Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

We develop a consistent closure procedure for the calculation of the scaling exponents ζ _n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ _n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ _n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.     

Scaling analysis of high temperature superconductors under pressure

The scaling relation of single parameter scaling hypothesis is applied to the study of the scaling behavior of high temperature superconductors under pressure. The data of resistance and specific heat coefficient under various pressures are scaled onto a universal curve according to this scaling relation. The scaling parameters are pressure dependent while temperature independent. It is found that the controlling parameter Bi equals to the relative critical temperature tcP, which indicates that the superconducting energy gap at the zero temperature 2Δs0 is the controlling parameter in this scaling.     

Non-scaling and longitudinal effects in neutrino γ-distributions

Effects of scaling breakdown in deep inelastic νN and _ν^?N y-distributions are assessed for both charged and neutral currents. Asymptotically free gauge models predict dramatic changes from scaling parton models, which invalidate conventional discussions of y-dependence. Non-scaling effects on Weinberg angle determinations are discussed. An additional complication is the presence of a longitudinal structure function, absent from the simple parton model.     

Scaling properties of target excitation cross sections in collisions between atoms and multiply charged ions

The scaling properties of one-electron target excitation cross sections in collisions between atoms and multiply charged ions are investigated. For the example of bare-ion (Z)-hydrogen collisions with various charge numbers Z it is shown that, in many-state calculations for Z ≥ 2 and energies E/Z ≈ 15–100 keV/amu, calculated 2p excitation cross sections lie approximately on a universal curve. This curve deviates from the one which has been derived earlier by Janev and Presnyakov for all Z on the basis of a simplified three-state model, and in fact results from a consistent three-state treatment show no scaling. The scaling properties of the excitation cross sections are easily understood in a purely classical model of distant collisions.     

Electron-positron scaling in bound state models

We investigate scaling assuming a generalized vector meson dominance picture. The vector mesons are described as relativistic quark-antiquark bound states by a Bethe-Salpeter equation which yields the mass spectrum and the coupling to e⁺e^? pairs. We discuss the spin structure and find that scaling can occur only for a γ_μ type amplitude. We solve the BS equation using a generalized WKB approximation and find scaling, independent of the detailed shape of the interaction. This means that scaling in e⁺e^? annihilation does not select a particular “confinement potential”. The scaling constant depends on the current renormalization constant and on the details of the relativistic spin structure.     

Advection of passive magnetic field by the Gaussian velocity field with finite correlations in time and spatial parity violation

Using the field theoretic renormalization group technique the model of passively advected weak magnetic field by an incompressible isotropic helical turbulent flow is investigated up to the second order of the perturbation theory (two-loop approximation) in the framework of an extended Kazantsev-Kraichnan model of kinematic magnetohydrodynamics. Statistical fluctuations of the velocity field are taken in the form of a Gaussian distribution with zero mean and defined noise with finite correlations in time. The two-loop analysis of all possible scaling regimes is done and the influence of helicity on the stability of scaling regimes is discussed and shown in the plane of exponents ? ? η, where ? characterizes the energy spectrum of the velocity field in the inertial range E ∞ k ^{1 ? 2ε}, and η is related to the correlation time at the wave number k which is scaled as k ^{?2 + η}. It is shown that in non-helical case the scaling regimes of the present vector model are completely identical and have also the same properties as those obtained in the corresponding model of passively advected scalar field. Besides, it is also shown that when the turbulent environment under consideration is helical then the properties of the scaling regimes in models of passively advected scalar and vector (magnetic) fields are essentially different. The results demonstrate the importance of the presence of a symmetry breaking in a given turbulent environment for investigation of the influence of an internal tensor structure of the advected field on the inertial range scaling properties of the model under consideration and will be used in the analysis of the influence of helicity on the anomalous scaling of correlation functions of passively advected magnetic field.     

Scaling distribution of large transverse momenta

We compute the structure function of large k_T scaling law in the framework of a multiperipheral or parton model. This function depends on two scaling variables. We show that recent NAL data are in perfect agreement with the same 1/k_T⁸ law observed at ISR. The observed apparent change of the scaling power is faked by the neglect of the dependence on one of the scaling variables.     

