Publications

The Eshelby problem refers to the response of a two-dimensional elastic sheet to cutting away a circle, deforming it into an ellipse, and pushing it back. The resulting response is dominated by the so-called Eshelby kernel, which was derived for purely elastic (infinite) material, but has been employed extensively to model the redistribution of stress after plastic events in amorphous solids with finite boundaries. Here, we discuss and solve the Eshelby problem directly for amorphous solids, taking into account possible screening effects and realistic boundary conditions. We find major modifications compared to the classical Eshelby solution. These modifications are needed for modeling correctly the spatial responses to plastic events in amorphous solids.

The response of amorphous solids to mechanical loads is accompanied by plasticity that is generically associated with \u201cnon-affine\u201d quadrupolar events seen in the resulting displacement field. To develop a continuum theory, one needs to assess when these quadrupolar events have a finite density, allowing the development of a field theory. Is there a transition, as a function of the material parameters and the nature of the loads, from isolated plastic events whose density is zero to a regime governed by a finite density? And if so, what is the nature of this transition? The aim of the paper is to explore this issue. The motivation for the present study stems from recent research in which it was shown that gradients of the quadrupolar fields act as dipole charges that can screen elasticity. Analytically soluble examples of mechanical loading that lead to screening and emergent length scales (that are absent in classical elasticity) have been analyzed and tested. However, \u201cgradients of quadrupolar fields\u201d make sense only when the density of quadrupoles is finite, and hence, the issue is central to this article. The article introduces a notion of polarizability under the strain of Eshelby quadrupoles and concludes that the onset of a density of such quadrupoles with random orientations can only appear when the polarizability is finite.

A significant amount of attention was dedicated in recent years to the phenomenon of jamming of athermal amorphous solids by increasing the volume fraction of the microscopic constituents. At a critical value of the volume fraction, pressure shoots up from zero to finite values with a host of critical exponents discovered and discussed. In this paper, we advance evidence for the existence of a second transition, within the jammed state of two-dimensional granular systems, that separates two regimes of different mechanical responses. Explicitly, highly packed systems are quasielastic with quadrupole screening, and more loosely jammed systems exhibit anomalous mechanics with dipole screening. Evidence is given for a clear transition between these two regimes, reminiscent of the intermediate hexatic phase of crystal melting in two-dimensional crystals. Theoretical estimates of the screening parameters and the pressure where transition takes place are provided.

Applying very small purely radial strains on amorphous solids in radial geometry one observes elastic responses that break the radial symmetry. Without any plasticity involved, the responses indicate mode coupling contributions even for minute strains. We show that these symmetry-breaking responses are due to disorder, typical to amorphous configurations. The symmetry-breaking responses are quantitatively explained using the classical Michell solutions which are excited by mode coupling.

Applying constant tensile stress to a piece of amorphous solid results in a slow extension, followed by an eventual rapid mechanical collapse. This "creep"process is of paramount engineering concern, and as such was the subject of study in a variety of materials, for more than a century. Predictive theories for τw, the expected time of collapse, are incomplete, mainly due to its dependence on a bewildering variety of parameters, including temperature, system size, tensile force, but also the detailed microscopic interactions between constituents. The complex dependence of the collapse time on all the parameters is discussed below, using simulations of strip of amorphous material. Different scenarios are observed for ductile and brittle materials, resulting in serious difficulties in creating an all-encompassing theory that could offer safety measures for given conditions. A central aim of this paper is to employ scaling concepts, to achieve data collapse for the probability distribution function (pdf) of lnτw. The scaling ideas result in a universal function which provides a prediction of the pdf of lnτw for out-of-sample systems, from measurements at other values of these parameters. The predictive power of the scaling theory is demonstrated for both ductile and brittle systems. Finally, we present a derivation of universal scaling function for brittle materials. The ductile case appears to be due to a plastic necking instability and is left for future research.

The theoretical understanding of the low-frequency modes in amorphous solids at finite temperature is still incomplete. The study of the relevant modes is obscured by the dressing of interparticle forces by collision-induced momentum transfer that is unavoidable at finite temperatures. Recently, it was proposed that low-frequency modes of vibrations around the thermally averaged configurations deserve special attention. In simple model glasses with bare binary interactions, these included quasilocalized modes whose density of states appears to be universal, depending on the frequencies as D(ω)∼ω4, in agreement with the similar law that is obtained with bare forces at zero temperature. In this paper, we report investigations of a model of silica glass at finite temperature; here the bare forces include binary and ternary interactions. Nevertheless, we can establish the validity of the universal law of the density of quasilocalized modes also in this richer and more realistic model glass.

Recent progress in studying the physics of amorphous solids has revealed that mechanical strains can be strongly screened by the formation of plastic events that are typically quadrupolar in nature. The theory stipulates that gradients in the density of the quadrupoles act as emergent dipole sources, leading to strong screening and to qualitative changes in the mechanical response, as seen, for example, in the displacement field. In this Letter we first offer direct measurements of the dipole field, independently of any theoretical assumptions, and second we demonstrate detailed agreement with the recently proposed theory. These two goals are achieved by using data from both simulations and experiments. Finally, we show how measurements of the dipole fields pinpoint the theory parameters that determine the profile of the displacement field.

A careful examination of the Quasi-Localized Modes (QLMs) that exist in a generic atomistic model of a glass former reveals at least two types of them, each exhibiting a different density of states, one depending on the frequency as and the other as . The properties of the glassy energy landscape that is responsible for the two types of modes is examined here, explaining the analytic feature responsible for the creations of (at least) two families of QLMs. It is argued that the QLMs that are revealed by diagonalizing the Hessian matrix are not related to possibly existing Two-Level Systems (TLS).

Diffusion-limited aggregation (DLA) has served for 40 years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references, no exact result for the fractal dimension D of DLA is known. In this Letter we announce an exact result for off-lattice DLA grown on a line embedded in the plane D = 3/2. The result relies on representing DLA with iterated conformal maps, allowing one to prove self-affinity, a proper scaling limit, and a well-defined fractal dimension. Mathematical proofs of the main results are available in N. Berger, E. B. Procaccia, and A. Turner, Growth of stationary Hastings-Levitov, arXiv:2008.05792.

It has been established that the low frequency quasilocalized modes of amorphous solids at zero temperature exhibit universal density of states, depending on the frequencies as D(omega) similar to omega(4). It remains an open question whether this universal law extends to finite temperatures. In this Letter we show that well quenched model glasses at temperatures as high as T-g/3 possess the same universal density of states. The only condition required is that average particle positions stabilize before thermal diffusion destroys the cage structure of the material. The universal density of quasilocalized low frequency modes refers then to vibrations around the thermally averaged configuration of the material.

Low-frequency quasilocalized modes of amorphous glass appear to exhibit universal density of states, depending on the frequencies as D(omega) similar to omega(4). To date, various models of glass formers with short-range binary interaction and network glass with both binary and ternary interactions were shown to conform with this law. In this paper, we examine granular amorphous solids with long-range electrostatic interactions and find that they exhibit the same law. To rationalize this wide universality class, we return to a model proposed by Gurevich, Parshin, and Schober (GPS) [Phys. Rev. B 67, 094203 (2003)] and analyze its predictions for interaction laws with varying spatial decay, exploring this wider-than-expected universality class. Numerical and analytic results are provided for both the actual system with long-range interaction and for the GPS model.

Catastrophic events in nature can be often triggered by small perturbations, with "remote triggering" of earthquakes being an important example. Here we present a mechanism for the giant amplification of small perturbations that is expected to be generic in systems whose dynamics is not derivable from a Hamiltonian. We offer a general discussion of the typical instabilities involved (being oscillatory with an exponential increase of noise) and examine in detail the normal forms that determine the relevant dynamics. The high sensitivity to external perturbations is explained for systems with and without dissipation. Numerical examples are provided using the dynamics of frictional granular matter. Finally, we point out the relationship of the presently discussed phenomenon to the highly topical issue of "exceptional points" in quantum models with non-Hermitian Hamiltonians.

Electrostatic theory preserves charges, but allows dipolar excitations. Elasticity theory preserves dipoles, but allows quadrupolar (Eshelby-like) plastic events. Charged amorphous granular systems are interesting in their own right; here we focus on their plastic instabilities and examine their mechanical response to external strain and to an external electric field, to expose the competition between elasticity and electrostatics. In this paper a generic model is offered, its mechanical instabilities are examined, and a theoretical analysis is presented. Plastic instabilities are discussed as saddle-node bifurcations that can be fully understood in terms of eigenvalues and eigenfunctions of the relevant Hessian matrix. This system exhibits moduli that describe how electric polarization and stress are influenced by strain and the electric field. Theoretical expression for these moduli are offered and compared to the measurements in numerical simulations.

Plastic instabilities in amorphous materials are often studied using idealized models of binary mixtures that do not capture accurately molecular interactions and bonding present in real glasses. Here we study atomic-scale plastic instabilities in a three-dimensional molecular dynamics model of silica glass under quasistatic shear. We identify two distinct types of elementary plastic events, one is a standard quasilocalized atomic rearrangement while the second is a bond-breaking event that is absent in simplified models of fragile glass formers. Our results show that both plastic events can be predicted by a drop of the lowest nonzero eigenvalue of the Hessian matrix that vanishes at a critical strain. Remarkably, we find very high correlation between the associated eigenvectors and the nonaffine displacement fields accompanying the bond-breaking event, predicting the locus of structural failure. Both eigenvectors and nonaffine displacement fields display an Eshelby-like quadrupolar structure for both failure modes, rearrangement, and bond breaking Our results thus clarify the nature of atomic-scale plastic instabilities in silica glasses, providing useful information for the development of mesoscale models of amorphous plasticity.

