Scale-invariant Regime Research Articles

Scaling regimes of the one-dimensional phase turbulence in the deterministic complex Ginzburg-Landau equation.

We consider the one-dimensional deterministic complex Ginzburg-Landau equationin the regime of phase turbulence, where the order parameter displays a defect-free chaotic phase dynamics, which maps to the Kuramoto-Sivashinsky equation, characterized by negative viscosity and a modulational instability at linear level. In this regime, the dynamical behavior of the large wavelength modes is captured by the Kardar-Parisi-Zhang (KPZ) universality class, determining their universal scaling and their statistical properties. These modes exhibit the characteristic KPZ superdiffusive scaling with the dynamical critical exponent z=3/2. We present numerical evidence of the existence of an additional scale-invariant regime, with the dynamical exponent z=1, emerging at scales which are intermediate between the microscopic ones, intrinsic to the modulational instability, and the macroscopic ones. We argue that this new scaling regime belongs to the universality class corresponding to the inviscid limit of the KPZ equation.

Front propagation in cellular flows: scale dependence versus scale invariance

We experimentally study front propagation in a vortex lattice providing closed steady cellular flows and no mean flow. To this end, we trigger an autocatalytic reaction in a solution stirred by magnetohydrodynamic flows in a Hele-Shaw cell. We evidence a scale-invariant regime below some flow magnitude and a scale-dependent regime above, the scales referring here to the vortex scale and the front thickness. The transition between these regimes corresponds to a unitary Damköhler number $Da$ : $Da=1$ . The enhancement of the mean front velocity with the flow magnitude nicely agrees with the literature on numerical simulations and theoretical analyses in the scale-invariant regime $Da>1$ , but displays noticeable discrepancies in the scale-dependent one $Da<1$ . This shows that the transition between regimes is qualitatively sharp but quantitatively smooth.

Primordial power spectrum from a matter-ekpyrotic bounce scenario in loop quantum cosmology

A union of matter bounce and Ekpyrotic scenarios is often studied in an attempt to combine the most promising features of these two models. Since non-perturbative quantum geometric effects in loop quantum cosmology (LQC) result in natural bouncing scenarios without any violation of energy conditions or fine tuning, an investigation of matter-Ekpyrotic bounce scenario is interesting to explore in this quantum gravitational setting. In this work, we explore this unified phenomenological model for a spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe in LQC filled with dust and a scalar field in an Ekpyrotic scenario like negative potential. Background dynamics and the power spectrum of the comoving curvature perturbations are numerically analyzed with various initial conditions and a suitable choice of the initial states. By varying the initial conditions we consider different cases of dust and Ekpyrotic field domination in the contracting phase. We use the dressed metric approach to numerically compute the primordial power spectrum of the comoving curvature perturbations which turns out to be almost scale invariant for the modes which exit the horizon in the matter-dominated phase. But, in contrast with a constant magnitude power spectrum obtained under approximation of a constant Ekpyrotic equation of state using deformed algebra approach in an earlier work, we find that the magnitude of power spectrum changes during evolution. Our analysis shows that the bouncing regime only leaves imprints on the modes outside the scale-invariant regime. However, an analysis of the spectral index shows inconsistency with the observational data, thus making further improvements in such a model necessary.

S-PFM model for ideal grain growth

The phase-field method is increasingly used for studying grain growth in metallic alloys. Such an approach offers a thermodynamically consistent framework for studying microstructure evolution during grain growth without the need to explicitly track the interface positions. However, the numerical solution of the models requires a grid spacing much smaller than the interface width. This leads to computationally very intensive simulations, especially when a large number of grains is required to accurately measure statistical quantities. The recently proposed S-PFM approach provides a new inherently discrete formulation of phase field models where the interface width can be as small as the grid spacing, thus drastically improving the numerical performances of the method. Here, we show that this approach can be extended to a multi-phase field model for ideal grain growth. Then, we perform two dimensional simulations to analyse in detail the kinetics of both grain boundaries and triple junctions. We compare our model to a classical phase field formulation and we demonstrate that, for a prescribed accuracy, the memory requirement and simulation times are both reduced by a factor of 4D, where D is the space dimension. Finally, we perform a large-scale simulation of grain growth in two dimensions and show that the method is able to reveal the specific shape of the grain size distribution in the scale-invariant regime displaying two peaks around the mean grain size and quantitative measurements of the underlying topological classes are performed.

