General Scaling Relations Research Articles

Kinetic roughening on rough substrates

We study the kinetic roughening described by a linear diffusion equation on an initially rough substrate. We show analytically that it is not possible to write a general scaling relation valid for rough substrates. However, for a particular substrate generated by the same linear growth process, we can write a scaling relation similar to the relation valid for flat substrates, with the same growth and roughness exponents. Numerical simulations in a solid-on-solid model with surface relaxation, for $d=1,$ supports the scaling relation and the exponents.

Exact results for spatiotemporal correlations in a self-organized critical model of punctuated equilibrium.

We introduce a self-organized critical model of punctuated equilibrium with many internal degrees of freedom ({ital M}) per site. We find exact solutions for {ital M}{r_arrow}{infinity} of cascade equations describing avalanche dynamics in the steady state. This proves the existence of simple power laws with critical exponents that verify general scaling relations for nonequilibrium phenomena. Punctuated equilibrium is described by a devil{close_quote}s staircase with a characteristic exponent {tau}{sub first}=2{minus}{ital d}/4 where {ital d} is the spatial dimension. {copyright} {ital 1996 The American Physical Society.}

Landau theory of quantum spin glasses of rotors and Ising spins.

We consider quantum rotors or Ising spins in a transverse field on a $d$-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin glass and quantum paramagnet phases and a zero temperature quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions can be reproduced by minimization of a Landau effective-action functional for the model in finite $d$ with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the zero temperature critical point is mapped out as a function of temperature, strength of the quantum coupling, and applied fields. The spin glass ground state is found to be replica symmetric, with replica symmetry breaking appearing only at finite temperatures. Next we examine the consequences of fluctuations about mean-field for the critical properties. Above $d=8$, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above $d=8$, and for all physical couplings below $d=8$, we find runaway renormalization group flows to strong coupling. General scaling relations that should be valid even at the strong coupling fixed point are proposed and compared with Monte Carlo simulations.

Conductivity and dielectric constant in a wedge-shaped Pt-film percolation system

A wedge-shaped Pt-film percolation system was prepared by the magnetron sputtering method. A new nonlinear I- Vbehavior was found and can be well explained by the location-dependent hopping effect. Power-law behavior, σ( ω) α ω x and ϵ( ω) α ω − y , is observed near the percolation threshold. The exponents x and y are found to be 0.87±0.04 and 0.08±0.03, respectively, in agreement with the general scaling relation x + y = 1.

Magnetic anisotropy in single-crystal rare-earth-doped RBa2Cu4O8 (R=Y, Dy, Gd, Ho, Er, Tm, Yb).

We measured magnetic susceptibility of single-crystal rare-earth-doped R${\mathrm{Ba}}_{2}$${\mathrm{Cu}}_{4}$${\mathrm{O}}_{8}$ (where R=Y, Dy, Gd, Ho, Er, Tm, Yb) in the normal state along the three principal axes. The anisotropy measured fits well to calculated values obtained from a crystal electric field (CEF) analysis. From our results we developed a general scaling relation for the CEF parameters across the rare-earth series.

Critical behaviors in a Pt-film percolation system deposited on fracture surfaces of alpha -Al2O3 ceramics.