We report on a combined theoretical and numerical study of counterflow turbulence in superfluid He-4 in a wide range of parameters. The energy spectra of the velocity fluctuations of both the normal-fluid and superfluid components are strongly anisotropic. The angular dependence of the correlation between velocity fluctuations of the two components plays the key role. A selective energy dissipation intensifies as scales decrease, with the streamwise velocity fluctuations becoming dominant. Most of the flow energy is concentrated in a wave-vector plane which is orthogonal to the direction of the counterflow. The phenomenon becomes more prominent at higher temperatures as the coupling between the components depends on the temperature and the direction with respect to the counterflow velocity.

We study agitated frictional disks in two dimensions with the aim of
developing a scaling theory for their diffusion over time. As a function
of the area fraction ϕ and mean-square velocity fluctuations ⟨v2⟩ the mean-square displacement of the disks ⟨d2⟩
spans four to five orders of magnitude. The motion evolves from a
subdiffusive form to a complex diffusive behavior at long times. The
statistics of ⟨dn⟩
at all times are multiscaling, since the probability distribution
function (PDF) of displacements has very broad wings. Even where a
diffusion constant can be identified it is a complex function of ϕ and ⟨v2⟩.
By identifying the relevant length and time scales and their
interdependence one can rescale the data for the mean-square
displacement and the PDF of displacements into collapsed scaling
functions for all ϕ and ⟨v2⟩. These scaling functions provide a predictive tool, allowing one to infer from one set of measurements (at a given ϕ and ⟨v2⟩) what are the expected results at any value of ϕ and ⟨v2⟩ within the scaling range.

Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to the vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e., the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations that are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.

Three-dimensional anisotropic turbulence in classical fluids tends towards isotropy and homogeneity with decreasing scales, allowing-eventually-the abstract model of homogeneous and isotropic turbulence to be relevant. We show here that the opposite is true for superfluid He-4 turbulence in three-dimensional counterflow channel geometry. This flow becomes less isotropic upon decreasing scales, becoming eventually quasi-two-dimensional. The physical reason for this unusual phenomenon is elucidated and supported by theory and simulations.

Numerical Simulations are employed to create amorphous nano-films of a chosen thickness on a crystalline substrate which induces strain on the film. The films are grown by a vapor deposition technique. Using the exact relations between the Hessian matrix and the shear and bulk moduli we explore the mechanical properties of the nano-films as a function of the density of the substrate and the film thickness. The existence of the substrate dominates the mechanical properties of the combined substrate-film system. Scaling concepts are then employed to achieve data collapse in a wide range of densities and film thicknesses.

We report a joint experimental and theoretical investigation of the probability distribution functions (PDFs) of the normal and tangential (frictional) forces in amorphous frictional media. We consider both the joint PDF of normal and tangential forces together, and the marginal PDFs of normal forces separately and tangential forces separately. A maximum entropy formalism is utilized for all these cases after identifying the appropriate constraints. Excellent agreements with both experimental and simulation data are reported. The proposed joint PDF predicts giant slip events at low pressures, again in agreement with observations.

Standard approaches to magnetomechanical interactions in thermal magnetic crystalline solids involve Landau functionals in which the lattice anisotropy and the resulting magnetization easy axes are taken explicitly into account. In glassy systems one needs to develop a theory in which the amorphous structure precludes the existence of an easy axis, and in which the constituent particles are free to respond to their local amorphous surroundings and the resulting forces. We present a theory of all the mixed responses of an amorphous solid to mechanical strains and magnetic fields. Atomistic models are proposed in which we test the predictions of magnetostriction for both bulk and nanofilm amorphous samples in the paramagnetic phase. The application to nanofilms with emergent self-affine free interfaces requires a careful definition of the film "width" and its change due to the magnetostriction effect.

The large-scale turbulent statistics of mechanically driven superfluid He4 was shown experimentally to follow the classical counterpart. In this paper, we use direct numerical simulations to study the whole range of scales in a range of temperatures T [1.3,2.1] K. The numerics employ self-consistent and nonlinearly coupled normal and superfluid components. The main results are that (i) the velocity fluctuations of normal and super components are well correlated in the inertial range of scales, but decorrelate at small scales. (ii) The energy transfer by mutual friction between components is particulary efficient in the temperature range between 1.8 and 2 K, leading to enhancement of small-scale intermittency for these temperatures. (iii) At low T and close to Tλ, the scaling properties of the energy spectra and structure functions of the two components are approaching those of classical hydrodynamic turbulence.

We revisit the problem of the stress distribution in a frictional sandpile with both normal and tangential (frictional) inter-granular forces, under gravity, equipped with a new numerical method of generating such assemblies. Numerical simulations allow a determination of the spatial dependence of all the components of the stress field, principle stress axis, angle of repose, as a function of systems size, the coefficient of static friction and the frictional interaction with the bottom surface. We compare these results with the predictions of a theory based on continuum equilibrium mechanics. Basic to the theory of sandpiles are assumptions about the form of scaling solutions and constitutive relations for cohesiveless hard grains for which no typical scale is available. We find that these constitutive relations must be modified; moreover for smaller friction coefficients and smaller piles these scaling assumptions break down in the bulk of the sandpile due to the presence of length scales that must be carefully identified. Fortunately, for larger friction coefficient and for larger piles the breaking of scaling is weak in the bulk, allowing an approximate analytic theory which agrees well with the observations. After identifying the crucial scale, triggering the breaking of scaling, we provide a predictive theory to when scaling solutions are expected to break down. At the bottom of the pile the scaling assumption breaks always, due to the different interactions with the bottom surface. The consequences for measurable quantities like the pressure distribution and shear stress at the bottom of the pile are discussed. For example one can have a transition from no dip in the base-pressure to a dip at the center of the pile as friction increases.

Amorphous solids increase their stress as a function of an applied strain until a mechanical yield point whereupon the stress cannot increase anymore, afterward exhibiting a steady state with a constant mean stress. In stress-controlled experiments, the system simply breaks when pushed beyond this mean stress. The ubiquity of this phenomenon over a huge variety of amorphous solids calls for a generic theory that is free of microscopic details. Here, we offer such a theory: The mechanical yield is a thermodynamic phase transition, where yield occurs as a spinodal phenomenon. At the spinodal point, there exists a divergent correlation length that is associated with the system-spanning instabilities (also known as shear bands), which are typical to the mechanical yield. The theory, the order parameter used, and the correlation functions that exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solids that undergo mechanical yield.

We consider the problem of how to determine the force laws in an amorphous system of interacting particles. Given the positions of the centers of mass of the constituent particles we propose an algorithm to determine the interparticle force laws. Having n different types of constituents we determine the coefficients in the Laurent polynomials for the n(n+1)/2 possibly different force laws. A visual providing the particle positions in addition to a measurement of the pressure is all that is required. The algorithm proposed includes a part that can correct for experimental errors in the positions of the particles. Such a correction of unavoidable measurement errors is expected to benefit many experiments in the field.

Amorphous solids yield in strain-controlled protocols at a critical value of the strain. For larger strains the stress and energy display a generic complex serrated signal with elastic segments punctuated by sharp energy and stress plastic drops having a wide range of magnitudes. Here we provide a theory of the scaling properties of such serrated signals taking into account the system-size dependence. We show that the statistics are not homogeneous: they separate sharply to a regime of "small" and "large" drops, each endowed with its own scaling properties. A scaling theory is first derived solely by data analysis, showing a somewhat complex picture. But after considering the physical interpretation one discovers that the scaling behavior and the scaling exponents are in fact very simple and universal.

The mechanical properties of amorphous solids like metallic glasses can be dramatically changed by adding small concentrations (as low as 0.1%) of foreign elements. The glass-forming-ability, the ductility, the yield stress and the elastic moduli can all be greatly effected. This paper presents theoretical considerations with the aim of explaining the magnitude of these changes in light of the small concentrations involved. The theory is built around the experimental evidence that the microalloying elements organise around them a neighbourhood that differs from both the crystalline and the glassy phases of the material in the absence of the additional elements. These regions act as isotropic defects that in unstressed systems modify the shear moduli. When strained, these defects interact with the incipient plastic responses which are quadrupolar in nature. It will be shown that this interaction interferes with the creation of system-spanning shear bands and increases the yield strain. We offer experimentally testable estimates of the lengths of nano-shear bands in the presence of the additional elements.

The determination of the normal and transverse (frictional) interparticle forces within a granular medium is a long-standing, daunting, and yet unresolved problem. We present a new formalism that employs the knowledge of the external forces and the orientations of contacts between particles (of any given size), to compute all the interparticle forces. Having solved this problem, we exemplify the efficacy of the formalism showing that the force chains in such systems are determined by an expansion in the eigenfunctions of a newly defined operator.

Amorphous solids yield at a critical value of the strain (in strain-controlled experiments); for larger strains, the average stress can no longer increase - the system displays an elastoplastic steady state. A long-standing riddle in the materials community is what the difference is between the microscopic states of the material before and after yield. Explanations in the literature are material specific, but the universality of the phenomenon begs a universal answer. We argue here that there is no fundamental difference in the states of matter before and after yield, but the yield is a bona fide first-order phase transition between a highly restricted set of possible configurations residing in a small region of phase space to a vastly rich set of configurations which include many marginally stable ones. To show this, we employ an order parameter of universal applicability, independent of the microscopic interactions, that is successful in quantifying the transition in an unambiguous manner.

