboctahedral and truncated decahedral clusters, the simulations show that the movement of the adatom is confined to a central region, and the adatom may escape from the center region only at elevated temperatures. In addition, we also find that the truncated decahedral structure is more favorable over the cuboctahedral structure for smaller clusters. The cluster growth experiments support our results.     

Anomalies of the electronic properties of alloys in the presence of variation of the local magnetic moments

The concentration dependences of the electronic properties (residual electrical resistivity, diffusion thermoelectric power, normal Hall effect, and low-temperature specific heat) and the magnetic characteristics (magnetization and paramagnetic susceptibility) of quasibinary (Pd_xPt_1−x )₃Fe, Pt₃Mn_xFe_1−x , (Pd_xAu_1−x )₃Fe, and Sc_xTi_1−x Fe₂ alloys are investigated. A relationship is established between the anomalous behavior of the kinetic properties and the variation of the local magnetic moments. The absence of corresponding anomalies in the concentration dependence of the specific heat indicates that the density of states at the Fermi level does not change significantly and, therefore, that the conventional Mott two-band model cannot be used to describe the anomalies in the properties of the alloys in question. A single interpretation of the sum-total of the experimental results is given on the basis of the theory of local fluctuations of the electron spin density in metal magnets. Fiz. Tverd. Tela (St. Petersburg) 39, 1257–1262 (July 1997)     

Quantum diffusion of muonium in GaAs

The diffusion rate of muonium in the III–V compound semiconductor GaAs has been determined from measurements of muon spinT ₁ relaxation induced by motion in the presence of nuclear hyperfine interactions. It is shown for the first time in a semiconductor that (a) there is a crossover of the transport mechanism at about 90 K from stochastic to zero-phonon hopping, as evidenced by a steep rise in the hop rate at lower temperatures, and that (b) the muonium diffuses at the hop rate of 10¹⁰ s⁻¹ (corresponding diffusion constantD≈10⁻⁶ cm²s⁻¹) at lower temperatures as well as at room temperature.     

Francium sources and traps for fundamental interaction studies

Francium is one of the best candidates for atomic parity nonconservation (APNC) and for the search of permanent electric dipole moments (EDMs). APNC measurements test the weak force between electrons and nucleons at very low momentum transfers. They also represent a unique way to detect weak nucleon-nucleon interactions. EDMs are instead related to the time-reversal symmetry. Preliminary to these fundamental measurements are precision studies in atomic spectroscopy and the development of magneto-optical traps (MOT), which partially compensate for the lack of stable Fr isotopes. At LNL Legnaro, francium is produced by fusion of 100-MeV ¹⁸O with ¹⁹⁷Au in a thick target, followed by evaporation of neutrons from the compound nucleus. Francium diffuses inside the hot target (1200 K) and is surface ionized for injection at 3 keV in an electrostatic beamline. Typically, we produce 1×10⁶ (²¹⁰Fr ions)/s for a primary flux of 1.5×10¹² particles/s. We have studied Fr yields as a function of primary beam energy, intensity, and target temperature. Information on the efficiency of bulk diffusion, surface desorption and ionization is deduced. The beam then enters a Dryfilm-coated cell, where it is neutralized on a heated yttrium plate. The escape time of neutral Fr (diffusion + desorption) is approximately 20 s at 950 K, as measured with a dedicated setup. In the MOT, we use 6 orthogonal Ti:sapphire laser beams for the main pumping transition and 6 beams from a stabilized diode repumper. Fluorescence from trapped atoms is observed with a cooled CCD camera, in order to reach noise levels from stray light equivalent to approximately 50 atoms. Systematic tests are being done to improve the trapping efficiency. We plan to further develop Fr traps at LNL; in parallel, we will study APNC and EDM techniques and systematics with stable alkalis at Pisa, Siena, and Ferrara.     

Hyperfine fields and magnetic moments in intermetallic ScFe2 with cubic and hexagonal structures

The Sc, Fe and Al NMR spectra have been measured. The spectra and hyperfine fields are dependent on the crystal structures. The magnetic moments of Fe sites were estimated. The magnetic moments of Sc sites μ(Sc) in ScFe₂ (MgCu₂) were evaluated from the magnetic moment per formula μ_∫ as well. μ_∫ in ScFe₂ with hexagonal structures are sharply dependent on the technological conditions and the impurities in the samples.     

Analytical representations of non-Gaussian laws of random walks

An analytical representation of a random process with independent increments in some space (random walks introduced by Pearson) is considered. The law of random walk distribution in space is derived from the general representation of stochastic elementary hops (distribution law of hop probability) using Kadanoff’s concept of the unit increment as one hop. For limited hop laws and laws of hop distributions with all moments there naturally arises Chandrasekhar’s result that describes ordinary physical diffusion. For laws of hop distributions without the second and highest moments there also arise known Lévy walks (flights) sometimes treated as superdiffusion. For the intermediate case, where the distributions of hops have at least the second moment and not all finite moments (these hops are sometimes called truncated Lévy walks), the asymptotic form of the random walk distribution was obtained for the first time. The results obtained are compared with the experimental laws known in econophysics. Satisfactory agreement is observed between the developed theory and the empirical data for insufficiently studied truncated Lévy walks.