Multifractal Power Law Distributions: Negative and Critical Dimensions and Other “Anomalies,” Explained by a Simple Example

Divergence of high moments and dimension of the carrier is the subtitle of Mandelbrot's 1974 seed paper on random multifractals. The key words divergence and dimension met very different fates. Dimension expanded into a multifractal formalism based on an exponent and a function f(). An excellent exposition in Halsey et al. 1986 helped this formalism flourish. But it does not allow divergent high moments and the related inequalities f()<0 and <0. As a result, those possibilities did not flourish. Now their time has come for diverse reasons. The broad 1974 definitions of and f allow <0 and f()<0, but the original presentation demanded to be both developed and simplified. This paper shows that both multifractal anomalies occur in a very simple example, which has been crafted for this purpose. This example predicts the power law distribution. It generalizes and f() beyond their usual roles of being a Hölder exponent and a Hausdorff dimension. The effect is to allow either f or both f and to be negative, and the apparent anomalies are made into sources of new important information. In addition, this paper substantially clarifies the subtle way in which randomness manifests itself in multifractals.     

Über dieKα-Satelliten von Fluor in Lithiumfluorid

Using a focussing crystal spectrometer with photoelectric registration theK-emission spectrum of fluorine in solid lithium fluoride (=18.3 Å) was studied. If the spectrum is excited by Co or CuL-radiation or bremsstrahlung of a W-target the satellitesK ₃ andK ₄ appear with considerable intensity, whereas they appear only faintly ifL-radiation of Fe is used for excitation. In agreement with these observations the calculated threshold energies of the satellites are near the photon energy of FeL _{1, 2}. The measured energies of the components agree with values obtained from the energies of the singly and doubly ionized F^– ion as calculated by means of optical data. The results show that the satellitesK ₃ andK ₄ in the spectrum of fluorine in solid LiF correspond to transitions in the doubly ionized F^–-ion.     

Multifractality and scaling in disordered mesoscopic systems

We suggest a new method for investigating scaling properties of mesoscopic observables and their distributions in disordered systems showing metal-insulator transition. In such systems quantum interference effects lead to multifractal structure of eigenstates on scales much smaller than the correlation length of the transition which can be described by a set of exponents, thef() spectrum. The analysis off() spectra can be extended to any scaling variable. Multifractality is an indication for broad distributions of these variables. If the transition is governed by one correlation length only then thef() spectra of normalized scaling variables must be universal. The critical exponentv of the correlation length is determined by the value (0) wheref() takes its maximum and the scaling exponent of normalizationxv ^–1=(0)+x. As an illustrative example we calculate numerically thef() spectra of eigenstates in the critical regime of 2d disordered electron systems in high magnetic fields. We find similarf() spectra indicating universal log-normal distributions of scaling variables.Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-Jülich     

Some results for the exponential interaction in two or more dimensions

We show that for the regularized exponential interaction :e : ind space-time dimensions the Schwinger functions converge to the Schwinger functions for the free field ifd>2 for all or ifd=2 for all such that ||>₀.Partially sponsored by the I.H.E.S. through the Stiftung Volkswagenwerk     

Low-temperature and long-time anomalies of a damped quantum particle

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI() at low frequencies, where > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on . For > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for < 1, an initially localized state remains localized for all times. For 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for 2 and at zero temperature for 1 2.     

Solution of coupled differential equations of soliton theory

We present the complete set of solutions of the coupled differential equations of the form ()²=(), ² =(). Equations of this form appear in several physical situations.     

Finite-size effects in a field-theoretic model with long-range exchange interaction

We present a systematic approach to the calculation of finite-size (FS) effects for anO(n) field-theoretic model with both short-range (SR) and long-range (LR) exchange interactions. The LR exchange interaction decays at large distances as 1/r ^d+2–2,0⁺,0⁺. Renormalization group calculations ind=d _u – are performed for a system with a fully finite (block) geometry under periodic boundary conditions. We calculate the FS shift of the critical temperature and the FS renormalized coupling constant of the model to one-loop order. The universal scaling variable is obtained and the FS scaling hypothesis is verified.     