Long-ranged dipole-dipole interactions in magnetic glasses give rise to magnetic domains having labyrinthine patterns on the scale of about 1 micron. Barkhausen Noise then results from the movement of domain boundaries which is modeled by the motion of elastic membranes with random pinning. Here we propose that on the nanoscale new sources of Barkhausen Noise can arise. We propose an atomistic model of magnetic glasses in which we measure the Barkhausen Noise which results from the creation of new domains and the movement of domain boundaries on the nanoscale. The statistics of the Barkhausen Noise found in our simulations is in striking disagreement with the expectations in the literature. In fact we find exponential statistics without any power law, stressing the fact that Barkhausen Noise can belong to very different universality classes. In the present model the essence of the phenomenon is the fact that the spin response Green's function is decaying too rapidly for having sufficiently large magnetic jumps. A theory is offered in excellent agreement with the measured data without any free parameter.

Highly accurate numerical simulations are employed to highlight the subtle but important differences in the mechanical stability of perfect crystalline solids versus amorphous solids. We stress the difference between strain values at which the shear modulus vanishes and strain values at which a plastic instability ensues. The temperature dependence of the yield strain is computed for the two types of solids, showing different scaling laws: γY≃γY0-C1T1/3 for crystals versus γY≃γY0-C2T2/3 for amorphous solids.

We focus on the observed reduction in shear modulus when the stress on an amorphous solid is increased beyond the initial linear region. Careful numerical quasi-static simulations reveal an intimate relation between plastic failure and the reduction in shear modulus. The attainment of the smallest value of the shear modulus is identified with spreading of the regions that underwent a plastic event. We present an elementary "two-state" model that interpolates between failed and virgin regions and provides a simple and effective characterization of the phenomenon.

In this paper we focus on the mechanical properties of oligomeric glasses (waxes), employing a microscopic model that provides, via numerical simulations, information about the shear modulus of such materials, the failure mechanism via plastic instabilities, and the geometric responses of the oligomers themselves to a mechanical load. We present a microscopic theory that explains the numerically observed phenomena, including an exact theory of the shear modulus and of the plastic instabilities, both local and system spanning. In addition we present a model to explain the geometric changes in the oligomeric chains under increasing strains.

The study of metallic glasses which exhibit magnetic interactions was mainly restricted to the amorphous solid regime and its properties. Here we study the glass transition in fluids where particles are endowed with spins, such that magnetic and positional degrees of freedom are coupled. Results for slowing down in the spin time-correlation functions are described, and the effects of magnetic fields on the glass transition are studied. Aging effects in such systems and the corresponding data collapse are presented and discussed.

A nuclear capture reaction of a single neutron by ultracold superfluid He3 results in a rapid overheating followed by the expansion and subsequent cooling of the hot subregion, in a certain analogy with the big bang of the early universe. It was shown in a Grenoble experiment that a significant part of the energy released during the nuclear reaction was not converted into heat even after several seconds. It was thought that the missing energy was stored in a tangle of quantized vortex lines. This explanation, however, contradicts the expected lifetime of a bulk vortex tangle, 10-5-10-4 s, which is much shorter than the observed time delay of seconds. In this paper we propose a scenario that resolves the contradiction: the vortex tangle, created by the hot spot, emits isolated vortex loops that take with them a significant part of the tangle's energy. These loops quickly reach the container walls. The dilute ensemble of vortex loops attached to the walls can survive for a long time, while the remaining bulk vortex tangle decays quickly.

"Microalloying", which refers to the addition of small concentrations of a foreign metal to a given metallic glass, has been used extensively in recent years in attempts to improve the mechanical properties of these glasses. The results are haphazard and nonsystematic. In this paper we provide a microscopic theory of the effect of microalloying, exposing the delicate consequences of this procedure and the large parameter space which needs to be controlled. In particular we consider two very similar models which exhibit opposite trends for the change of the shear modulus, and explain the origins of this difference as displayed in the various microscopic structures and properties.

The dramatic dynamic slowing down associated with the glass transition is considered by many to be related to the existence of a static length scale that grows when temperature decreases. Defining, identifying, and measuring such a length is a subtle problem. Recently, two proposals, based on very different insights regarding the relevant physics, were put forward. One approach is based on the point-to-set correlation technique and the other on the scale where the lowest eigenvalue of the Hessian matrix becomes sensitive to disorder. Here we present numerical evidence that the two approaches might result in the same identical length scale. This provides mutual support for their relevance and, at the same time, raises interesting theoretical questions, discussed in the conclusion.

In this paper we extend the recent theory of shear localization in two-dimensional (2D) amorphous solids to three dimensions. In two dimensions the fundamental instability of shear localization is related to the appearance of a line of displacement quadrupoles that makes an angle of 45 ^â̂̃ with the principal stress axis. In three dimensions the fundamental plastic instability is also explained by the formation of a lattice of anisotropic elastic inclusions. In the case of pure external shear stress, we demonstrate that this is a 2D triangular lattice of similar elementary events. It is shown that this lattice is arranged on a plane that, similarly to the 2D case, makes an angle of 45 ^â̂̃ with the principal stress axis. This solution is energetically favorable only if the external strain exceeds a yield-strain value that is determined by the strain parameters of the elementary events and the Poisson ratio. The predictions of the theory are compared to numerical simulations and very good agreement is observed.

The usefulness of glasses, and particularly of metallic glasses, in technological applications is often limited by their toughness, (which is defined as the area under the stress vs. strain curve before plastic yielding). Recently, toughness was found to increase significantly by the addition of small concentrations of foreign atoms that act as pinning centers. We model this phenomenon at zero temperature and quasi-static straining with randomly positioned particles that participate in the elastic deformation, but are pinned in the non-affine return to mechanical equilibrium. We find a very strong effect on toughness via the increase of both the shear modulus and the yield stress as a function of the density of pinned particles. Understanding the results calls for analyzing separately the elastic, or Born term, and the contributions of the excess modes that result from glassy disorder. Finally, we present a scaling theory that collapses the data on one universal curve as a function of rescaled variables.

In recent research it was found that the fundamental shear-localizing instability of amorphous solids under external strain, which eventually results in a shear band and failure, consists of a highly correlated array of Eshelby quadrupoles all having the same orientation and some density rho. In this paper we calculate analytically the energy E(rho, gamma) associated with such highly correlated structures as a function of the density rho and the external strain gamma. We show that for strains smaller than a characteristic strain gamma(Y) the total strain energy initially increases as the quadrupole density increases, but that for strains larger than gamma(Y) the energy monotonically decreases with quadrupole density. We identify gamma(Y) as the yield strain. Its value, derived from values of the qudrupole strength based on the atomistic model, agrees with that from the computed stress-strain curves and broadly with experimental results. DOI: 10.1103/PhysRevE.87.022810

We present a short review of the theory of shear localization which results in shear bands in amorphous solids. As this is the main mechanism for the failure of metallic glasses, understanding the instability is invaluable in finding how to stabilize such materials against the tendency to shear localize. We explain the mechanism for shear localization under external shear-strain, which in 2-dimensions is the appearance of highly correlated lines of Eshelby-like quadrupolar singularities which organize the non-affine plastic flow of the amorphous solid into a shear band. We prove analytically that such highly correlated solutions in which N equi-distant quadrupoles are aligned with equal orientations are minimum energy states when the strain is high enough. The line lies at 45 degrees to the compressive stress. We use the theory to first predict the yield strain at zero temperature and quasi-static conditions, but later generalize to the case of finite temperature and finite shear rates, deriving the Johnson-Samwer T-2/3 law.

We employ a recently developed model that allows the study of two-dimensional brittle crack propagation under fixed grip boundary conditions. The crack development highlights the importance of voids which appear ahead of the crack as observed in experiments on the nanoscale. The appearance of these voids is responsible for roughening the crack path on small scales, in agreement with theoretical expectations. With increasing speed of propagation one observes the branching instabilities that were reported in experiments. The simulations allow understanding the phenomena by analyzing the elastic stress field that accompanies the crack dynamics.

Over the past decade, computer simulations have had an increasing role in shedding light on difficult statistical physical phenomena, and in particular on the ubiquitous problem of the glass transition. Here in a wide variety of materials the viscosity of a supercooled liquid increases by many orders of magnitude upon decreasing the temperature over a modest range. A natural concern in these computer simulations is the very small size of the simulated systems compared to experimental ones, raising the issue of how to assess the thermodynamic limit. Here we turn this limitation to our advantage by performing finite size scaling on the system size dependence of the relaxation time for supercooled liquids to emphasize the importance of a growing static length scale in the theory of glass transition. We demonstrate that the static length scale that was discovered by us in Physica A0378-437110.1016/j.physa.2011.11.020 391, 1001 (2012) fits the bill extremely well, allowing us to provide a finite-size scaling theory for the α-relaxation time of the glass transition, including predictions for the thermodynamic limit based on simulations in small systems.

Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length scale separating a quasi-inertial range of Kelvin-wave turbulence from a far-dissipation range. The letter culminates with analytical predictions for the energy spectrum of the Kelvin-wave turbulence in both of these regimes.

