Scaling and universality in turbulent convection

Anomalous correlation functions of the temperature field in two-dimensional turbulent convection are shown to be universal with respect to the choice of external sources. Moreover, they are equal to the anomalous correlations of the concentration field of a passive tracer advected by the convective flow itself. The statistics of velocity differences is found to be universal, self-similar, and close to Gaussian. These results point to the conclusion that temperature intermittency in two-dimensional turbulent convection may be traced back to the existence of statistically preserved structures, as it is in passive scalar turbulence.     

Scaling and universality in proportional elections

A most debated topic of the last years is whether simple statistical physics models can explain collective features of social dynamics. A necessary step in this line of endeavor is to find regularities in data referring to large-scale social phenomena, such as scaling and universality. We show that, in proportional elections, the distribution of the number of votes received by candidates is a universal scaling function, identical in different countries and years. This finding reveals the existence in the voting process of a general microscopic dynamics that does not depend on the historical, political, and/or economical context where voters operate. A simple dynamical model for the behavior of voters, similar to a branching process, reproduces the universal distribution.     

Scaling and universality in electrical failure of thin films

We describe the electrical failure of thin films as a percolation in two-dimensional random resistor networks. We show that the resistance evolution follows a scaling relation expressed as R approximately epsilon(-&mgr;) where epsilon = (1-t/tau), tau is the time of electrical failure of the film, and &mgr; is the same critical exponent appearing in the scaling relation between R and the defect concentration. For uniform degradation the value of &mgr; is universal. The validity of this scaling relation in the case of nonuniform degradation is proved by discussing the case in which the failure is due to a filamentary defect growth. The existence of this relation allows predictions of failure times from early time measurements of the resistance.     

Scaling effect on the mixed-mode fracture path of rock materials

In this paper, a new approach is presented to predict the crack growth path in the rock materials by taking into account the size effect. The proposed approach is an incremental method in which the crack initiation angle for each step is determined from the modified forms of the maximum tangential stress criterion. These modified maximum tangential stress criteria take into account the influence of the higher order terms of the stress series at the crack tip in addition to the singular terms. As an important parameter in the proposed method, the critical distance r _c is also assumed to be size dependent. Finally the incremental method is evaluated by experimental results obtained from Guiting limestone and CJhorveh marble specimens reported in the previous studies. It is shown that the proposed approach can predict the fracture trajectory of cracked specimens with different sizes in good agreement with the experimental results when three terms of Williams series expansion are considered for characterizing the stress field around the crack tip.     

Scaling and universality of critical fluctuations in granular gases

The total energy fluctuations of a low-density granular gas in the homogeneous cooling state near the threshold of the clustering instability are studied by means of molecular dynamics simulations. The relative dispersion of the fluctuations is shown to exhibit a power-law divergent behavior. Moreover, the probability distribution of the fluctuations presents data collapse as the system approaches the instability, for different values of the inelasticity. The function describing the collapse turns out to be the symmetric of the one found in several molecular equilibrium and nonequilibrium systems.     

Scaling and universality in the optics of disordered exciton chains

The joint probability distribution of exciton energies and transition dipole moments determines a variety of optical observables in disordered exciton systems. We demonstrate numerically that this distribution obeys a one-parameter scaling, originating from the fact that both the energy and the dipole moment are determined by the number of coherently bound molecules. A universal underlying distribution is found, which is identical for uncorrelated Gaussian disorder in the molecular transition energies or in the intermolecular transfer interactions. The universality breaks down for disorder in the transfer interactions resulting from variations in the molecular positions. We suggest the possibility to probe the joint distribution by means of single-molecule spectroscopy.     

Scaling and universality of the complexity of analog computation

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We numerically analyze various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.     

Scaling and universality of integer quantum Hall plateau-to-plateau transitions

We have investigated the integer quantum Hall plateau-to-plateau transition in two-dimensional electrons confined to AlxGa(1-x)As-Al0.33Ga0.67As heterostructures over a broad range of Al concentration x. For x between 0.65% and 1.6%, where the dominant contribution to disorder is from the short-range alloy potential fluctuations, we observe a perfect power-law scaling in the temperature range from 30 mK to 1 K with a critical exponent kappa = 0.42 +/- 0.01.     

Scaling behavior of the directed percolation universality class

In this work we consider five different lattice models which exhibit continuous phase transitions into absorbing states. By measuring certain universal functions, which characterize the steady state as well as the dynamical scaling behavior, we present clear numerical evidence that all models belong to the universality class of directed percolation. Since the considered models are characterized by different interaction details the obtained universal scaling plots are an impressive manifestation of the universality of directed percolation.     

Scaling and universality of self-organized patterns on unstable vicinal surfaces

We propose a unified treatment of the step bunching instability during epitaxial growth. The scaling properties of the self-organized surface pattern are shown to depend on a single parameter, the leading power in the expansion of the biased diffusion current in powers of the local surface slope. We demonstrate the existence of universality classes for the self-organized patterning appearing in models and experiments.     

