Multiplicative Lévy noise in bistable systems

Stochastic motion in a bistable, periodically modulated potential is discussed. Thesystem is stimulated by a white noise increments of which have a symmetric stable Lévydistribution. The noise is multiplicative: its intensity depends on the process variablelike |x|^?θ . The Stratonovich and Itôinterpretations of the stochastic integral are taken into account. The mean first passagetime is calculated as a function of θ for different values of thestability index α and size of the barrier. Dependence of the outputamplitude on the noise intensity reveals a pattern typical for the stochastic resonance.Properties of the resonance as a function of α, θ andsize of the barrier are discussed. Both height and position of the peak strongly dependson θ and on a specific interpretation of the stochastic integral.     

The structure of a vortex line and the lower critical field in superconducting alloys

The structure of an isolated vortex line, and the lower critical fieldH _c 1, is calculated by means of the generalized Ginzburg-Landau (GL) theory for arbitrary values of the GL-parameterk(≧1/√2) and the mean free pathl at temperaturesT in the vicinity ofT _c . The free energy functional including the corrections of order [1?(T/T _c )] to the GL-functional is derived exactly. The corresponding Euler-Lagrange equations determining the zero-order (GL) contributions and the corrections of order [1?(T/T _c )] to the order parameter,f(r), and the superfluid velocity,v(r), have been solved numerically. The shapes of the first-order corrections off(r), v(r), and the magnetic field,h(r) are found to depend markedly, for a given value ofκ, on a second parameter,α=0.882(ξ ₀ /l) (whereξ ₀ is theBCS-coherence-distance). The deviations from the GL-solutions become largest forh(r) at parameter valuesk≈ 1 andα ≈ 0(the deviation ofh(0) is about 6% atT=0.9T _c forκ=1 andα=0). The ratioH _c1/H _c (where the thermodynamic criticalH _c has the BCS-temperature-dependence) is found to increase slightly in the “clean” limit (α=0), and to decrease slightly in the “dirty” limit (α=∞) asT decreases (the variation ofH _c 1/H _c is always less than 3% for arbitrary values ofκ andα asT decreases fromT _c to 0.9T _c ).     

Topologic distance in the Lucena network

The Lucena network (LN) is the dual of a multifractal partition of the square. We analyzethe relation between the typical topologic distance l and the number ofvertices Nof the LN. The multifractal partition has one parameter ρ which controls thegeometrical asymmetry of the multifractal. In the limit of ρ → 1 the blocks of thepartition are squared, the connections amont the blocks are short range, the LN is moreregular and the relation l ∝ √N is observed. For the limit ρ → 0 the blocks arestrongly asymmetric, long range connections appear, and the topologic distance followsl ∝(log?N)^α, a weak smallworld phenomenon. For any network size we calculate analytically the size of the minimumdistance l_min (ρ → 0) and the maximaldistance l_max (ρ → 1). The distance in theweak small world regime is calculated using the number of vertices inside a radius oflength land taking into account the network average connectivity and the exponent α.     

Catastrophe in diffusion-controlled annihilation dynamics: general scaling properties

We present a systematic analytical and numerical study of the annihilation catastrophephenomenon which develops in an open system, where species A and B diffuse from the bulk ofrestricted medium and die on its surface (desorb) by the reaction A + B → 0.This phenomenon arises in the diffusion-controlled limit as a result of self-organizingexplosive growth (drop) of the surface concentrations of, respectively, slow and fastparticles (concentration explosion) and manifests itself in the form of an abrupt singularjump of the desorption flux relaxation rate. In the recent work [B.M. Shipilevsky, Phys.Rev. E 76, 031126 (2007)] a closed scaling theory of catastrophe developmenthas been given for the asymptotic limit when the characteristic time scale of explosionbecomes much less than the characteristic time scales of diffusion of slow and fastparticles at an arbitrary ratio of their diffusivities 0< p <1. In this paper we consider the behavior of the system at strongdifference of species diffusivities p ? 1 and reveal a rich general pattern ofcatastrophe development for an arbitrary ratio of the characteristic time scales ofexplosion and fast particle diffusion. As striking results we find remarkable scalingproperties of catastrophe evolution at the crossover between two limiting regimes withradically different dynamics.     

