Breakdown Of Self-similarity Research Articles

Non-self-similar light transport in scattering media

Transport processes underpin a wide variety of phenomena, ranging from chemistry, to physics and ecology. Despite their pervasiveness, however, several distinctive features of these processes are still elusive, making it difficult to recognize and classify the associated transport regimes. Using light scattering as a probe to explore different propagation regimes, we report on the experimental observation of non-self-similar light transport through turbid membranes. Our results show that a breakdown of self-similarity can arise for light waves even in the presence of isotropic and homogeneous disorder, and can be tuned by varying the turbidity of the system. By introducing the concept of self-similarity for light propagation, we provide a unified framework for the classification of light transport regimes—overcoming the dichotomy between normal and anomalous diffusion—and show that non-self-similar propagation is a common and experimentally accessible phenomenon. This insight can help to understand and model other scenarios where light transport is dominated by rare propagation events, such as in nonlinear and active media, but also in other fields of research beyond optics. Published by the American Physical Society 2024

Progressive polymer deformation induced by polar activity and the influence of inertia

Polar activity is shown here to induce a progressive local deformation of linear polymer chains, making a clear distinction between head and tail, while the overall chain conformation gets more compact. This breakdown of self-similarity, provoked by the accumulated tension on the polymer backbone induced by the activity, is shown to occur both in the absence and presence of inertia, although it is stronger in the latter case. Other properties like the relaxation time and diffusion are also largely influenced by the presence of a polar activity. Published by the American Physical Society 2024

Characterizing the Solar Cycle Variability Using Nonlinear Time Series Analysis at Different Amounts of Dynamo Supercriticality: Solar Dynamo is Not Highly Supercritical

The solar dynamo is essentially a cyclic process in which the toroidal component of the magnetic field is converted into the poloidal one and vice versa. This cyclic loop is disturbed by some nonlinear and stochastic processes mainly operating in the toroidal to poloidal part. Hence, the memory of the polar field decreases in every cycle. On the other hand, the dynamo efficiency and, thus, the supercriticality of the dynamo decreases with the Sun’s age. Previous studies have shown that the memory of the polar magnetic field decreases with the increase of supercriticality of the dynamo. In this study, we employ popular techniques of time series analysis, namely, compute Higuchi’s fractal dimension, Hurst exponent, and Multi-Fractal Detrended Fluctuation Analysis to the amplitude of the solar magnetic cycle obtained from dynamo models operating at near-critical and supercritical regimes. We show that the magnetic field in the near-critical regime is governed by strong memory, less stochasticity, intermittency, and breakdown of self-similarity. On the contrary, the magnetic field in the supercritical region has less memory, strong stochasticity, and shows a good amount of self-similarity. Finally, applying the same time series analysis techniques in the reconstructed sunspot data of 85 cycles and comparing their results with that from models, we conclude that the solar dynamo is possibly operating near the critical regime and not too much supercritical regime. Thus the Sun may not be too far from the critical dynamo transition.

Entanglement spectrum and quantum phase diagram of the long-range XXZ chain

Entanglement is a central feature of many-body quantum systems and plays a unique role in quantum phase transitions. In many cases, the entanglement spectrum, which represents the spectrum of the density matrix of a bipartite system, contains valuable information beyond the sole entanglement entropy. Here we investigate the entanglement spectrum of the long-range XXZ model. We show that within the critical phase it exhibits a remarkable self-similarity. The breakdown of self-similarity and the transition away from a Luttinger liquid is consistent with renormalization group theory. Combining the two, we are able to determine the quantum phase diagram of the model and locate the corresponding phase transitions. Our results are confirmed by numerically-exact calculations using tensor-network techniques. Moreover, we show that the self-similar rescaling extends to the geometrical entanglement as well as the Luttinger parameter in the critical phase. Our results pave the way to further studies of entanglement properties in long-range quantum models.