Solving the Bethe-Salpeter equation in Minkowski space for a fermion-scalar system

The ladder Bethe-Salpeter equation of a bound $(1/2{)}^{+}$ system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: (i) a massive scalar exchange and (ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully nonperturbative framework, as well as the onset of a scale-invariant regime.

Entanglement spectrum and Rényi entropies of nonrelativistic conformal fermions

We characterize non-perturbatively the R\'enyi entropies of degree n=2,3,4, and 5 of three-dimensional, strongly coupled many-fermion systems in the scale-invariant regime of short interaction range and large scattering length, i.e. in the unitary limit. We carry out our calculations using lattice methods devised recently by us. Our results show the effect of strong pairing correlations on the entanglement entropy, which modify the sub-leading behavior for large subsystem sizes (as characterized by the dimensionless parameter x=kF L_A, where kF is the Fermi momentum and L_A the linear subsystem size), but leave the leading order unchanged relative to the non-interacting case. Moreover, we find that the onset of the sub-leading asymptotic regime is at surprisingly small x=2-4. We provide further insight into the entanglement properties of this system by analyzing the spectrum of the entanglement Hamiltonian of the two-body problem from weak to strong coupling. The low-lying entanglement spectrum displays clear features as the strength of the coupling is varied, such as eigenvalue crossing, a sharp change in the Schmidt gap, and scale invariance at unitarity. Beyond the low-lying component, the spectrum appears as a quasi-continuum distribution, for which we present a statistical characterization; we find, in particular, that the mean shifts to infinity as the coupling is turned off, which indicates that that part of the spectrum represents non-perturbative contributions to the entanglement Hamiltonian. In contrast, the low-lying entanglement spectrum evolves to finite values in the noninteracting limit. The scale invariance of the unitary regime guarantees that our results are universal features intrinsic to 3D quantum mechanics and represent a well-defined prediction for ultracold atom experiments, which were recently shown to have direct access to the entanglement entropy.

Effect of inclusion density on ductile fracture toughness and roughness

Three dimensional calculations of ductile fracture under mode I plane strain, small scale yielding conditions are carried out using an elastic-viscoplastic constitutive relation for a progressively cavitating solid with two populations of void nucleating second phase particles. Larger inclusions that result in void nucleation at an early stage are modeled discretely while smaller particles that require large strains to nucleate voids are homogeneously distributed. Full field solutions are obtained for eight volume fractions, ranging from 1% to 19%, of randomly distributed larger inclusions. For each volume fraction calculations are carried out for seven random distributions of inclusion centers. Crack growth resistance curves and fracture surface roughness statistics are calculated using standard procedures. The crack growth resistance is characterized in terms of both JIC and the tearing modulus TR. For all volume fractions considered, the computed fracture surfaces are self-affine over a size range of nearly two orders of magnitude with a microstructure independent roughness exponent of 0.53 with a standard error of 0.0023. The cut-off length of the scale invariant regime is found to depend on the inclusion volume fraction. Consideration of the full statistics of the fracture surface roughness revealed other parameters that vary with inclusion volume fraction. For smaller values of the discretely modeled inclusion volume fraction (≤7%), there is a linear correlation between several measures of fracture surface roughness and both JIC and TR. In this regime crack growth is dominated by a void-by-void process. For greater values of the discretely modeled inclusion volume fraction, crack growth mainly involves multiple void interactions and no such correlation is found.

Spike avalanches exhibit universal dynamics across the sleep-wake cycle.