The dc resistance R, critical current ${\mathit{I}}_{\mathit{c}}$, ac dielectric constant \ensuremath{\epsilon}(\ensuremath{\omega}), and ac conductivity \ensuremath{\sigma}(\ensuremath{\omega}) have been studied in a Pt-film percolation system deposited on \ensuremath{\alpha}-${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$ fracture surfaces by dc- and rf-magnetron sputtering methods. The fractal dimension of the surfaces is found to be 2.20\ifmmode\pm\else\textpm\fi{}0.06. Although the percolation system is entirely different from both the lattice-percolation system and the continuum-percolation system, the power-law behaviors, ${\mathit{I}}_{\mathit{c}}$\ensuremath{\propto}${\mathit{R}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, \ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathit{x}}$, and \ensuremath{\epsilon}(\ensuremath{\omega})\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}\mathit{y}}$, are observed near the percolation threshold. The exponent \ensuremath{\alpha} is determined to be 1.12\ifmmode\pm\else\textpm\fi{}0.05, which is significantly different from both predictions of two-dimensional (2D) percolation theories and the recent experimental results on Au-film and Ag-film continuum-percolation systems. The exponents x and y are found to be 0.89\ifmmode\pm\else\textpm\fi{}0.06 and 0.05\ifmmode\pm\else\textpm\fi{}0.02, respectively, in agreement with the general scaling relation x+y=1. These results suggest that the power-law behaviors as well as the general scaling relation in percolation systems could not be affected by the statistical fractal structure of the substrates. The experimental results also support the prediction that the exponents x and y are universal.

AC CHARACTERISTICS OF A THIN Ag FILM PERCOLATION SYSTEM DEPOSITED ON A STATISTICAL FRACTAL SUBSTRATE

Ac characteristics of a thin Ag film percolation system deposited on a broken surface of α-Al2O3 ceramics are measured. We prove that the ac conductivity and ac dielectric constant obey the power law and the general scaling relation near the percolation threshold. The ac capacitance data as a function of the dc resistivity are obtained and its role is discussed.

Percolative, self-affine, and faceted domain growth in random three-dimensional magnets.

We study the advance of an interface separating two magnetic domains in a three-dimensional random-field Ising model. An external magnetic field causes one domain to grow. Three types of growth are found. When disorder in the medium is large, the interface forms a self-similar pattern with large scale structure characteristic of percolation. As the disorder decreases, there is a critical transition to compact growth with a self-affine interface. Finally, for sufficiently weak disorder, simple faceted growth occurs. The transitions between these growth morphologies are related to results for bootstrap percolation. In the self-similar and self-affine growth regimes there are also critical transitions at the onset of steady-state motion. These are studied numerically, and found to lie in different universality classes. The critical exponents obtained from our simulations obey general scaling relations derived in the context of fluid invasion.

General theory of the metal-insulator transition.

We present a general theory of the metal-insulator transition based on the formal introduction of an auxiliary irrotational gauge field and the subsequent behavior of the system in response to it. Correspondingly general scaling relations are derived which determine the point of transition at a critical value of the controlling density in terms of a series expansion with respect to a coupling constant. Functional integral techniques are also used to show the relation of this transition to a density-fluctuation instability. Finally, an interpretation is given of the transition in terms either of condensation of gauge bosons or, alternatively, in terms of the appearance of an additional geometric phase, an interpretation that is valid for quite general condensed-matter systems.

Strike‐slip earthquakes on quasi‐vertical transcurrent faults: Inferences for general scaling relations

The recent occurrence of several large strike‐slip earthquakes provides the opportunity to review and complement available data on the scaling of seismic moment (MO) with length of rupture (L) for large earthquakes, depending on their tectonic setting and mechanism. For strike‐slip earthquakes on quasi‐vertical transcurrent faults, the MO versus L relation has a significant change of slope around MO ∼(0.6–0.8)*1020 N‐m, and for larger earthquakes, MO scales linearly with L. This is compatible with models where slip is controlled by the width of the fault. Also, it appears to be easier to categorize large earthquakes by their mechanism (strike‐slip on vertical transcurrent fault, versus pure thrust/normal) than their tectonic setting (interplate/intraplate).

Scaling relations for interface motion through disordered media: Application to two-dimensional fluid invasion.

We consider the critical transitions that occur as the force driving an interface through a random medium is increased. The total displacement of the interface, and the incremental advance after a small increase in force, diverge as the force approaches a critical depinning threshold. At the critical force there is a power-law distribution of growth sizes. General scaling relations are derived between the critical exponents associated with such transitions. These scaling relations are tested on a model system---fluid invasion of a two-dimensional porous medium. Critical exponents are determined from simulations using finite-size-scaling techniques. Two universality classes are identified: percolation and depinning. In both cases the calculated exponents obey the scaling relations.