We developed a model for the enhancement of the heat flux by spherical and elongated nanoparticles in sheared laminar flows of nanofluids. Besides the heat flux carried by the nanoparticles, the model accounts for the contribution of their rotation to the heat flux inside and outside the particles. The rotation of the nanoparticles has a twofold effect: it induces a fluid advection around the particle and it strongly influences the statistical distribution of particle orientations. These dynamical effects, which were not included in existing thermal models, are responsible for changing the thermal properties of flowing fluids as compared to quiescent fluids. The proposed model is strongly supported by extensive numerical simulations, demonstrating a potential increase of the heat flux far beyond the Maxwell-Garnett limit for the spherical nanoparticles. The road ahead, which should lead toward robust predictive models of heat flux enhancement, is discussed.

In superfluid 3He-B, turbulence is carried predominantly by the superfluid component. To explore the statistical properties of this quantum turbulence and its differences from the classical counterpart, we adopt the time-honored approach of shell models. Using this approach, we provide numerical simulations of a Sabra shell model that allows us to uncover the nature of the energy spectrum in the relevant hydrodynamic regimes. These results are in qualitative agreement with analytical expressions for the superfluid turbulent energy spectra that were found using a differential approximation for the energy flux.

By comparing the response to external strains in metallic glasses and in Lennard-Jones glasses we find a quantitative universality of the fundamental plastic instabilities in the athermal, quasistatic limit. Microscopically these two types of glasses are as different as one can imagine, the latter being determined by binary interactions, whereas the former is determined by multiple interactions due to the effect of the electron gas that cannot be disregarded. In spite of this enormous difference the plastic instability is the same saddle-node bifurcation. As a result, the statistics of stress and energy drops in the elastoplastic steady state are universal, sharing the same system-size exponents.

Glasses are liquids whose viscosity has increased so much that they cannot flow. Accordingly, there have been many attempts to define a static length-scale associated with the dramatic slowing down of supercooled liquid with decreasing temperature. Here we present a simple method to extract the desired length-scale which is highly accessible both for experiments and for numerical simulations. The fundamental new idea is that low lying vibrational frequencies come in two types, those related to elastic response and those determined by plastic instabilities. The minimal observed frequency is determined by one or the other, crossing at a typical length-scale which is growing with the approach of the glass transition. This length-scale characterizes the correlated disorder in the system: on longer length-scales the details of the disorder become irrelevant, dominated by the Debye model of elastic modes. To connect the newly defined length-scale to relaxation dynamics near the glass transition, we show that supercooled liquids in which there exist random pinning sites of density ^ρim∼1ξsd exhibit complete jamming of all dynamics. This is a direct demonstration that the proposed length scale is indeed the static length that was long sought-after.

Brittle materials exhibit sharp dynamical fractures when meeting Griffith's criterion, whereas ductile materials blunt a sharp crack by plastic responses. Upon continuous pulling, ductile materials exhibit a necking instability that is dominated by a plastic flow. Usually one discusses the brittle to ductile transition as a function of increasing temperature. We introduce an athermal brittle to ductile transition as a function of the cutoff length of the interparticle potential. On the basis of extensive numerical simulations of the response to pulling the material boundaries at a constant speed we offer an explanation of the onset of ductility via the increase in the density of plastic modes as a function of the potential cutoff length. Finally we can resolve an old riddle: In experiments brittle materials can be strained under grip boundary conditions and exhibit a dynamic crack when cut with a sufficiently long initial slot. Mysteriously, in molecular dynamics simulations it appeared that cracks refused to propagate dynamically under grip boundary conditions, and continuous pulling was necessary to achieve fracture. We argue that this mystery is removed when one understands the distinction between brittle and ductile athermal amorphous materials.

We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient B₂ has anomalous fluctuations and the second nonlinear coefficient B₃ and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.

The effect of finite temperature T and finite strain rate γ on the statistical physics of plastic deformations in amorphous solids made of N particles is investigated. We recognize three regimes of temperature where the statistics are qualitatively different. In the first regime the temperature is very low, T

Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner "remember" that direction also when relaxed back to a state with zero mean stress. We address the question "what is the order parameter that is responsible for this memory?" and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elastoplasticity.

We propose a method to predict the value of the external strain where a generic amorphous solid will fail by a plastic response (i.e., an irreversible deformation), solely on the basis of measurements of the nonlinear elastic moduli. While usually considered fundamentally different, with the elastic properties describing reversible phenomena and plastic failure epitomizing irreversible behavior, we show that the knowledge of some nonlinear elastic moduli is enough to predict where plasticity sets in.

An effective temperature Teff which differs from the bath temperature is believed to play an essential role in the theory of elastoplasticity of amorphous solids. Here, we introduce a natural definition of Teff appearing naturally in a Boltzmann-like distribution of measurable structural features without recourse to any questionable assumption. The value of Teff is connected, using theory and scaling concepts, to the flow stress and the mean energy that characterize the elastoplastic flow.

We address the now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (sub- or super-diffusion) at longer times. Using the standard non-Markovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the probability distribution function obtained within the continuous random walk approach but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.

Much of the discussion in the literature of the low-frequency part of the density of states of amorphous solids was dominated for years by comparing measured or simulated density of states to the classical Debye model. Since this model is hardly appropriate for the materials at hand, this created some amount of confusion regarding the existence and universality of the so-called "boson peak" which results from such comparisons. We propose that one should pay attention to the different roles played by different aspects of disorder, the first being disorder in the interaction strengths, the second positional disorder, and the third coordination disorder. These have different effects on the low-frequency part of the density of states. We examine the density of states of a number of tractable models in one and two dimensions and reach a clearer picture of the softening and redistribution of frequencies in such materials. We discuss the effects of disorder on the elastic moduli and the relation of the latter to frequency softening, reaching the final conclusion that the boson peak is not universal at all.

Strongly correlated amorphous solids are a class of glass formers whose interparticle potential admits an approximate inverse power-law form in a relevant range of interparticle distances. We study the steady-state plastic flow of such systems, first in the athermal quasistatic limit and second at finite temperatures and strain rates. In all cases we demonstrate the usefulness of scaling concepts to reduce the data to universal scaling functions where the scaling exponents are determined a priori from the interparticle potential. In particular we show that the steady plastic flow at finite temperatures with efficient heat extraction is uniquely characterized by two scaled variables; equivalently, the steady-state displays an equation of state that relates one scaled variable to the other two. We discuss the range of applicability of the scaling theory, and the connection to density scaling in supercooled liquid dynamics. We explain that the description of transient states calls for additional state variables whose identity is still far from obvious.

The shear modulus and yield stress of amorphous solids are important material parameters, with the former determining the rate of increase in stress under external strain and the latter being the stress value at which the material flows in a plastic manner. It is therefore important to understand how these parameters can be related to the interparticle potential. Here a scaling theory is presented such that given the interparticle potential, the dependence of the yield stress, and the shear modulus on the density of the solid can be predicted in the athermal limit. It is explained when such prediction is possible at all densities and when it is only applicable at high densities.

Volume alteration in solid materials is a common cause of material failure. Here we investigate the crack formation in thin elastic layers attached to a substrate. We show that small variations in the volume contraction and substrate restraint can produce widely different crack patterns ranging from spirals to complex hierarchical networks. The networks are formed when there is no prevailing gradient in material contraction, whereas spirals are formed in the presence of a radial gradient in the contraction of a thin elastic layer.

In the context of a classical example of glass formation in three dimensions, we exemplify how to construct a statistical-mechanical theory of the glass transition. At the heart of the approach is a simple criterion for verifying a proper choice of upscaled quasispecies that allow the construction of a theory with a finite number of "states." Once constructed, the theory identifies a typical scale ξ that increases rapidly with lowering the temperature and which determines the α-relaxation time τα as τα∼exp(mu;ξ/T), with μ a typical chemical potential. The theory can predict relaxation times at temperatures that are inaccessible to numerical simulations.

Experimental measurements of the specific heat in glass-forming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called "specific heat peak" in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same α -relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formers-unfortunately, one needs to understand each case by itself.

We present a quantitative theory for a relaxation function in a simple glass-forming model (binary mixture of particles with different interaction parameters). It is shown that the slowing down is caused by the competition between locally favored regions (clusters) that are long-lived but each of which relaxes as a simple function of time. Without the clusters, the relaxation of the background is simply determined by one typical length, which we deduce from an elementary statistical mechanical argument. The total relaxation function (which depends on time in a nontrivial manner) is quantitatively determined as a weighted sum over the clusters and the background. The "fragility" in this system can be understood quantitatively since it is determined by the temperature dependence of the number fractions of the locally favored regions.

We ask what determines the (small) angle of turbulent jets. To answer this question we first construct a deterministic vortex-street model representing the large-scale structure in a self-similar plane turbulent jet. Without adjustable parameters the model reproduces the mean velocity profiles and the transverse positions of the large-scale structures, including their mean sweeping velocities, in a quantitative agreement with experiments. Nevertheless, the exact self-similar arrangement of the vortices (or any other deterministic model) necessarily leads to a collapse of the jet angle. The observed (small) angle results from a competition between vortex sweeping tending to strongly collapse the jet and randomness in the vortex structure, with the latter resulting in a weak spreading of the jet.

The aim of this paper is to discuss some basic notions regarding generic glass-forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction, we discuss the so-called glass transition in which a supercooled amorphous state is formed, accompanied by a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length scales, we show the following. (i) The system remains ergodic at all temperatures. (ii) The number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the small world property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0.