Superconformal Symmetry and HyperKähler Manifolds with Torsion

The geometry arising from Michelson & Strominger's study of =4B supersymmetric quantum mechanics with superconformal D(2, 1; )-symmetry is a hyperKähler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters are related via changes in potentials for the HKT target spaces. For 0, –1, we describe how each such HKT manifold M ^4m is derived from a space N ^4m–4 which is quaternionic Kähler with torsion and carries an Abelian instanton.     

Periodic Motion of Atoms Near a Charged Wire

We study the classical motion of an atom in the vicinity of an infinite straight wire which carries an oscillating uniform charge. This system has been proposed as a mechanism for trapping cold neutral atoms. The parameters of the problem are the magnitude Q and frequency of oscillation of the charge, the mass M and polarizability of the atom, and the angular momentum L of the atom about the wire. For 0 and 2MQ ² greater than, but close to, L ², we prove that the atom's radial motion is periodic (with period 2/), and that the atom moves in a helical path around the wire. For 2MQ ² L ² we prove that the atom must either collide with the wire or else escape to infinity in the radial direction.     

Lattice vibrations in ordered and disordered thiourea

The critical dynamics of the Syozi model for dilute ferromagnetism is considered by the use of master equations. The dynamics is soluble as it is assumed that the time scale of motion on the sublattice on which the impurities move is so much faster than on the other sublattice that fast relaxing variables may be adiabatically eliminated, leaving a new soluble master equation. It is found that the linear and non-linear relaxation of magnetization exponents ^(l) and ^(nl) increase on dilution to ^(l)/(1–) and ^(nl)/(1–) respectively ( is the specific heat exponent for the pure system, which itself changes on dilution to –/(1–)). Thus if the exponents for the pure system obey the scaling law of Rácz and Fisher ^(nl)= ^(l)– ( is the magnetization exponent which changes on dilution to /(1–)) then so do the exponents for the diluted system. Similarly the exponent for spin diffusion changes on dilution to /(1–).     

Relativistic Hypernuclei: Old Problems and New Prospects

The first experiments with relativistic hypernuclei (Bowen; Khorozov and Lukstins) were done many years ago. They demonstrated not only that such extremely difficult experiments are manageable but also their great advantage, a possibility of observing and studying independently the production and weak decay of hypernuclei: the points of the production and decay of relativistic hypernuclei are separated by tens of centimeters instead of some microns in classical experiments. At the same time these first experiments revealed a huge problem with selection of the proper trigger.Recently we proposed to explore a unique feature of the ⁹Be nucleus: after removing a neutron from its ground state several groups of alpha-particles appear from different excited states of a residual nucleus ⁸Be. Detection of the correlated pair of -particles produced in a vacuum volume at a distance of some 40 cm from target is an unambiguous signal of nonmesonic decay of hypernucleus ¹⁰ Be(++n+n) or ¹⁰ Be(++n+p). In this particular case it is possible to take exclusive decay rates (on different excited states of ⁸Be^*). This open a way for a phenomenological analysis of matrix elements of the four-baryon weak interaction.The experiment is approved for new accelerator Nuclotron at JINR, Dubna.     

Correlations between clusterization and electrical properties within fluorite-type anions excess solid solutions: Setting of a model

We proposed a model to correlate, in a continuous manner, the composition dependence of electrical properties and the progressive extension of clusterization when the substitution rate increases in a fluoride anion excess CaF₂-type solid solution of M_1–xM _x ²⁺ F_2+x(=1,2,3). A new classification of clusters is given based on the presence or absence of coexistence between two types of interstitial fluoride ions.The second part of the paper is devoted to the representation of the sum of interstitial fluoride ionsn _{F int} and the sum of vacancies in normal sitesn according to the general equationy=(mx ³+qx)/(x ²+q). This model allows us to correlate the structural and electrical properties of a large number of solid solutions with fluorite-type structure.     