Dual phase continuity in polymer blends

AbstractThe literature on formation of immiscible polymer blends with dual phase continuity is shortly reviewed. When the volume fraction of one phase is low, nearly spherical discrete domains may be formed in flow fields. The critical volume fraction for formation of infinite structures: φcr = 0.156, predicted by percolation theory for monodisperse spherical domains, is in reasonable accordance with experimental data for samples with spherical domains. In simple shear flow and low volume fraction the coalescence and break up processes may lead to the coexistence of long nearly cylindrical domains (large) oriented in the flow direction and (small) spherical domains. A hypothesis predicting a decrease of the percolation threshold value when φL* · φS* · P · 4 is large, is shown to be in accordance with experimental observations. φL* and φS* are the relative volume fraction of large and small domains, respectively, and P is the aspect ratio of the elongated (cylindrical) drops. The applicability of general (scaling) relations based on percolation theory are discussed.

Radial pulsations of strange stars and the internal composition of pulsars

We present calculations of radial oscillations of homogeneous strange stars, showing that the particular form of the equation of state allows some simple and general scaling relations which may prove to be very useful for the search of these objects.

Superdiffusive transport due to random velocity fields

The motion of a random walk in a medium containing random, but spatially correlated velocity fields is discussed. This type of disorder generally leads to superdiffusive behavior in which the mean-square displacement, 〈x2(t)〉, grows faster than linearly with time. For a two-dimensional medium with layers of random velocities in the x-direction, 〈x2(t)〉 is found to increase as t2v with 2v=32. The probability distribution of displacements appears to fit the form P(x,t)∼t−34exp[−(x/t34)δ], with δ≲1.7. However,a Lifshitz argument for the tail of the distribution suggests that δ≤43. This discrepancy is yet to be fully resolved. We also discuss some intriguing properties in a model with isotropic velocity fields. In two dimensions, we find v=43, while δ=3. These values obey the general scaling relation δ=(1−v)−1.

THEORY OF BEAM REFLECTION, TRANSMISSION, TRAPPING AND BREAKUP AT NONLINEAR OPTICAL INTERFACES

We show with an extensive numerical study, that the global reflection and transmission properties of a finite width optical self-focussed channel incident at an oblique angle to a nonlinear dielectric interface, can be categorized into three distinct regimes of behavior as the incidence angle is varied through the angle for total internal reflection. The largest regime in parameter space is the nonlinear one, where a channel either undergoes total internal reflection or transmission, in marked contrast to the well known linear Snell's law behavior. The beam asymptotics in this latter region are quantitatively explained by a recent equivalent particle theory. 1 INTRODUCTION In this work we study the behavior of propagating self-focussed channels incident at at oblique angle to a nonlinear dielectric interface [I]. General scaling relations allow our analysis to be applied in widely different physical contexts. Our main results are: (1) the reflection and transmission properties of the incident channel can be divided into three distinct categories as a function of scaled incident beam power ( v j o l a ) and angle (vela) [ see below for definitions]. For fixed incident power, low amplitude, wide beams show the well-known linear behavior of partial reflection and transmission in agreement with Snell's laws and the Fresnel relations [2]. In other words, a high frequency spatial interference pattern appears on the envelope of the incident beam as it interacts with the interface and the individual reflected and transmitted components spread due to diffraction as they separate and propagate away from the interface. Higher amplitude, narrower beams show qualitatively the same behavior except that the partially reflected and transmitted components are spatial solitons (self-focussed channels) appropriate to the medium in which they are localized. By far the most extensive region in parameter space is the fully nonlinear regime where the incident channel is sufficiently narrow that it either reflects or transmits as one tunes through the critical angle. This regime will be discussed at the end of the paper. (2) The important physical parameter which defines the dividing boundary between the intermediate and fully nonlinear behavior is the ratio of linear to nonlinear refractive index mismatches, divided by the incident power S (defined below). An analytic expression for one of the boundaries separating the intermediate from the nonlinear region in ( ~ j o l a , vela) parameter space is given in terms of S whose value at the critical angle S,,i, is shown to be close to 4 for all cases studied. (3) The boundary separating the fully nonlinear from the intermediate regime in ( v j 0 / a , v o / a ) parameter space is symmetric and wedge shaped about the critical angle for a uniform nonlinearity ( a = ao/al = 1.0). As the nonlinear mismatch increases ( a 0; (ni, a;, i = 0 , l are the linear refractive indices and nonlinear coefficients for the left and right media respectively). We assume that no > nl and a,, < al. Let us now rescale this (permanent address : Physics Department. Heriot-Watt University. Riccarton. GB-Edinburgh EH14 4AS. Scotland, Great-Britain Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988275 C2-320 JOURNAL DE PHYSIQUE equation in order to display the above parameter dependence explicitly. Consider the following set of transformations,