We propose that there exists a generic class of glass-forming systems that have competing states (of crystalline order or not) which are locally close in energy to the ground state (which is typically unique). Upon cooling, such systems exhibit patches (or clusters) of these competing states which become locally stable in the sense of having a relatively high local shear modulus. It is in between these clusters where aging, relaxation, and plasticity under strain can take place. We demonstrate explicitly that relaxation events that lead to aging occur where the local shear modulus is low (even negative) and result in an increase in the size of local patches of relative order. We examine the aging events closely from two points of view. On the one hand we show that they are very localized in real space, taking place outside the patches of relative order, and from the other point of view we show that they represent transitions from one local minimum in the potential surface to another. This picture offers a direct relation between structure and dynamics, ascribing the slowing down in glass-forming systems to the reduction in relative volume of the amorphous material which is liquidlike. While we agree with the well-known Adam-Gibbs proposition that the slowing down is due to an entropic squeeze (a dramatic decrease in the number of available configurations), we do not agree with the Adam-Gibbs (or the Volger-Fulcher) formulas that predict an infinite relaxation time at a finite temperature. Rather, we propose that generically there should be no singular crisis at any finite temperature: the relaxation time and the associated correlation length (average cluster size) increase at most superexponentially when the temperature is lowered.

We address the relaxation dynamics in hydrogen-bonded supercooled liquids near (but above) the glass transition, measured via broadband dielectric spectroscopy (BDS). We propose a theory based on decomposing the relaxation of the macroscopic dipole moment into contributions from hydrogen-bonded clusters of s molecules, with smin ≤s≤ smax. The existence of smax is translated into a sum rule on the concentrations of clusters of size s. We construct the statistical mechanics of the supercooled liquid subject to this sum rule as a constraint, to estimate the temperature-dependent density of clusters of size s. With a theoretical estimate of the relaxation time of each cluster, we provide predictions for the real and imaginary parts of the frequency-dependent dielectric response. The predicted spectra and their temperature dependence are in accord with measurements, explaining a host of phenomenological fits like the Vogel-Fulcher fit and the stretched exponential fit. Using glycerol as a particular example, we demonstrate quantitative correspondence between theory and experiments. The theory also demonstrates that the α peak and the "excess wing" stem from the same physics in this material. The theory also shows that in other hydrogen-bonded glass formers the excess wing can develop into a β peak, depending on the molecular material parameters (predominantly the surface energy of the clusters). We thus argue that α and β peaks can stem from the same physics. We address the BDS in constrained geometries (pores) and explain why recent experiments on glycerol did not show a deviation from bulk spectra. Finally, we discuss the dc part of the BDS spectrum and argue why it scales with the frequency of the α peak, providing an explanation for the remarkable data collapse observed in experiments.

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a semianalytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.

We propose an approach to study the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We achieve this by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by 'Statistically Preserved Structures' which are associated with statistical conservation laws. The latter can be used for example to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations, and it demonstrates that the scaling exponents of these nonlinear models are indeed anomalous. In order to adress the universality of these nonlinear model we study the statistical properties of a semi-infinite chain of passive vectors advecting each other and study the scaling exponents at the fixed point of this chain.

We study a recently introduced model of one-component glass-forming liquids whose constituents interact with an anisotropic potential. This system is interesting per se and as a model of liquids such as glycerol (interacting via hydrogen bonds) which are excellent glass formers. We work out the statistical mechanics of this system, encoding the liquid and glass disorder using appropriate quasiparticles (36 of them). The theory provides a full explanation of the glass transition phenomenology, including the identification of a diverging length scale and a relation between the structural changes and the diverging relaxation times.

We develop an athermal shear-transformation-zone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, time-dependent tractions. The theory is illustrated here for the case of a circular hole in an infinite two-dimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elastoplastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundary-layer theory that might be useful in less symmetric situations.

The statistical mechanics of simple glass forming systems in two dimensions is worked out. The glass disorder is encoded via a Voronoi tesselation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, Tg and Tc, the first associated with jamming and the second associated with crystallization at very low temperatures.

Understanding the mechanical properties of glasses remains elusive since the glass transition itself is not fully understood, even in well-studied examples of glass formers in two dimensions. In this context we demonstrate here: i) a direct evidence for a diverging length scale at the glass transition ii) an identification of the glass transition with the disappearance of fluid-like regions and iii) the appearance in the glass state of fluid-like regions when mechanical strain is applied. These fluid-like regions are associated with the onset of plasticity in the amorphous solid. The relaxation times which diverge upon the approach to the glass transition are related quantitatively to the diverging length scale.

In the preceding paper, we developed an athermal shear-transformation-zone (STZ) theory of amorphous plasticity. Here we use this theory in an analysis of numerical simulations of plasticity in amorphous silicon by Demkowicz and Argon (DA). In addition to bulk mechanical properties, those authors observed internal features of their deforming system that challenge our theory in important ways. We propose a quasithermodynamic interpretation of their observations in which the effective disorder temperature, generated by mechanical deformation well below the glass temperature, governs the behavior of other state variables that fall in and out of equilibrium with it. Our analysis points to a limitation of either the step-strain procedure used by DA in their simulations, or the STZ theory in its ability to describe rapid transients in stress-strain curves, or perhaps to both. Once we allow for this limitation, we are able to bring our theoretical predictions into accurate agreement with the simulations.

We investigate whether fractal viscous fingering and diffusion-limited aggregates are in the same scaling universality class. We bring together the largest available viscous fingering patterns and a novel technique for obtaining the conformal map from the unit circle to an arbitrary singly connected domain in two dimensions. These two Laplacian fractals appear different to the eye; in addition, viscous fingering is grown in parallel and the aggregates by a serial algorithm. Nevertheless, the data strongly indicate that these two fractal growth patterns are in the same universality class.

We employ the full FENE-P model of the hydrodynamics of a dilute polymer solution to derive a theoretical approach to drag reduction in wall-bounded turbulence. We recapture the results of a recent simplified theory which derived the universal maximum drag reduction (MDR) asymptote, and complement that theory with a discussion of the cross-over from the MDR to the Newtonian plug when the drag reduction saturates. The FENE-P model gives rise to a rather complex theory due to the interaction of the velocity field with the polymeric conformation tensor, making analytic estimates quite taxing. To overcome this we develop the theory in a computer-assisted manner, checking at each point the analytic estimates by direct numerical simulations (DNS) of viscoelastic turbulence in a channel.

Dynamic fracture in a wide class of materials reveals a "fracture energy" Γ much larger than the expected nominal surface energy due to the formation of two fresh surfaces. Moreover, the fracture energy depends on the crack velocity, Γ=Γ(v). We show that a simple dynamical theory of viscoplasticity coupled to asymptotic pure linear elasticity provides a possible explanation to the above phenomena. The theory predicts tip blunting characterized by a dynamically determined crack tip radius of curvature. In addition, we demonstrate velocity selection for cracks in fixed-grip strip geometry accompanied by the identification of Γ and its velocity dependence.

Fracture paths in quasi-two-dimensional (2D) media (e.g., thin layers of materials or paper) are analyzed as self-affine graphs h(x) of height h as a function of length x. We show that these are multiscaling, in the sense that nth order moments of the height fluctuations across any distance â.," scale with a characteristic exponent that depends nonlinearly on the order of the moment. Having demonstrated this, one rules out a widely held conjecture that fracture in 2D belongs to the universality class of directed polymers in random media. In fact, 2D fracture does not belong to any of the known kinetic roughening models. The presence of multiscaling offers a stringent test for any theoretical model; we show that a recently introduced model of quasistatic fracture passes this test.

Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

Structure functions of rough fracture surfaces in isotropic materials exhibit complicated scaling properties due to the broken isotropy in the fracture plane generated by a preferred propagation direction. Decomposing the structure functions into the even order irreducible representations of the SO(2) symmetry group indexed by (m=0,2,4,...) results in a lucid and quickly convergent description. The scaling exponent of the isotropic sector (m=0) dominates at small length scales. One can reconstruct the anisotropic structure functions using only the isotropic and the first nonvanishing anisotropic sector (m=2) [or at most the next one (m=4)]. The scaling exponent of the isotropic sector should be observed in a proposed, yet unperformed, experiment.

A celebrated universal aspect of wall-bounded turbulent flows is the von Kármán log-law-of-the-wall, describing how the mean velocity in the stream-wise direction depends on the distance from the wall. Although the log-law is known for more than 75 years, the von Kármán constant governing the slope of the log-law was not determined theoretically. In this letter we show that the von Kármán constant can be estimated from homogeneous turbulent data, i. e, without information from wall-bounded flows.

The problem of anisotropy and its effects on the statistical theory of high Reynolds number (Re) turbulence (and turbulent transport) is intimately related and intermingled with the problem of the universality of the (anomalous) scaling exponents of structure functions. Both problems had seen tremendous progress in the last 5 years. In this review we present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations. Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO (d) symmetry group (with d being the dimension, d = 2 or 3). This device allows a discussion of the scaling properties of the statistical objects in well-defined sectors of the symmetry group, each of which is determined by the "angular momenta" sector numbers (j, m). For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made: (i) the scaling exponents are universal, (ii) the isotropic scaling exponents are always leading, (iii) the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of j. This emerging picture offers a complete understanding of the decay of anisotropy upon going to smaller and smaller scales. Next, we explain how to apply the SO (3) decomposition to the statistical Navier-Stokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite. A systematic analysis of direct numerical simulations (DNS) of the Navier-Stokes equations and of experiments provides a strong support to the proposition that also for the non-linear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of j. An approximate calculation of the anisotropic exponents based on a closure theory is reviewed. The conflicting experimental measurements on the rate of decay of anisotropy upon reducing the scales are explained and systematized, showing that isotropy is eventually recovered at small scales.

Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the Îm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent ζâ0.8 in 3 dimensions and ζâ 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

We propose a theoretical model for branching instabilities in 2-dimensional fracture, offering predictions for when crack branching occurs, how multiple cracks develop, and what is the geometry of multiple branches. The model is based on equations of motion for crack tips which depend only on the time dependent stress intensity factors. The latter are obtained by invoking an approximate relation between static and dynamic stress intensity factors, together with an essentially exact calculation of the static ones. The results of this model are in qualitative agreement with a number of experiments in the literature.

Drag reduction by polymers in turbulent flows raises an apparent contradiction: the stretching of the polymers must increase the viscosity, so why is the drag reduced? A recent theory proposed that drag reduction, in agreement with experiments, is consistent with the effective viscosity growing linearly with the distance from the wall. With this self-consistent solution the reduction in the Reynolds stress overwhelms the increase in viscous drag. In this Rapid Communication we show, using direct numerical simulations, that a linear viscosity profile indeed reduces the drag in agreement with the theory and in close correspondence with direct simulations of the FENE-P model at the same flow conditions.

We address the effect of polymer additives on two-dimensional turbulence, an issue that was studied recently in experiments and direct numerical simulations. We show that the same simple shell model that reproduced drag reduction in three-dimensional turbulence reproduces all the reported effects in the two-dimensional case. The simplicity of the model offers a straightforward simulation of all the major effects under consideration.

We demonstrate, by using suitable shell models, that drag reduction in homogeneous turbulence is usefully discussed in terms of a scale-dependent effective viscosity. The essence of the phenomenon of drag reduction found in models that couple the velocity field to the polymers can be recaptured by an \u201cequivalent\u201d equation of motion for the velocity field alone, with a judiciously chosen scale-dependent effective viscosity that succinctly summarizes the important aspects of the interaction between the velocity and the polymer fields. Finally, we clarify the differences between drag reduction in homogeneous and in wall bounded flows.

Post mortem analysis of fracture surfaces of ductile and brittle materials on the mum-mm and the nm scales, respectively, reveal self-affine cracks with anomalous scaling exponent zetaapproximate to0.8 in three dimensions and zetaapproximate to0.65 in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent zetaapproximate to0.5 up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.

The stress intensity factors for arbitrarily shaped cracks in an infinite two-dimensional elastic medium were investigated. The conformal map was constructed from the exterior of the unit circle to exterior of the arbitrary crack using the method of iterated conformal maps. The method, when compared with the Schwartz-Cristoffel transformation, was found to be better. The conformal method allowed an accurate estimate of the stress intensity factors or of the subleading terms such as the T-stress and simplified the calculation of the full stress field.

An exact integro-differential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2-dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.

Recent direct numerical simulations of the finite-extensibility nonlinear elastic dumbbell model with the Peterlin approximation of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We use here a simple shell model for homogeneous viscoelastic flows, which recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on large scales. Understanding the mechanism allows us to predict how the amount of drag reduction depends on the various parameters in the model. The main conclusion is that drag reduction is not a universal phenomenon; it peaks in a window of parameters such as the Reynolds number and the relaxation rate of the polymer.

We have recently proposed that the statistics of active fields (which affect the velocity field itself) in well-developed turbulence are also dominated by the statistically preserved structures of auxiliary passive fields which are advected by the same velocity field. The statistically preserved structures are eigenmodes of eigenvalue 1 of an appropriate propagator of the decaying (unforced) passive field, or equivalently, the zero modes of a related operator. In this paper we investigate further this surprising finding via two examples of shell models, one akin to turbulent convection in which the temperature is the active scalar, and the other akin to magnetohydrodynamics in which the magnetic field is the active vector. In the first example, all the even correlation functions of the active and passive fields exhibit identical scaling behavior. The second example appears at first sight to be a counterexample: the statistical objects of the active and passive fields have entirely different scaling exponents. We demonstrate, nevertheless, that the statistically preserved structures of the passive vector dominate again the statistics of the active field, except that due to a dynamical conservation law the amplitude of the leading zero mode cancels exactly. The active vector is then dominated by the subleading zero mode of the passive vector. Our work thus suggests that the statistical properties of active fields in turbulence can be understood with the same generality as those of passive fields.

The weak version of universality in turbulence refers to the independence of the scaling exponents of the nth order structure functions from the statistics of the forcing. The strong version includes universality of the coefficients of the structure functions in the isotropic sector, once normalized by the mean energy flux. We demonstrate that shell models of turbulence exhibit strong universality for both forced and decaying turbulence. The exponents and the normalized coefficients are time independent in decaying turbulence, forcing independent in forced turbulence, and equal for decaying and forced turbulence. We conjecture that this is also the case for Navier-Stokes turbulence.

An early (and influential) scaling relation in the multifractal theory of diffusion limited aggregation (DLA) is the Turkevich-Scher conjecture that relates the exponent [Formula presented] that characterizes the \u201chottest\u201d region of the harmonic measure and the fractal dimension D of the cluster, i.e., [Formula presented] Due to lack of accurate direct measurements of both D and [Formula presented] this conjecture could never be put to a serious test. Using the method of iterated conformal maps, D was recently determined as [Formula presented] In this paper, we determine [Formula presented] accurately with the result [Formula presented] We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.

The anomalous scaling of correlation functions in the turbulent statistics of active scalars (like temperature in turbulent convection) is understood in terms of an auxiliary passive scalar which is advected by the same turbulent velocity field. The even-order correlation functions of the two fields are the same to leading order (up to a trivial multiplicative factor). The leading correlation functions are statistically preserved structures of the passive scalar decaying problem; thus the universality of the scaling exponents of the even-order correlations of the active scalar is demonstrated.

The statistics of two-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian-like in equilibrium. The skewness Sequivalent toS(3)(R)/S-2(3/2)(R) was measured as S(exp)approximate to0.03. This contradiction is lifted by understanding that two-dimensional turbulence is not far from a situation with equipartition of enstrophy, which exists as true thermodynamic equilibrium with K41 exponents in space dimension of d=4/3. We evaluate the skewness S(d) for 4/3 less than or equal todless than or equal to2, showing that S(d)=0 at d=4/3, and that it remains as small as S-exp in two dimensions.

We study the geometrical characteristic of quasistatic fractures in brittle media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern, it crosses over from about 0.5 for small scales to about 0.75 for large scales. We propose that this crossover reflects the increased ramification of the fracture pattern.

The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10 ^-35, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D_q are infinite for q

The method of iterated conformal maps for the study of diffusion limited aggregates (DLA) is generalized to the study of Laplacian growth patterns and related processes. We emphasize the fundamental difference between these processes: DLA is grown serially with constant size particles, while Laplacian patterns are grown by advancing each boundary point in parallel, proportional to the gradient of the Laplacian field. We introduce a two-parameter family of growth patterns that interpolates between DLA and a discrete version of Laplacian growth. The ultraviolet putative finite-time singularities are regularized here by a minimal tip size, equivalently for all the models in this family. With this we stress that the difference between DLA and Laplacian growth is not in the manner of ultraviolet regularization, but rather in their deeply different growth rules. The fractal dimensions of the asymptotic patterns depend continuously on the two parameters of the family, giving rise to a \u201cphase diagram\u201d in which DLA and discretized Laplacian growth are at the extreme ends. In particular, we show that the fractal dimension of Laplacian growth patterns is higher than the fractal dimension of DLA, with the possibility of dimension 2 for the former not excluded.

Motivated by turbulent drag reduction by minute concentrations of polymers we study the effects of minor viscosity contrasts on the stability of hydrodynamic flows. The key player is a localized region where fluctuations are produced by interactions with the mean flow (the \u201ccritical layer\u201d). We show that a layer of weakly space-dependent viscosity placed near the critical layer has a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. The effect is not due to a modified dissipation (as is assumed in theories of drag reduction) but is due to reduced energy intake from the mean flow to the fluctuations. Similar physics may act in turbulent drag reduction.

We study the dynamics of "finger" formation in Laplacian growth without surface tension in a channel geometry (the Saffman Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic "finger" whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.

Traditional devices for measuring turbulence have been unable to keep up with the latest developments in theory. But detectors derived from high-energy physics may narrow the gap between experiment and theory.

We discuss the scaling exponents characterizing the power-law behavior of the anisotropic components of correlation functions in turbulent systems with pressure. The anisotropic components are conveniently labeled by the angular momentum index (Formula presented) of the irreducible representation of the SO(3) symmetry group. Such exponents govern the rate of decay of anisotropy with decreasing scales. It is a fundamental question whether they ever increase as (Formula presented) increases, or they are bounded from above. The equations of motion in systems with pressure contain nonlocal integrals over all space. One could argue that the requirement of convergence of these integrals bounds the exponents from above. It is shown here on the basis of a solvable model (the \u201clinear pressure model\u201d) that this is not necessarily the case. The model introduced here is of a passive vector advection by a rapidly varying velocity field. The advected vector field is divergent free and the equation contains a pressure term that maintains this condition. The zero modes of the second-order correlation function are found in all the sectors of the symmetry group. We show that the spectrum of scaling exponents can increase with (Formula presented) without bounds while preserving finite integrals. The conclusion is that contributions from higher and higher anisotropic sectors can disappear faster and faster upon decreasing the scales also in systems with pressure.

A conformal theory of fractal growth processes in two dimensions was considered, including diffusion limited aggregation (DLA) as a special case. The computation of the dimension from early stage dynamics was demonstrated. Further, a novel notion was introduced that the fractal dimension appears as a dynamical exponent already at early stages of the growth in which the cluster has not yet developed a fractal geometry.