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

Phase Diagram of a Generalized ABC Model on the Interval

We study the equilibrium phase diagram of a generalized ABC model on an interval of the one-dimensional lattice: each site i=1,…,N is occupied by a particle of type α=A,B,C, with the average density of each particle species N _α /N=r _α fixed. These particles interact via a mean field nonreflection-symmetric pair interaction. The interaction need not be invariant under cyclic permutation of the particle species as in the standard ABC model studied earlier. We prove in some cases and conjecture in others that the scaled infinite system N→∞, i/N→x∈[0,1] has a unique density profile ρ _α (x) except for some special values of the r _α for which the system undergoes a second order phase transition from a uniform to a nonuniform periodic profile at a critical temperature \(T_{c}=3\sqrt{r_{A} r_{B} r_{C}}/2\pi\).     

Feynman formulas generated by self-adjoint extensions of the Laplace operator

The Feynman formulas give a representation of a solution of the Cauchy problem for a Schrödinger-type equation (in a special case, for a heat-type equation) using the limit of integrals of finite multiplicity over Cartesian powers of the phase space (in the special case of the configuration space). The limit thus obtained, defining an explicit representation of a one-parameter unitary group e ^it? or a similar object (in our case, this concerns the semigroup e ^t? , which is often referred to in the literature as the Schrödinger semigroup) by integral operators, is interpreted by using Feynman integrals, whereas the expression thus obtained is referred in turn as the Feynman formula. As a rule, the Chernoff theorem, which is a generalization of the well known Trotter formula, is used in the derivation of the Feynman formula.In the paper, Feynman formulas for Schrödinger semigroups e ^t? are obtained, where the role of ? is played by the operator ? _a +V which is a perturbation of the self-adjoint extension of the Laplace operator (parametrized by some a ∈ (?∞, ∞]).     

Transient bi-fractional diffusion: space-time coupling inducing the coexistence of two fractional diffusions

Anomalous diffusion is researched within the framework of the coupled continuous time random walk model, in which the space-time coupling is considered through the correlated function g(t) ~ t ^γ , 0 ≤ γ< 2, and the probability density function ω(t) of a particle’s transition time t follows a power law for large t: ω(t) ~ t ^{? (1 + α)},1 <α< 2. The bi-fractional generalized master equation is derived analytically which can be applied to describe the transient bi-fractional diffusion phenomenon which is induced by the space-time coupling and the asymptotic behavior of ω(t). Numerical results show that for the transient bi-fractional diffusion, there is a transition from one fractional diffusion to another one in the diffusive process.     

An epidemic model of rumor diffusion in online social networks

So far, in some standard rumor spreading models, the transition probability from ignorants to spreaders is always treated as a constant. However, from a practical perspective, the case that individual whether or not be infected by the neighbor spreader greatly depends on the trustiness of ties between them. In order to solve this problem, we introduce a stochastic epidemic model of the rumor diffusion, in which the infectious probability is defined as a function of the strength of ties. Moreover, we investigate numerically the behavior of the model on a real scale-free social site with the exponent γ = 2.2. We verify that the strength of ties plays a critical role in the rumor diffusion process. Specially, selecting weak ties preferentially cannot make rumor spread faster and wider, but the efficiency of diffusion will be greatly affected after removing them. Another significant finding is that the maximum number of spreaders max(S) is very sensitive to the immune probability μ and the decay probability v. We show that a smaller μ or v leads to a larger spreading of the rumor, and their relationships can be described as the function ln(max(S)) = Av + B, in which the intercept B and the slope A can be fitted perfectly as power-law functions of μ. Our findings may offer some useful insights, helping guide the application in practice and reduce the damage brought by the rumor.     

Solutions of a Particle with Fractional <Emphasis Type="Italic">δ</Emphasis>-Potential in a Fractional Dimensional Space

A Fourier transformation in a fractional dimensional space of order λ (0<λ≤1) is defined to solve the Schrödinger equation with Riesz fractional derivatives of order α. This new method is applied for a particle in a fractional δ-potential well defined by V(x)=?γ δ ^λ (x), where γ>0  and δ ^λ (x) is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if 0<λ<α. The results for λ=1 and α=2 are in exact agreement with those presented in the standard quantum mechanics.     