Finite size scaling for O(N) φ4-theory at the upper critical dimension

A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N) φ⁴-theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ⁴₄-theories for all N.     

KNO scaling in terms of partial coherence and frequencies separation

A KNO scaling function is presented which takes into account besides the partial coherence also the difference of frequencies characterizing coherent and chaotic field components. The scaling function has its maximum at the values of the KNO scaling variablez=n/<n> which is not higher than unity; this property is revealed at high energies by presently available charged hadronic multiplicity data obtained froml-l, l-h as well ash-h collisions. A practical procedure for application of that scaling function is suggested. In the present contribution it is successfully applied to the KNO scaling data obtained from thee ⁺e^? annihilations (Tasso Collab.) in the 14–34 GeV cms energy range as well as from thepp collisions at the ISR cms energies 53 and 63 GeV completed by \(\bar pp\) data at 540 GeV.     

Electric multipole modes in the fluid-dynamical approximation

Fluid-dynamical equations obtained from a generalized scaling approximation are solved for (T = 0) oscillations with multipolarities q = 0, 2, 3 for Fermi systems with particle numbers A = 90, 208, 1000. Results are compared to the standard phenomenological scaling model.     

On the velocity and dissipation signature of vortex tubes in isotropic turbulence

The properties of vortex tubes are extracted and analyzed from a DNS database at various Re_λ, with the objective to characterize the associated distributions of induced velocity and kinetic energy dissipation. The induced velocity exhibits an inverse power-law scaling in the far field, different from Burgers’ r⁻¹ scaling, supporting the interpretation that tubes are the remnants of vortex sheets after roll-up, and suggesting a possible link with the Kolmogorov k^−5/3 spectral scaling. The energy dissipation signature is characterized by a local maximum near the edge of the vortex core, and an absolute peak at its center, which can be tentatively explained appealing to the occurrence of a bi-axial configuration of the strain-rate tensor.     

Efficiency of heteronuclear broadband decoupling and homonuclear J cross polarization analyzed on two time scales

Decoupling sequences can be evaluated, as shown in Waugh's theory, with a J scaling factor on a long time scale. The efficiency of low-power decoupling, however, must be determined by cycling sidebands, as well as by the J scaling factor, when the sampling is not synchronized with the decoupling cycle. We introduced, therefore, another scaling factor which characterizes the decoupling on a short time scale. It is also shown that these scaling factors are useful for evaluating the efficiency of homonuclear J cross polarization. We clarified criteria of the factors suitable for various J coupling constants and chemical-shift ranges. Typical decoupling sequences were analyzed using the two scaling factors.     

Long-range correlations in two-dimensional spatio-temporal seismic fluctuations

We analysed the scaling behaviour of the two-dimensional (2-D) sequence (Δs, Δt) of the 1981–1998 southern California seismicity, where Δs is the distance between two consecutive earthquakes (jump) and Δt is their interevent interval. The 2-D seismic spatio-temporal fluctuations were investigated by means of the detrended fluctuation analysis (DFA), well-known methodology used to detect scaling behaviour in observational time series possibly affected by nonstationarities. The estimated scaling exponents α_DFA, larger than 0.5, indicate the presence of persistent long-range correlations in the 2-D sequence analysed. The variation of the scaling exponent with the increase of threshold magnitude shows a two-fold behaviour: in the range between 1.5 (the completeness magnitude of the catalog) and 3.0, the scaling exponent is quite constant and denoting a flicker-noise dynamics; while for magnitudes larger than 3.0 it decreases with the increase of magnitude, indicating a tendency toward a 2-D space–time Poissonian process for large events.     

On the origin of approximate KNO scaling ine + e − annihilation

KNO scaling of the multiplicity distribution in hadronic final states was originally derived as a consequence of Feynman scaling. We show that in iterative models of hadron production in jets, incorporating Feynman scaling, KNO scaling obtains only in the limit when the width of the multiplicity distribution tends to zero. Within the context of the models currently employed to describee ⁺ e ^? annihilation into hadrons, the apparent KNO scaling observed is an accidental consequence of effects which violate Feynman scaling.     