Two Empirical Double-Corner-Frequency Source Spectra and Their Physical Implications

ABSTRACTWe introduce double-corner-frequency (DCF) source spectral models JA19 and JA19_2S, which, in conjunction with a stochastic ground-motion model, can reproduce the mean peak ground acceleration (PGA) and mean peak ground velocity (PGV) of the Next Generation Attenuation-West 2 database for magnitudes 3.3–7.3. Their displacement amplitude spectrum remains constant for frequencies less than fc1, decays as f−1 between fc1 and fc2, and decays as f−2 for frequencies greater than fc2. The model JA19 is self-similar. Its two corner frequencies fc1 and fc2 scale with moment magnitude (M) as (1) log(fc1(M))=1.754−0.5M and (2) log(fc2(M))=3.250−0.5M. We find that relation (1) is consistent with the known self-similar scaling relations of the rupture duration (Td), in which Td=1/(πfc1). Relation (2) may reflect the scaling relation of the average rise time (TR), where TR∼0.8/(fc2). Stochastic simulations of ground motion using JA19 cannot reproduce the sharp change in magnitude dependence of PGA and PGV at M 5.3, suggesting a breakdown of self-similarity. The magnitude dependence of PGA and PGV and this change in slope is well explained by JA19_2S, which results from perturbing the fc1 scaling relationship in JA19. For JA19_2S: log(fc1(M))=1.474−0.415M for M≤5.3; log(fc1(M))=2.375−0.585M for M>5.3. The scaling relation for fc2 is unchanged. When fc1≪fc2, the scaled energy (ratio of radiated energy and seismic moment) scales with M0fc12fc2. The scaled energy of JA19 is 2.2×10−5, independent of magnitude. Because JA19_2S is not self-similar, its scaled energy is 2.2–4.7×10−5, increasing 2.2 times, when magnitude increases from 3.3 to 5.3, and, subsequently decreasing 2.2 times, as magnitude further increases from 5.3 to 7.3. Both agree with the global average (∼3×10−5) reported previously. Using our proposed empirical models, the standard deviation of average static stress drop from seismological studies can be significantly greater than the standard deviation of the stress parameter used to estimate PGA and PGV.

The breakdown of self-similarity in electrified counterflow diffusion flames

This work addresses the question of the validity of self-similar formulations in describing the structures of methane/air laminar counterflow diffusion flames subjected to incident sub-breakdown DC electric fields. The electric field is induced by two flat porous electrodes located on the oxidizer and fuel sides of the burner and arranged parallel to the mixing layer. Both experiments and numerical simulations of this configuration in recent work suggest the presence of a strong coupling effect between the aerothermochemical and electric fields whereby the velocity field is significantly modified by the momentum carried by a bi-directional ionic wind directed axially outwards from the diffusion flame. However, as shown in this study, such strong coupling is incompatible with standard self-similar formulations of the problem. An a-priori analysis of the steady axisymmetric numerical simulations results in Di Renzo et al. (2018) [1], which employ multi-component transport and detailed chemical kinetics, is presented in this study in order to address the suitability of self-similar descriptions in the present configuration. It is shown that, while self-similarity is preserved in unelectrified conditions along radial distances similar to one orifice radius, it breaks down profusely in electrified conditions as the applied voltage increases and nears saturation conditions, where the electric force field becomes two-dimensional and non-conservative in the close vicinity of the burner axis. As a result, for the purposes of self-similarity, increasing electrification counteracts the slenderness of the counterflow burner and decreases its effective aspect ratio. Counterflow burners should therefore be extra slender if preservation of self-similar conditions is sought under incident electric fields.

Experimental measurement of residual stress and distortion in additively manufactured stainless steel components with various dimensions

Disk-shaped 316L stainless steel parts with various diameters and heights were additively manufactured using a direct metal laser sintering (DMLS) technique. Neutron diffraction was used to profile the residual stresses in the samples before and after removal of the build plate and support structures. Moreover, distortion level of the parts before and after the removal was quantified using a coordinate measuring machine (CMM). Large tensile in-plane stresses (up to ≈ 400MPa) were measured near the as-built disk top surfaces, where the stress magnitude decreased from the disk center to the edges. The stress gradient was steeper for the disks with smaller diameters and heights. Following the removal of the build plate and support structures, the magnitude of the in-plane residual stresses decreased dramatically (up to 330MPa) whereas the axial stress magnitude did not change significantly. The stress relaxation caused the disks to distort, where the distortion metric was higher for the disks with smaller diameters and heights. The distribution of the residual stresses revealed a marked breakdown of self-similarity in their distribution even comparing disk-shaped samples that were fabricated under identical printing parameters; the stress field profiles were not linearly scaled as a function of height and diameter.