On the energy transport velocity in a dissipative medium

An exact definition of the group velocity v _g is proposed for a wave process with arbitrary dispersion relation ω = ω′(k) + iω″(k). For the monochromatic approximation, a limit expression v _g (k) is obtained. A condition under which v _g (k) takes the form of the Kuzelev–Rukhadze expression [1] dω′(k)/dk is found. In the general case, it appears that v _g (k) is defined not only by the dispersion relation ω(k), but also by other elements of the initial problem. As applied to the dissipative medium, it is shown that v _g (k) defines the field energy transfer velocity, and this velocity does not exceed thee light speed in vacuum. An expression for the energy transfer velocity is also obtained for the case where the dispersion relation is given in the form k = k′(ω) + ik″(ω) which corresponds to the boundary problem.     

Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 &lt; <Emphasis Type="Italic">α</Emphasis> &lt; 5/2

Fractional difference operators with discrete-Mittag-Leffler kernels of order α > 1 are defined and their corresponding fractional sum operators are confirmed. We prove existence and uniqueness theorems for the discrete fractional initial value problems in the frame of discrete Caputo (ABC) and Riemann (ABR) operators by using Banach contraction theorem. Then, we prove Lyapunov type inequality for a Riemann type fractional difference boundary value problem of order 2 < α < 5∕2 within discrete Mittag-Leffler kernels, where the limiting case α → 2⁺ results in the ordinary difference Lyapunov inequality. Examples are given to clarify the applicability of our results and an application about the discrete fractional Sturm-Liouville eigenvalue problem is analyzed.     

Renormalization group functions of the φ<Superscript>4</Superscript> theory in the strong coupling limit: Analytical results

The previous attempts of reconstructing the Gell-Mann-Low function β(g) of the φ⁴ theory by summing perturbation series give the asymptotic behavior β(g) = β_∞ g in the limit g→∞, where α = 1 for the space dimensions d = 2, 3, 4. It can be hypothesized that the asymptotic behavior is β(g) ～ g for all d values. The consideration of the zero-dimensional case supports this hypothesis and reveals the mechanism of its appearance: it is associated with vanishing of one of the functional integrals. The generalization of the analysis confirms the asymptotic behavior β(g) ～ g in the general d-dimensional case. The asymptotic behaviors of other renormalization group functions are constant. The connection with the zero-charge problem and triviality of the φ⁴ theory is discussed.     

Die Trägervermehrung einer Lawine mit Eigenraumladung

From oscillograms of avalanches of high amplification (ether,p=370 Torr,d=0,3 cm,E/p=77) one can deduce that the number of carriers (n) increases less thane ^αx , ifn overpasses 10⁶. It is the space charge field of the positive ions which reduces the ionisation effect of electrons.     

High-temperature dynamic behavior in bulk liquid water: A molecular dynamics simulation study using the OPC and TIP4P-Ew potentials

Classical molecular dynamics simulations were performed to study the high-temperature (above 300 K) dynamic behavior of bulk water, specifically the behavior of the diffusion coefficient, hydrogen bond, and nearest-neighbor lifetimes. Two water potentials were compared: the recently proposed “globally optimal” point charge (OPC) model and the well-known TIP4P-Ew model. By considering the Arrhenius plots of the computed inverse diffusion coefficient and rotational relaxation constants, a crossover from Vogel–Fulcher–Tammann behavior to a linear trend with increasing temperature was detected at T* ≈ 309 and T* ≈ 285 K for the OPC and TIP4P-Ew models, respectively. Experimentally, the crossover point was previously observed at T* ± 315–5 K. We also verified that for the coefficient of thermal expansion α _P (T, P), the isobaric α _P (T) curves cross at about the same T* as in the experiment. The lifetimes of water hydrogen bonds and of the nearest neighbors were evaluated and were found to cross near T*, where the lifetimes are about 1 ps. For T < T*, hydrogen bonds persist longer than nearest neighbors, suggesting that the hydrogen bonding network dominates the water structure at T < T*, whereas for T > T*, water behaves more like a simple liquid. The fact that T* falls within the biologically relevant temperature range is a strong motivation for further analysis of the phenomenon and its possible consequences for biomolecular systems.     

Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials

We consider a quantum particle interacting with N obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form N²V (Nx) (Gross-Pitaevskii potential). We show convergence of the N dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for N large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.     