Observation of a correlation length finite-size effect in Rayleigh scattering from thin critical fluid films.

Rayleigh linewidth data gathered at small scattering angles on critical-mixture films of thickness 2L are found to collapse about a single universal curve, ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{*}}$(K${\ensuremath{\xi}}^{\mathrm{eff}}$), where ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{*}}$(x) is the 3D reduced Rayleigh linewidth function, and ${\ensuremath{\xi}}^{\mathrm{eff}}$ is an effective correlation length obeying a general finite-size scaling relation, ${\ensuremath{\xi}}^{\mathrm{eff}/\mathrm{L}=\mathrm{F}(\ensuremath{\xi}/\mathrm{L})}$. The latter reveals that ${\ensuremath{\xi}}^{\mathrm{eff}}$ grows with \ensuremath{\xi} until \ensuremath{\xi}\ensuremath{\approxeq}L, when its growth terminates, implying the existence of surface layers of thickness \ensuremath{\xi} formed by preferential wall-fluid interactions.

Strain scaling law for flux pinning in practical superconductors. Part 1: Basic relationship and application to Nb 3Sn conductors

Critical current and flux pinning densities have been determined for a series of Nb 3Sn, V 3Ga, Nb 3Ge, and NbTi conductors as a function of uniaxial tensile strain in magnetic fields ranging from 4 to 19 T. An empirical relationship has been found at 4.2 K that describes these data over the entire range of field under both compressive and tensile strain. The pinning force F has been found to obey a scaling law of the form F = [B c2∗(ϵ)] nf(b) , where B c2 ∗ is the strain-dependent upper-critical field determined from high-field critical-current measurements and f(b) is a function only of the reduced magnetic field b  B/B c2∗ . The detailed shape of f(b) depends on the super-conducting material and reaction conditions, but n was found to be nearly constant for a given type of superconductor. For Nb 3Sn conductors n = 1 ± 0.3, for multifilamentary V 3 Ga n≅1.3, for CVD-Nb 3Ge tape n≅1.6, and for multifilamentary NbTi n≅3.3. The importance of this relationship is that, for these conductors at least, it is possible to measure F at one strain and then immediately be able to predict F (and thus the critical current) at other strain levels simply by scaling the results by [B c2∗(ϵ)] n. Part I of this paper presents the basic uniaxial-strain scaling relationship and focuses on its application to Nb 3Sn conductors. The strain scaling law with n = 1 ± 0.3 was found to hold for all Nb-Sn based conductors examined thus far, including commercial-multifilamentary conductors, extremely fine-filament composites, partially-reacted specimens, ‘insitu’ conductors, and Nb-Hf/Cu-Sn-Ga conductors. The detailed dependence of B c2∗ on strain was-found to be nearly universal for highly-reacted commercial Nb 3Sn specimens, greatly simplifying the application of the scaling law to this group of practical superconductors. These results are discussed within the context of flux pinning models and a general scaling relation is proposed which unifies the usual temperature-scaling relation with this strain-scaling relation.