Source Parameters of Earthquakes Recorded Near the Itoiz Dam (Northern Spain)

We calculate the source parameters and attenuation from earthquakes recorded near the Itoiz dam, from 2004 to 2009, with magnitudes ranging between 1.2 and 5.2. We use a Genetic Algorithm in order to fit the three-component P-wave spectra with the spectral level, corner frequency, and attenuation factor as searching parameters. The obtained moments range from 1.72 × 1011 to 2.65 × 1015 Nm, the radii span from 0.09 to 1.00 km, and the stress drops vary from 0.006 to 29.462 MPa. The maximum value for the Q attenuation factor is 794, and the minimum value is 53. We find a good agreement between empirical and theoretical relationships between moment and magnitude. There seems to be a breakdown of self-similarity, but it could be due to the method used. We group the data by means of a Self-Organizing Map and the clusters found are related by their magnitude, and not by other considerations.

Overdetermined shooting methods for computing standing water waves with spectral accuracy

A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counter-propagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth. In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high order Runge-Kutta or spectral deferred correction method in time, and quadruple-precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for GPU acceleration are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.

On non-self-similar regimes in homogeneous isotropic turbulence decay

AbstractBoth theoretical analysis and eddy-damped quasi-normal Markovian (EDQNM) simulations are carried out to investigate the different decay regimes of an initially non-self-similar isotropic turbulence. Breakdown of self-similarity is due to the consideration of a composite three-range energy spectrum, with two different slopes at scales larger than the integral length scale. It is shown that, depending on the initial conditions, the solution can bifurcate towards a true self-similar decay regime, or sustain a non-self-similar state over an arbitrarily long time. It is observed that these non-self-similar regimes cannot be detected, restricting the observation to time exponents of global quantities such as kinetic energy or dissipation. The actual reason is that the decay is controlled by large scales close to the energy spectrum peak. This theoretical prediction is assessed by a detailed analysis of triadic energy transfers, which show that the largest scales have a negligible impact on the total transfers. Therefore, it is concluded that details of the energy spectrum near the peak, which may be related to the turbulence production mechanisms, are important. Since these mechanisms are certainly not universal, this may at least partially explain the significant discrepancies that exist between experimental data and theoretical predictions. Another conclusion is that classical self-similarity theories, which connect the asymptotic behaviour of either the energy spectrum $E(k\ensuremath{\rightarrow} 0)$ or the velocity correlation function $f(r\ensuremath{\rightarrow} + \infty )$ and the turbulence decay exponent, are not particularly relevant when the large-scale spectrum shape exhibits more than one range.

Breakdown of Self-Similarity at the Crests of Large-Amplitude Standing Water Waves

We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes's sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.

Self-similarity and finite-size effects in nano-indentation of highly cross-linkedpolymers

The scalability of thermomechanical polymer deformations in the sub-10 nm regime is ofparticular importance for nano-imprint techniques, hardness measurements of thin films bynano-indentations, and scanning-probe-based thermomechanical data storage. Weinvestigate nano-indentation in the sub-10 nm regime performed on highly cross-linkedpolymer films of different thicknesses. It is shown that the lateral and vertical geometriccharacteristics of the indents independently scale down to an indent depth of1 nm and that the scaling parameters are functions of the film thickness and thetemperature of the indenter. However, in the limit of shallow indents the scaling ofthe cross-coupling between lateral and vertical dimensions is lost. It is arguedthat the breakdown of self-similarity is due to a minimum strain requirementoriginating from the co-operative nature of the polymer response induced byα transitions which lock the indent in the deformed state. The results shed new light on thefundamental processes and size effects involved in nanoscale plastic replication, in general.

Cluster–cluster aggregation as an analogue of a turbulent cascade: Kolmogorov phenomenology, scaling laws and the breakdown of self-similarity

We present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers. We emphasize the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean-field description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterized by a constant flux of mass directed from small to large masses. It therefore admits a phenomenological description based on the assumption of self-similarity and constant mass flux analogous to the Kolmogorov’s 1941 theory of turbulence. Unlike turbulence, the aggregation problem is analytically tractable using powerful methods of statistical field theory. We explain in detail how these methods should be adapted to study the far-from-equilibrium, flux-dominated states characteristic of turbulent systems. We consider multipoint correlation functions of the mass density. By an exact evaluation of the scaling exponents for the one and two-point correlation functions, we show that the assumption of self-similiarity breaks down at large masses for spatial dimensions, d ≤ 2 . We calculate non-rigorously the exponents of the higher order correlation functions as an ϵ -expansion where ϵ = 2 − d . We show that the mass distribution exhibits non-trivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in d ≤ 2 . The analytic methods developed here will have more general applicability beyond the study of this specific problem.