Scaling and memory in the return intervals of realized volatility

We perform return interval analysis of 1-min realized volatility defined by the sum of absolute high-frequency intraday returns for the Shanghai Stock Exchange Composite Index (SSEC) and 22 constituent stocks of SSEC. The scaling behavior and memory effect of the return intervals between successive realized volatilities above a certain threshold q are carefully investigated. In comparison with the volatility defined by the closest tick prices to the minute marks, the return interval distribution for the realized volatility shows a better scaling behavior since 20 stocks (out of 22 stocks) and the SSEC pass the Kolmogorov-Smirnov (KS) test and exhibit scaling behaviors, among which the scaling function for 8 stocks could be approximated well by a stretched exponential distribution revealed by the KS goodness-of-fit test under the significance level of 5%. The improved scaling behavior is further confirmed by the relation between the fitted exponent γ and the threshold q. In addition, the similarity of the return interval distributions for different stocks is also observed for the realized volatility. The investigation of the conditional probability distribution and the detrended fluctuation analysis (DFA) show that both short-term and long-term memory exists in the return intervals of realized volatility.     

Critical behavior studies in ferromagnetic (Nd, K)-Mn-O compounds

The critical scaling behavior of K-doped Nd-Mn-O based double-exchange ferromagnetic compounds was studied by measuring isothermal magnetization of Nd_0.84K_0.16MnO₃ and Nd_0.77K_0.23MnO₃ samples. The critical exponents β, γ and δ corresponding to the spontaneous magnetization, initial susceptibility and isothermal magnetization, respectively, were determined by analyzing the magnetization data in terms of the modified Arrott plot method. The critical exponent values of both samples are found to be comparable to values predicted by a mean field model. The role of ferromagnetic clusters on the scaling behavior is discussed. The critical exponent values are found to be consistent with the Widom scaling relation and the universal scaling hypothesis.     

Reducibility and thermal scaling in nuclear multifragmentation

Recent studies have revealed the existence of a number of reducibility and thermal scaling properties in nuclear multifragmentation. The probability of emitting n-fragments is found to be reducible to the probability of emitting a single fragment through the binomial expression. The resulting one fragment probability shows thermal scaling by producing linear Arrhenius plots. Similarly, the charge distributions associated with n-fragment emission are reducible to the one-fragment charge distribution. Thermal scaling is also observed. The reducibility equation contains a constant whose value, zero or positive, can be related to a univariant (two phase) or bivariant (one-phase) regime. The light fragment particle-particle angular correlations also show reducibility to the single-particle angular distributions as well as thermal scaling. A mass scaling associated with the angular correlations suggests emission from several small sources (A ≈ 20). The limits of applicability of scaling and reducibility are discussed as well as their implications for the mechanism of multifragmentation.     

Diffusion in a random medium: A renormalization group approach

We investigate through a continuous random diffusion equation the long-distance properties of the general non-symmetric hopping model. The lower and upper critical dimensionalities are d = 1 and d = 2 respectively. A renormalization group analysis shows that the velocity and the diffusion constant obey scaling laws with non-classical exponents, which are computed to first order in ε = 2 ? d. Similar scaling laws, based on heuristic arguments, are conjectured for the AC conductivity.     

Electroproduction and annihilation scaling laws in a dual model

Scaling laws for large virtual photon mass (q²) in electroproduction and annihilation are studied in the framework of a simple planar dual model. We find, as has recently been conjectured, that the scaling behaviour depends on the number of space-time dimensions spanned by large momenta. In particular, for a certain range of parameters in the model, we find that the annihilation cross section is dominated by the one-dimensional configuration and increases with q² relative to its canonical behaviour while the electroproduction total cross section is dominated by the two-dimensional configuration and has the canonical Bjorken scaling behavior. In general the scaling laws and therefore the structure of events in the model are distinctively different from the conventional parton model. The problem of consistency of planar dual tree diagrams with unitarity sum rules is discussed.