A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions

We rigorously derive a linear kinetic equation of Fokker–Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obstacle of the Lorentz gas generates a potential V( ), where V is a smooth radially symmetric function with compact support, and >0. The density of obstacles diverges as ^– , where >0. We prove that when 0< <1/8 and =2+1, the probability density of a test particle converges as 0 to a solution of our kinetic equation.     

Relativistic alpha-particles from inelastic collisions of nuclei of24Mg in photoemulsion at 4.5 GeV/(c· nucleon)

The transverse momentum spectra and correlation characteristics of secondary -particles (fragments of the²⁴Mg nuclei) are studied. An excessive number of high-momentum -particles are found, which cannot be explained by the statistical model. It is shown that the azimuth correlations of -particles can be due to a large transverse momentum of the fragmenting residual magnesium nucleus.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–23, May, 1991.     

On the cauchy problem for the discrete Boltzmann equation with initial values inL 1 + (ℝ)

We prove a global existence theorem for a discrete velocity model of the Boltzmann equation when the initial values _i (x) have finite entropy and, for some constant>0, (1+|x|) _i (x)L ₁ ⁺ ().     

Scalar particle with polarizability in a uniform external magnetic field

The model of a scalar structured particle is considered, which possesses polarizability in an external electromagnetic field. The expression for the 4-dimensional current density is found. The exact solution of the equations describing a scalar particle with polarizability in a uniform external magnetic field is obtained. Up to the terms of order O(H²), the energy spectrum can be formally obtained by the substitution of the particle mass in the expression for a pointlike scalar particle: mm–H²/2, where is the magnetic polarizability of the particle. It is shown that the rms radius of a trajectory can be obtained by the substitution of the charge in the well-known formula for a structureless scalar particle: ee(1{-H²/m)^1/2 (where is the electric polarizability).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 91–94, January, 1991.I thank A. I. L'vov for discussions.     

First passage time densities for random walk spans

A general expression is derived for the Laplace transform of the probability density of the first passage time for the span of a symmetric continuous-time random walk to reach levelS. We show that when the mean time between steps is finite, the mean first passage time toS is proportional toS ₂. When the pausing time density is asymptotic to a stable density we show that the first passage density is also asymptotically stable. Finally when the jump distribution of the random walk has the asymptotic formp(j)A/|j| ₊₁, 0 < < 2 it is shown that the mean first passage time toS goes likeS .     

Microstructure of fiber-like SiC/Si3N4 composite particulate

The microstructure of fiber-like SiC/Si₃N₄ composite particulate was investigated using high-resolution transmission electron microscopy techniques. The SiC/Si₃N₄ composite particulate consisted of a-SiC core and a -Si₃N₄ outer shell. Two kinds of composite particulate were distinguished when the observed orientation of the SiC core was <110>. In one type of the SiC/Si₃N₄ composite particulate, a crystal relationship of (111)-SiC | (102) -Si₃N₄ and (111)-SiC (114) -Si₃N₄ was identified; in the other type of the SiC/Si₃N₄ composite particulate, a crystal relationship of (111)-SiC (001) -Si₃N₄, and (111)-SiC (101) -Si₃N₄ was observed.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

Zero value of the Schwarzschild mass for an asymptotically Euclidian system of gravitational waves

A class of the asymptotically Euclidian space-times is shown to exist for which the Schwarzschild mass is equal to zero. The coordinate atlases of these space-times satisfy two additional conditions: _k (-gg ^0k )=0 and _ik ⁰ _0g ^ik - _ik ^k _0g ⁰ⁱ =0. In aT-orthogonal metricgs ²=g ₀₀ dt ² -g dx dx these conditions take a simple form: ₀(detg )=0 and (₀ g )(₀ g )=0.