Species turnover at small scales in dune slack plant communities

Summary Patterns of both species accumulation with increasing area and of individual species occurrences depend on the scale level considered. This study investigated community diversity and individual species turnover patterns between four scale levels within 2×2 m 2 nested plots situated in a dune slack plant community. The number of species increased with plot area following a log–log function, with a slope of 0.23. However, species turnover was higher between the lowest scale levels, indicating limitations on species occurrences at the 25×25 cm 2 scale level. Alpha diversity in rectangular plots was significantly higher than in square plots of the same area. There were strong differences between individual species turnover patterns. Most species occurrence patterns had a box-counting fractal dimension value between 0.8 and 1.6, which is rather low compared with other studies on larger scale levels. Analyses of occurrence probabilities and scale area plots showed that there is a systematic deviation from self-similarity at the smallest scale level. Species had a lower frequency than expected from a fractal distribution, suggesting a higher level of species aggregation. The higher species diversity turnover at the smallest scale level can be linked to a higher spatial aggregation of individual species, due to biotic or abiotic limitations on their occurrence. These results confirm the general nature of the pattern of break-down of self-similarity at the smallest scale level considered.

Free decay of turbulence and breakdown of self-similarity

It has been generally assumed, since the work of von Kármán and Howarth in 1938, that free decay of fully-developed turbulence is self-similar. Here we present a simple phenomenological model of the decay of three-dimensional incompressible turbulence, which predicts breakdown of self-similarity for low-wavenumber spectral exponents n in the range nc<n<4, where nc is some threshold wavenumber. Calculations with the eddy-damped quasi-normal Markovian approximation give the value as nc≈3.45. The energy spectrum for this range of exponents develops two length-scales, separating three distinct wavenumber ranges.

Self-Similar Decay in the Kraichnan Model of a Passive Scalar

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0<ζ<2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent a≤(d/γ)+1 are statistically realizable, where d is space dimension and γ=2−ζ. An infinite sequence of invariants Jp, p=0, 1, 2,..., is pointed out, where J0 is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J0 or J1 must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being Jp—converges at long times to a scaling solution Φp with a=(d/γ)+p, p=0, 1. The latter belongs to an exceptional series of self-similar solutions with stretched-exponential decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a “two-scale” decay with breakdown of self-similarity for a range of exponents (d+γ)/γ<a<(d+2)/γ, analogous to what has recently been found in the decay of Burgers turbulence.

Self-similarity and coarsening of three dimensional particles on a one or two dimensional matrix

We examine the validity of the hypothesis of self-similarity in systems coarsening under the driving force of interface energy reduction in which three dimensional particles are intersected by a one or two dimensional diffusion matrix. In both cases, solute fluxes onto the surface of the particles, assumed spherical, depend on both particle radius and interparticle distance. We argue that overall mass conservation requires independent scalings for particle sizes and interparticle distances under magnification of the structure, and predict power law growth for the average particle size in the case of a one dimensional matrix (3D/1D), and a weak breakdown of self-similarity in the two dimensional case (3D/2D). Numerical calculations confirm our predictions regarding self-similarity and power law growth of average particle size with an exponent 1/7 for the 3D/1D case, and provide evidence for the existence of logarithmic factors in the laws of boundary motion for the 3D/2D case. The latter indicate a weak breakdown of self-similarity.

Remarks on the renormalization group in statistical fluid dynamics

A variant of the renormalization group is applied to the problem of randomly forced fluids studied by Forster, Nelson, and Stephen [Phys. Rev. A 16, 732 (1977)] and others. Amplitude factors (thought to be nonuniversal by some authors) are evaluated and shown to have universal values. Comparisons with closures are made. The possibility of a breakdown of self-similarity and/or universality due to intermittency effects is discussed.