Scaling and universality in city space syntax: Between Zipf and Matthew

We report about the universality of rank-integration distributions of open spaces in city space syntax similar to the famous rank-size distributions of cities (Zipf’s law). We also demonstrate that the degree of choice an open space represents for other spaces directly linked to it in a city follows a power-law statistic. Universal statistical behavior of space syntax measures uncovers the universality of the city creation mechanism. We suggest that the observed universality may help to establish the international definition of a city as a specific land use pattern.     

Scaling and universality of roughening in thermal oxidation of Si(001)

By analyzing atomic force microscopy images, we derive a continuum equation that quantitatively explains the roughening at the Si(001)-SiO2 interface during thermal oxidation at the temperature at 1200 degrees C in an Ar atmosphere containing a small fraction of O2. We also show that there is a phase transition in the universality class from a disordered to step-terrace structure at the interface at oxidation temperatures between 1150 and 1380 degrees C with the miscut angle of the substrate as the scaling parameter.     

Scaling behaviour in the fracture of fibrous materials

We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density scales with the system size L, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation. Received 24 June 1999     

Scaling of fracture strength in disordered quasi-brittle materials

The European Physical Journal B - This paper presents two main results. The first result indicates that in materials with broadly distributed microscopic heterogeneities, the fracture strength...     

Scaling of crack surfaces and implications for fracture mechanics

The scaling laws describing the roughness development of crack surfaces are incorporated into the Griffith criterion. We show that, in the case of a Family-Vicsek scaling, the energy balance leads to a purely elastic brittle behavior. On the contrary, it appears that an anomalous scaling reflects an R-curve behavior associated with a size effect of the critical resistance to crack growth in agreement with the fracture process of heterogeneous brittle materials exhibiting a microcracking damage.     

Scaling exponents for fracture surfaces in homogeneous glass and glassy ceramics

We investigate the scaling properties of postmortem fracture surfaces in silica glass and glassy ceramics. In both cases, the 2D height-height correlation function is found to obey Family-Viseck scaling properties, but with two sets of critical exponents, in particular, a roughness exponent zeta approximately 0.75 in homogeneous glass and zeta approximately 0.4 in glassy ceramics. The ranges of length scales over which these two scalings are observed are shown to be below and above the size of the process zone, respectively. A model derived from linear elastic fracture mechanics in the quasistatic approximation succeeds to reproduce the scaling exponents observed in glassy ceramics. The critical exponents observed in homogeneous glass are conjectured to reflect the damage screening occurring for length scales below the size of the process zone.     

Dragon-Kings in rock fracturing: Insights gained from rock fracture tests in the laboratory

In order to shed some lights to the “dragon-kings” concept, this paper re-examines experimental results on rock fracture tests in the laboratory, obtained from acoustic emission monitoring. The fracture of intact rocks as well as rocks containing natural structures (joints, faults, foliations) under constant stress rate loading or creep conditions is generally characterized by typical stages with different underlying physics. The primary phase reflects the initial rupture of pre-existing microcrack population in the sample or in the fault zone. Sub-critical growth dominates the secondary phase. The third phases termed nucleation phase corresponds to the initiation and accelerated growth of the ultimate fracture. The secondary and nucleation phases in both intact rock and faulted rock show power-law (of time-to-failure) increasing event rate and moment release. Samples containing planar structures such as foliations and faults demonstrate very similar features to natural earthquakes including: 1) small number of immediate foreshocks by which fault nucleation zones could be mapped; 2) the critical nucleation zone size is normally a fraction of the sample dimension; 3) a lot of aftershocks concentrated on the fault ruptured during the main event; 4) stress drop due to the main rupture is of the order from a few tens to a few hundreds MPa; 5) b-value drops during foreshocks and recovers during the aftershocks. All these results agree with the suggestion that laboratory measurements require no scaling but can be applied directly to the Earth to represent local fault behavior. The ultimate failure of the sample, or fracture of major asperities on the fault surface, normally lead to extreme events, i.e., dragon-kings, which has a magnitude significantly greater than that expected by the Gutenberg-Richter power-law relation in the magnitude-frequency distribution for either foreshocks or aftershocks. There are at least two mechanisms that may lead to dragon-kings: 1) The power-law increasing event rate and moment release; and 2) Hierarchical fracturing behavior resulting from hierarchical inhomogeneities in the sample. In the 1st mechanism, the final failure corresponds to the end point of the progressive occurrence of events and thus the resulted dragon-king event can be interpreted as a superposition of many small events. While for the 2nd mechanism an event of extreme size is the result of fracture growth stepping from a lower hierarchy into a higher hierarchy on fault surface having asperities characterized by hierarchical distribution (of size or strength) rather than simple fractal distribution. In both mechanisms the underlying physics is that fracture in rocks is hard to stop beyond certain threshold corresponding to the critical nucleation zone size.     

Scaling of the acoustic emission characteristics during plastic deformation and fracture

The scaling of the amplitude and time distributions of acoustic emission pulses, which reflects the self-similarity of defect structures, is revealed. The possibility of separation of independent contributions to the flow of acoustic emission events, which have substantially different scaling exponents, is shown for porous materials. The differences in the scaling exponents are related to the development of plastic deformation and fracture of the materials. The developed approach to an analysis of acoustic emission can be used to describe its predominant mechanisms during deformation.