We present a model of hydrodynamic turbulence for which the program of computing the scaling exponents from first principles can be developed in a controlled fashion. The model consists of N suitably coupled copies of the "Sabra" shell model of turbulence. The couplings are chosen to include two components: random and deterministic, with a relative importance that is characterized by a parameter called ∈. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the N→∞ limit but for 0

Kraichnans model of passive scalar advection in which the driving (Gaussian) velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields. We show here that the solutions of the Kraichnan equation for the n-order correlation functions foliate into sectors that are classified by the irreducible representations of the [Formula Presented] symmetry group. We find a discrete spectrum of universal anomalous exponents, with a different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result. In the second method we compute the exponents from the zero modes of the Kraichnan equation for the correlation functions of the scalar field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer and inner scales as renormalization scales. In addition we derive exact fusion rules, which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions.

Diffusion limited aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (23 particles) to asymptotically large [Formula Presented] clusters. The computed dimension is [Formula Presented].

The Taylor hypothesis, which allows surrogating spatial measurements requiring many experimental probes by time series from one or two probes, is examined on the basis of a simple analytic model of turbulent statistics. The main points are as follows: (i) The Taylor hypothesis introduces systematic errors in the evaluation of scaling exponents. (ii) When the mean wind V̄₀ is not infinitely larger than the root-mean-square longitudinal turbulent fluctuations v_T, the effective Taylor advection velocity V_ad should take the latter into account. (iii) When two or more probes are employed the application of the Taylor hypothesis and the optimal choice of the effective advecting wind V_ad need extra care. We present practical considerations for minimizing the errors incurred in experiments using one or two probes. (iv) Analysis of the Taylor hypothesis when different probes experience different mean winds is offered.

We show that the Sabra shell model of turbulence, which was introduced recently, displays a Hamiltonian structure for given values of the parameters. The requirement of scale independence of the flux of this Hamiltonian allows us to compute exactly a one-parameter family of anomalous scaling exponents associated with 4th-order correlation functions.

We consider flame front propagation in channel geometries. The steady-state solution in this problem is space dependent and therefore the linear stability analysis is described by a partial integro-differential equation with a space-dependent coefficient. Accordingly, it involves complicated eigenfunctions. We show that the analysis can be performed using a finite-order dynamical system in terms of the dynamics of singularities in the complex plane, yielding a detailed understanding of the physics of the eigenfunctions and eigenvalues.

A recent theoretical development in the understanding of the small-scale structure of Navier-Stokes turbulence has been the proposition that the scales η_n(R) that separate inertial from viscous behavior of many-point correlation functions depend on the order n and on the typical separations R of points in the correlation. This is of fundamental significance in itself but it also has implications for the scaling behaviour of various correlation functions. This dependence has never been observed directly in laboratory experiments. In order to observe it, turbulence data which both display a well-developed scaling range with clean scaling behaviour and are well-resolved in the small scales to well within the viscous range is required. The data of the experiments performed in the laboratory of P. Tabeling of Navier-Stokes turbulence in a helium cell with counter-rotating disks approach these criteria, and provide supporting evidence for the existence of the predicted scaling of the viscous scale.

In this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents ζ_n of the nth order correlation functions in fully developed hydrodynamic 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 [V.S. L'vov and I. Procaccia, Physica A (1998), in press, chao-dyn/9707015]. The scaling exponents in this set of equations cannot be found from power counting. In this short paper we discuss in detail low order non-trivial closures of this infinite set of equations, and prove that these closures lead 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 analyze turbulent velocity signals in the atmospheric surface layer, obtained by pairs of probes separated by inertial-range distances parallel to the ground and (nominally) orthogonal to the mean wind. The Taylor microscale Reynolds number ranges up to 20 000. Choosing a suitable coordinate system with respect to the mean wind, we derive theoretical forms for second order structure functions and fit them to experimental data. The effect of flow anisotropy is small for the longitudinal component but significant for the transverse component. The data provide an estimate for a universal exponent from among a hierarchy that governs the decay of flow anisotropy with the scale size.

We show that both analytic and numerical evidence points to the existence of a critical angle of [formula presented][formula presented] in viscous fingers and diffusion-limited aggregates growing in a wedge. The significance of this angle is that it is the typical angular spread of a major finger. For wedges with an angle larger than [formula presented], two fingers can coexist. Thus a finger with this angular spread is a kind of building block for viscous fingering patterns and diffusion-limited aggregation clusters in radial geometry.

In a series of recent works it was proposed that shell models of turbulence exhibit inertial range scaling exponents that depend on the nature of the dissipative mechanism. If true, and if one could imply a similar phenomenon to Navier-Stokes turbulence, this finding would cast strong doubts on the universality of scaling in turbulence. In this Letter we propose that these \u201cnonuniversalities\u201d are just corrections to scaling that disappear when the Reynolds number goes to infinity.

The phenomenology of the scaling behavior of higher order structure functions of velocity differences across a scale R in turbulence should be built around the irreducible representations of the rotation symmetry group. Every irreducible representation is associated with a scalar function of R which may exhibit different scaling exponents. The common practice of using moments of longitudinal and transverse fluctuations mixes different scalar functions and therefore may mix different scaling exponents. It is shown explicitly how to extract pure scaling exponents for correlation functions of arbitrary orders.

On the basis of the Navier-Stokes equations, we develop the high Reynolds number statistical theory of different-time, many-point spatial correlation functions of velocity differences. We find that their time dependence is not scale invariant: n-order correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents. We derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents [Formula Presented] of the standard structure functions. We propose a representation of the time dependence using the Legendre-transform formalism of multifractals that automatically reproduces all the newly found bridge relationships.

All statistical models of turbulence take into account Kolmogorov's exact result known as the "4/5 law" which stems from energy conservation. This law states that the energy flux expressed as a spatial derivative of the 3rd order velocity correlator equals the rate of energy dissipation. We have found an additional exact result which stems from the conservation of helicity in turbulence without inversion symmetry. It equates the flux of helicity expressed as a second spatial derivative of the 3rd order velocity correlator with the rate of helicity dissipation. This exact result must be incorporated to all statistical theories of turbulence with helicity. After submitting this paper for publication we learned that the main result was independently found by Otto Chkhetiani in JETP Lett . 63, 808 (1996).

The anomalous scaling behavior of the nth order correlation functions [formula presented] of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation B-|[formula presented][formula presented]=0. Previous analysis found zero modes in perturbation theory with respect to a small parameter. We present a computer-assisted analysis of the simplest nontrivial case of n=3: we demonstrate nonperturbatively the existence of anomalous scaling, and compare the results with the perturbative predictions.

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws S-n(R)similar to R(zeta n), and the correlation of energy dissipation K-epsilon epsilon(R)similar to R(-mu). The goal is to understand from first principles what is the mechanism that is responsible for changing the exponents zeta(n) and mu from their classical Kolmogorov values. In paper II of this series [V. S. L'vov and I. Procaccia, Phys. Rev. E 52, 3858 (1995)] it was shown that the existence of an ultraviolet scale (the dissipation scale eta) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted Delta. The exact resummation of ladder diagrams resulted in a ''bridging relation,'' which determined Delta in terms of zeta(2): Delta = 2-zeta(2). In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e., zeta(n) not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from the classical Kolmogorov 1941 theory scaling of S-n(R) (zeta(n) not equal n/3) must appear if the correlation of dissipation is mixing (i.e., mu>0). We suggest possible scenarios for multiscaling, and discuss the implication of these scenarios on the values of the scaling exponents zeta(n) and their ''bridge'' with mu.

We consider turbulent advection of a scalar field [Formula Presented], passive or active, and focus on the statistics of gradient fields conditioned on scalar differences [Formula Presented] across a scale [Formula Presented]. In particular we focus on two conditional averages [Formula Presented] and [Formula Presented]. We find exact relations between these averages, and with the help of the fusion rules we propose a general representation for these objects in terms of the probability density function [Formula Presented] of [Formula Presented]. These results offer a way to analyze experimental data that is presented in this paper. The main question that we ask is whether the conditional average [Formula Presented] is linear in [Formula Presented]. We show that there exists a dimensionless parameter which governs the deviation from linearity. The data analysis indicates that this parameter is very small for passive scalar advection, and is generally a decreasing function of the Rayleigh number for the convection data.

The lectures presented by one of us (IP) at the Les Houches summer school dealt with the scaling properties of high Reynolds number turbulence in fluid flows. The results presented are available in the literature and there is no real need to reproduce them here. Quite on the contrary, some of the basic tools of the field and theoretical techniques are not available in a pedagogical format, and it seems worthwhile to present them here for the benefit of the interested student. We begin with a detailed exposition of the naive perturbation theory for the ensemble averages of hydrodynamic observables (the mean velocity, the response functions and the correlation functions). The effective expansion parameter in such a theory is the Reynolds number (Re); one needs therefore to perform infinite resummations to change the effective expansion parameter. We present in detail the Dyson-Wyld line resummation which allows one to dress the propagators, and to change the effective expansion parameter from Re to O(1). Next we develop the ``dressed vertex" representation of the diagrammatic series. Lastly we discuss in full detail the path-integral formulation of the statistical theory of turbulence, and show that it is equivalent order by order to the Dyson-Wyld theory. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field.