On the calculation of the two-point function for the thirring-model by the method of functional integration

By the method of functional integration the two-point functionS _F for the spinor model with the interaction\( - \lambda (\bar \psi \psi )^2 \) is calculated in a two-dimensional space-time. After Fourier-transformationS _F (p) results as a power series with respect to 1/√λ. If we change the order of terms, we get a series in powers of γp. Each coefficient is a series in powers of 1/√λ. The first terms of this series are considered as a good approximation for bigλ. By reasons of convergence of the integrals we must displace the expansion centre of the series in powers ofγ p fromp ²=0 top ²=a ².     

The electrical conductance growth of a metallic granular packing

We report on measurements of the electrical conductivity on a two-dimensional packing of metallic disks when a stable current of ~1 mA flows through the system. At low applied currents, the conductance σ is found to increase by a pattern σ(t) = σ _∞ ? Δσ E _α [ ? (t/τ) ^α ], where E _α denotes the Mittag-Leffler function of order α ∈ (0,1). By changing the inclination angle θ of the granular bed from horizontal, we have studied the impact of the effective gravitational acceleration g _eff = gsinθ on the relaxation features of the conductance σ(t). The characteristic timescale τ is found to grow when effective gravity g _eff decreases. By changing both the distance between the electrodes and the number of grains in the packing, we have shown that the long term resistance decay observed in the experiment is related to local micro-contacts rearrangements at each disk. By focusing on the electro-mechanical processes that allow both creation and breakdown of micro-contacts between two disks, we present an approach to granular conduction based on subordination of stochastic processes. In order to imitate, in a very simplified way, the conduction dynamics of granular material at low currents, we impose that the micro-contacts at the interface switch stochastically between two possible states, “on” and “off”, characterizing the conductivity of the micro-contact. We assume that the time intervals between the consecutive changes of state are governed by a certain waiting-time distribution. It is demonstrated how the microscopic random dynamics regarding the micro-contacts leads to the macroscopic observation of slow conductance growth, described by an exact fractional kinetic equations.     

Temperature Dependence of the Upper Critical Field in Disordered Hubbard Model with Attraction

We study disorder effects upon the temperature behavior of the upper critical magnetic field in an attractive Hubbard model within the generalized DMFT+Σ approach. We consider the wide range of attraction potentials U—from the weak coupling limit, where superconductivity is described by BCS model, up to the strong coupling limit, where superconducting transition is related to Bose–Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures significantly higher than superconducting transition temperature, as well as the wide range of disorder—from weak to strong, when the system is in the vicinity of Anderson transition. The growth of coupling strength leads to the rapid growth of H_c2(T), especially at low temperatures. In BEC limit and in the region of BCS–BEC crossover H_c2(T), dependence becomes practically linear. Disordering also leads to the general growth of H_c2(T). In BCS limit of weak coupling increasing disorder lead both to the growth of the slope of the upper critical field in the vicinity of the transition point and to the increase of H_c2(T) in the low temperature region. In the limit of strong disorder in the vicinity of the Anderson transition localization corrections lead to the additional growth of H_c2(T) at low temperatures, so that the H_c2(T) dependence becomes concave. In BCS–BEC crossover region and in BEC limit disorder only slightly influences the slope of the upper critical field close to T _c . However, in the low temperature region H_c2 (T may significantly grow with disorder in the vicinity of the Anderson transition, where localization corrections notably increase H_c2 (T = 0) also making H_c2(T) dependence concave.     

Komar energy and Smarr formula for noncommutative inspired Schwarzschild black hole

We calculate the Komar energy E for a noncommutative inspired Schwarzschild black hole. A deformation from the conventional identity E = 2ST _H is found in the next to leading order computation in the noncommutative parameter θ (i.e. \({\mathcal{O}(\sqrt{\theta}e^{-M^2/\theta})}\)) which is also consistent with the fact that the area law now breaks down. This deformation yields a nonvanishing Komar energy at the extremal point T _H  = 0 of these black holes. We then work out the Smarr formula, clearly elaborating the differences from the standard result M = 2ST _H , where the mass (M) of the black hole is identified with the asymptotic limit of the Komar energy. Similar conclusions are also shown to hold for a deSitter–Schwarzschild geometry.