Bond percolation problem in a semi-infinite medium. Landau-Ginzburg theory

The bond percolation problem in a lattice bound by a plane surface is defined by associating different occupation probabilities ${P}_{B}$, ${P}_{\ensuremath{\parallel}}$, or ${P}_{\ensuremath{\perp}}$ to bonds in the bulk, on the surface, or linking the bulk to the surface, respectively. The coordinate $z$ indicates the distance of parallel planes to the surface. The $z$-dependent percolation probability $\ensuremath{\phi}(z)$ is defined as the probability that a site on the plane $z$ belongs to an infinite cluster and I derive a rigorous expression for $\ensuremath{\phi}(z)$ by introducing $z$-dependent external fields ${h}_{z}$ in the $q$ states Potts Hamiltonian in the limit $q=1$. Gaussian integration techniques are used to derive a Landau-Ginzburg free-energy functional for arbitrary $q$. The resulting differential equations for the order parameter are explicitly solved for the bond percolation problem. We introduce the parameters $t=2n\mathrm{ln}[\frac{{q}_{B}}{{q}_{c}}]$ and $\ensuremath{\omega}=2\mathrm{ln}[\frac{{q}_{\ensuremath{\parallel}}^{\mathrm{ns}}{q}_{\ensuremath{\perp}}}{{q}_{c}^{n}}]$, where ${q}_{\ensuremath{\alpha}}=1\ensuremath{-}{P}_{\ensuremath{\alpha}}$ is the probability of a bond being absent, and $n({n}_{s})$ is the coordination number in the bulk (surface). Also ${P}_{c}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\frac{1}{n})$ is the mean-field value of the percolation concentration of bonds in the bulk. We obtain the following results: (a) for $t&gt;0$, ${\ensuremath{\omega}}_{s}&lt;\ensuremath{\omega}&lt;t$, where ${\ensuremath{\omega}}_{s}=\ensuremath{-}\frac{1}{2}{(\frac{t}{n})}^{\frac{1}{2}}$, all clusters are finite; (b) for $t&gt;0$, $\ensuremath{\omega}&lt;{\ensuremath{\omega}}_{s}$, all clusters are finite in the bulk but the probability for an infinite cluster to form at and near the surface is nonvanishing; (c) for $\ensuremath{\omega}&lt;t&lt;0$ an infinite cluster forms through the whole system, but it has a larger probability of being close to the surface; (d) for $0&lt;t&lt;\ensuremath{\omega}$ only finite clusters occur in the system; (e) for $\ensuremath{\omega}&gt;t$ and $t&lt;0$, an infinite cluster starts to form in the whole system, with a larger probability of being in the bulk than on the surface. Critical exponents are derived for the different transitions and they are shown to satisfy general scaling relations.

Scaling theory of the dependence of critical behaviour on dimensionality

General scaling relations are obtained for the dependence of several critical indices on dimensionality. This dependence is completely determined by a single, model dependent, parameter. The results agree exactly with rigorous ones on the spherical model and approximately with accepted Ising model indices.

Axial-vector form factors from current algebra Regge poles and Veneziano model

Considering the following processes axial current + nucleon → vector current + nucleon, axial current + pion → vector current + σ-meson, and making use of current algebra identities and the Regge model, an asymptotic expression is obtained for the axial-vector form factors of the nucleon and the pion. Though no definite power behaviour can be found by means of the Regge model alone, it is possible to find, under suitable assumptions, generalized scaling relations among vector and axial vector from factors. The use of the Veneziano model in the pion case, fixes the asymptotic power behaviour and thus, the pion scaling relations are obtained without further assumptions.