In this paper we address nonperturbative aspects of the analytic theory of hydrodynamic turbulence. Of paramount importance for this theory are the "fusion rules" that describe the asymptotic properties of [Formula Presented]-point correlation functions when some of the coordinates tend toward one other. We first derive here, on the basis of two fundamental assumptions, a set of fusion rules for correlations of velocity differences when all the separations are in the inertial interval. Using this set of fusion rules we consider the standard hierarchy of equations relating the [Formula Presented]-order correlations (originating from the viscous term in the Navier-Stokes equations) to [Formula Presented] order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain of equations is decoupled in the sense that the correlations of [Formula Presented] order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of equations when some separations go to zero we derive, on the basis of the Navier-Stokes equations, a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every [Formula Presented]-order correlation function of velocity differences [Formula Presented] exhibits its own crossover length [Formula Presented] to dissipative behavior as a function of, say, [Formula Presented]. This length depends on [Formula Presented] and on the remaining separations [Formula Presented] When all these separations are of the same order [Formula Presented] this length scales as [Formula Presented] with [Formula Presented], with [Formula Presented] being the scaling exponent of the [Formula Presented]-order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents [Formula Presented] of the [Formula Presented]-order structure functions. One of these relations is the well known "bridge relation" for the scaling exponent of dissipation fluctuations [Formula Presented].

The roughening of expanding flame fronts by the accretion of cusplike singularities is a fascinating example of the interplay between instability, noise, and nonlinear dynamics that is reminiscent of self-fractalization in Laplacian growth patterns. The nonlinear integro-differential equation that describes the dynamics of expanding flame fronts is amenable to analytic investigations using pole decomposition. This powerful technique allows the development of a satisfactory understanding of the qualitative and some quantitative aspects of the complex geometry that develops in expanding flame fronts.

Fusion rules in turbulence address the asymptotic properties of many-point correlation functions when some of the coordinates are very close to each other. Here we put to the experimental test some nontrivial consequences of the fusion rules for scalar correlations in turbulence. To this aim we examine passive turbulent advection as well as convective turbulence. Adding one assumption to the fusion rules, one obtains a prediction for universal conditional statistics of gradient fields. We examine the conditional average of the scalar dissipation field [Formula Presented] for [Formula Presented] in the inertial range and find that it is linear in [Formula Presented] with a fully determined proportionality constant. The implications of these findings for the general scaling theory of scalar turbulence are discussed.

In paper I of this series on fluid turbulence we showed that exact resummations of the perturbative theory of the structure functions of velocity differences result in a finite (order by order) theory. These findings exclude any known perturbative mechanism for anomalous scaling of the velocity structure functions. In this paper we continue to build the theory of turbulence and commence the analysis of nonperturbative effects that form the analytic basis of anomalous scaling. Starting from the Navier-Stokes equations (at high Reynolds number Re) we discuss the simplest examples of the appearance of anomalous exponents in fluid mechanics. These examples are the nonlinear (four-point) Green's function and related quantities. We show that the renormalized perturbation theory for these functions contains ''ladder'' diagrams with (convergent) logarithmic terms that sum up to anomalous exponents. Using a sum rule that is derived here we calculate the leading anomalous exponent and show that it is critical. This result opens up the possibility of multiscaling of the structure functions with the outer scale of turbulence as the renormalization length. This possibility will be discussed in detail in a concluding paper of this series.

We present a theoretical framework for the discussion of the scaling properties of interfaces advancing in systems with quenched disorder. In all such systems there are critical conditions at which the interface gains scale invariance for sufficiently slow growth. There are two fundamental concepts, the ''blocking surfaces'' and the ''associated processes,'' whose nature determines the scaling properties of the advancing interfaces at criticality. The associated processes define a network whose scaling properties determine all the exponents (static and dynamic) that characterize the critical growing interface via universal scaling relations. We point out in this paper that most of the physical rules that can be used to advance the interface also incorporate noncritical elements; as a result, the roughness exponent of the growing interface may deviate from that of the critical interface in a rule-dependent way. We illustrate the wide applicability of the universal scaling relations with diverse models, such as the Edwards-Wilkinson (EW) model with quenched noise, the random-field Ising model, and the Kardar-Parisi-Zhang (KPZ) model with quenched noise. It is shown that the last model is characterized by bounded slopes, whereas in the EW model the slopes are unbounded. This fact makes the KPZ model equivalent to the self-organized interface depinning model of Buldyrev and Sneppen.

We examine the possibility of a blowup of the vorticity due to self-stretching and mechanisms for its prevention. We first estimate directly from the Navier-Stokes equations the length scale of coherence in the direction of the vortex lines to be of the order of the Kolmogorov length. Alignment of vortex lines is seen to be a viscous phenomenon and may prevent some scenarios for blowup. Next we derive equations for the curvature and torsion of vortex lines. We show that the same stretching that amplifies the vorticity also tends to straighten out the vortex lines. Then we show that in well-aligned vortex tubes, the self-stretching rate of the vorticity is proportional to the ratio of the vorticity and the radius of curvature. Thus blowup of the vorticity in such tubes can be prevented by the growth of the vorticity being balanced by the straightening of the vortex lines. Implications for vorticity-strain alignment and the scaling theory of turbulence are noted. Finally, we examine the effects of viscous diffusion on the vorticity field and see how viscosity can lead to organization and alignment of vortex lines.

We present a theoretical analysis of a recently introduced interface growth model with a global search of optimal growth sites, realized by quenched random forces. The interest in this model lies in the fact that it yields a variety of long-ranged correlated phenomena, which are characterized by scaling exponents that differ from the Kardar-Parisi-Zhang universality class [Phys. Rev. Lett. 56, 889 (1986)], in closer correspondence with experimental observations. It is shown that all the phenomenological findings can be recovered from the theory, and all the exponents found can be computed from the knowledge of one exponent of the directed percolation problem.

A Comment on the Letter by C. Jayaprakash, F. Hayot, and R. Pandit, Phys. Rev. Lett. 71, 12 (1993).

Recent experiments on the dispersal of a dye by chaotic surface waves indicate that the iso-concentration curves exhibit fractal geometry on a significant range of scales. In this letter we offer a rigorous theory in which the dimension of the iso-concentration curves is calculated. It is shown that the results are universal for all types of surface waves without mean advection, and are in good agreement with the experimental measurements.

We develop a theory that is nonperturbative and free of uncontrolled approximations to understand scaling behavior in turbulence. The main tool is a connection between the dimension of the graphs of the hydrodynamic fields and the scaling exponents of their structure functions. The connection is developed in some generality for both scalar and vector fields, in terms of the geometric invariants of the gradient tensor. We show that fluid mechanics is consistent with fractal graphs for both the scalar and the vector fields, and explain how this leads to the scaling behavior of the structure functions. We derive scaling relations between various scaling exponents, and show that in the case of "strong scaling" (which is defined below) the Kolmogorov solution is unique. Our theory allows additional solutions in which a weaker version of scaling results in a spectrum of scaling exponents. In particular, we identify the dimensionless (but Reynolds-number-dependent) contributions which can lead to deviations from the Kolmogorov exponents (which are derived using dimensional analysis). Results for the dimensions of fractal level sets in hydrodynamic turbulence which are measured in experiments and simulations follow immediately from this theory.

It is shown that the scale invariant solutions of the KS and KPZ models of surface roughening are identical for dimensions d2 in the strong coupling limit. For d>2, these models posses two different scaling solutions, one with d-independent scaling exponents y=z=2, and the other with d-dependent nontrivial exponents. The first of these solutions is realizable in one of these models, but not the other. These conclusions are valid to all orders in renormalized perturbation theory.

The long-wavelength properties of the Kuramoto-Sivashinsky equation are studied in 2+1 dimensions using numerical and analytic techniques. It is shown that this equation is not in the universality class of the Kardar-Parisi-Zhang model. Its roughening exponents are (up to logarithmic corrections) like those of the free-field theory, with dimension 2 being the marginal dimension for roughening. Assuming that the solution has logarithmic corrections, we derive a scaling relation for the exponents of the logarithmic terms. This solution is consistent order by order with the Dyson-Wyld diagrams. We explain why previous renormalization-group treatments failed.

A mechanism for nucleating pairs of topological defects in non- equilibrium pattern-forming systems in two space dimensions plus time is studied. Its relation to the creation of phase slips in space-time plane is revealed, stressing its deterministric origin from smooth dynamics. The role of this mechanism in the context of spatiotemporal chaos is discussed.

By studying typical hydrodynamic systems in the large aspect ratio limit, the authors show that spatio-temporal chaos is a natural consequence of the availability of secondary hydrodynamic instabilities. The study of such systems beyond the onset of secondary instabilities shows that, within perturbation theory, there exist stationary, spatially biperiodic solutions. Going beyond perturbation theory, it is found that a KAM mechanism is responsible for creating stationary, spatially chaotic solutions. Second, these solutions are shown to have stable and unstable manifolds in function space. They therefore conjecture that, typically, the time evolution is attracted to one of these spatially chaotic states and is then repelled along an unstable direction. In this second stage, topological defects are created if the system is sufficiently wide and couples to two-dimensional modes. Thus, they identify spatio-temporal chaos with a mechanism of generating a random array of defects in an already spatially disordered system.

The scaling exponents of two-point correlation functions in nonisotropic convective turbulence are evaluated on the basis of the Boussinesq equations. The scaling indices differ significantly from Kolmogorovs scaling for isotropic turbulence. The physical reason for this difference is the length dependent buoyancy forcing in this case, in contrast to forcing on the largest scale only in Kolmogorovs picture. The calculation appears to be in agreement with preliminary experimental findings in convective turbulence with helium as the working fluid.