BackgroundScale-invariant neuronal avalanches have been observed in cell cultures and slices as well as anesthetized and awake brains, suggesting that the brain operates near criticality, i.e. within a narrow margin between avalanche propagation and extinction. In theory, criticality provides many desirable features for the behaving brain, optimizing computational capabilities, information transmission, sensitivity to sensory stimuli and size of memory repertoires. However, a thorough characterization of neuronal avalanches in freely-behaving (FB) animals is still missing, thus raising doubts about their relevance for brain function.Methodology/Principal FindingsTo address this issue, we employed chronically implanted multielectrode arrays (MEA) to record avalanches of action potentials (spikes) from the cerebral cortex and hippocampus of 14 rats, as they spontaneously traversed the wake-sleep cycle, explored novel objects or were subjected to anesthesia (AN). We then modeled spike avalanches to evaluate the impact of sparse MEA sampling on their statistics. We found that the size distribution of spike avalanches are well fit by lognormal distributions in FB animals, and by truncated power laws in the AN group. FB data surrogation markedly decreases the tail of the distribution, i.e. spike shuffling destroys the largest avalanches. The FB data are also characterized by multiple key features compatible with criticality in the temporal domain, such as 1/f spectra and long-term correlations as measured by detrended fluctuation analysis. These signatures are very stable across waking, slow-wave sleep and rapid-eye-movement sleep, but collapse during anesthesia. Likewise, waiting time distributions obey a single scaling function during all natural behavioral states, but not during anesthesia. Results are equivalent for neuronal ensembles recorded from visual and tactile areas of the cerebral cortex, as well as the hippocampus.Conclusions/SignificanceAltogether, the data provide a comprehensive link between behavior and brain criticality, revealing a unique scale-invariant regime of spike avalanches across all major behaviors.

Geometric scale invariance as a route to macroscopic degeneracy: Loading a toroidal trap with a Bose or Fermi gas

An easily scalable toroidal geometry presents an opportunity for creating large scale persistent currents in Bose-Einstein condensates, for studies of the Kibble-Zurek mechanism, and for investigations of toroidally trapped degenerate Fermi gases. We consider in detail the process of isentropic loading of a Bose or Fermi gas from a harmonic trap into the scale invariant toroidal regime that exhibits a high degree of system invariance when increasing the radius of the toroid. The heating involved in loading a Bose gas is evaluated analytically and numerically, both above and below the critical temperature. Our numerical calculations treat interactions within the Hartree-Fock-Bogoliubov-Popov theory. Minimal change in degeneracy is observed over a wide range of initial temperatures, and a regime of cooling is identified. The scale invariant property is further investigated analytically by studying the density of states of the system, revealing the robust nature of scale invariance in this trap, for both bosons and fermions. We give analytical results for a Thomas-Fermi treatment. We calculate the heating due to loading a spin-polarized Fermi gas and compare with analytical results for high and low temperature regimes. The Fermi gas is subjected to irreducible heating during loading, caused by the loss of one degree of freedom for thermalization.

Linguistic Complexity: English vs. Polish, Text vs. Corpus

We analyze the rank-frequency distributions of words in selected English and Polish texts. We show that for the lemmatized (basic) word forms the scale-invariant regime breaks after about two decades, while it might be consistent for the whole range of ranks for the inflected word forms. We also find that for a corpus consisting of texts written by dierent authors the basic scale-invariant regime is broken more strongly than in the case of comparable corpus consisting of texts written by the same author. Similarly, for a corpus consisting of texts translated into Polish from other languages the scale-invariant regime is broken more strongly than for a comparable corpus of native Polish texts. Moreover, we find that if the words are tagged with their proper part of speech, only verbs show rank-frequency distribution that is almost scale-invariant.

Scale-invariant thermodynamics of a toroidally trapped Bose gas

We consider a system of bosonic atoms in an axially symmetric harmonic trap augmented with a two-dimensional repulsive Gaussian optical potential. We find an expression for the grand free energy of the system for configurations ranging from the harmonic trap to the toroidal regime. For large tori we identify an accessible regime where the ideal-gas thermodynamics of the system are found to be independent of toroidal radius. This property is a consequence of an invariant extensive volume of the system that we identify analytically in the regime where the toroidal potential is radially harmonic. In considering corrections to the scale-invariant transition temperature, we find that the first-order interaction shift is the dominant effect in the thermodynamic limit and is also scale invariant. We also consider adiabatic loading from the harmonic-to-toroidal trap configuration, which we show to have only a small effect on the condensate fraction of the ideal gas, indicating that loading into the scale-invariant regime may be experimentally practical.

Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction

Motivated by the physics of graphene, we consider a model of $N$ species of $2+1$ dimensional four-component massless Dirac fermions interacting through a three dimensional instantaneous Coulomb interaction. We show that in the limit of infinitely strong Coulomb interaction, the system approaches a quantum critical point, at least for sufficiently large fermion degeneracy. In this regime, the system exhibits invariance under scale transformations in which time and space scale by different factors. The elementary excitations are fermions with dispersion relation $\ensuremath{\omega}\ensuremath{\sim}{p}^{z}$, where the dynamic critical exponent $z$ depends on $N$. In the limit of large $N$, we find $z=1\ensuremath{-}4∕({\ensuremath{\pi}}^{2}N)+O({N}^{\ensuremath{-}2})$. We argue that due to the numerically large Coulomb coupling, graphene (freely suspended) in a vacuum stays near the scale-invariant regime in a large momentum window, before eventually flowing to the trivial fixed point at very low momentum scales.

Coarsening of three-dimensional grains in crystals, or bubbles in dry foams, tends towards a universal, statistically scale-invariant regime

We perform extensive Potts model simulations of three-dimensional dry foam coarsening. Starting with 2.25 million bubbles, we have enough statistics to fulfill the three constraints required for the study of statistical scale invariance: first, enough time for the transient to end and reach the scaling state; then, enough time in the scaling state itself to characterize its properties; and finally, enough bubbles at the end to avoid spurious finite size effects. In the scaling state, we find that the average surface area of the bubbles increases linearly with time. The geometry (bubble shape and size) and topology (number of faces and edges), as well as their correlations, become constant in time. Their distributions agree with the data of the literature. We present an analytical model (universal, up to parameters extracted from the simulations) for a disordered foam minimizing its free energy, which agrees with the simulations. We discuss the limitations of the simulations and of the model.

Universal, statistically scale-invariant regime in 3D cellular systems

We perform extensive Potts model simulations of 3D dry foam coarsening and we report reaching a scaling state. Starting with 2.25 million bubbles, we have enough time for the transient to end and reach the scaling state; enough time in the scaling state to characterize its properties; and finally, enough bubbles at the end to avoid spurious finite size effects. In the scaling state, we find that the number of bubbles decreases as t - 1.5 and we obtain a time independent distribution in relative bubble size.

On the dynamical mechanism of cross-over from chaotic to turbulent states

The Portevin-Le Chatelier effect is one of the few examples of organization of defects. Here the spatio-temporal dynamics emerges from the cooperative behavior of the constituent defects, namely dislocations and point defects. Recent dynamical approach to the study of experimental time series reports an intriguing cross-over phenomenon from a low dimensional chaotic to an infinite dimensional scale invariant power-law regime of stress drops in experiments on CuAl single crystals and AlMg polycrystals, as a function of strain rate. We show that an extension of a dynamical model due to Ananthakrishna and coworkers for the Portevin-Le Chatelier effect reproduces this cross-over. At low and medium strain rates, the model shows chaos with the structure of the attractor resembling the reconstructed experimental attractor. At high strain rates, the model exhibits a power-law statistics for the magnitudes and durations of the stress drops as in experiments. Concomitantly, the largest Lyapunov exponent is zero. In this regime, there is a finite density of null exponents which itself follows a power law. This feature is similar to the Lyapunov spectrum of a shell model for turbulence. The marginal nature of this state is visualized through slow manifold approach.

Inflation in Realistic D-Brane Models

We find successful models of D-brane/anti-brane inflation within a string context. We work within the GKP-LT class of type IIB string vacua for which many moduli are stabilized through fluxes, as recently modified to include `realistic' orbifold sectors containing standard-model type particles. We allow all moduli to roll when searching for inflationary solutions and find that inflation is not generic inasmuch as special choices must be made for the parameters describing the vacuum. But given these choices inflation can occur for a reasonably wide range of initial conditions for the brane and antibrane. We find that D-terms associated with the orbifold blowing-up modes play an important role in the inflationary dynamics. Since the models contain a standard-model-like sector after inflation, they open up the possibility of addressing reheating issues. We calculate predictions for the CMB temperature fluctuations and find that these can be consistent with observations, but are generically not deep within the scale-invariant regime and so can allow appreciable values for dns/dln k as well as predicting a potentially observable gravity-wave signal. It is also possible to generate some admixture of isocurvature fluctuations.

Monte Carlo fixed scale transformation for nonlocal fractal growth models

The fixed scale transformation (FST) is a theoretical framework developed for the evaluation of scaling dimensions in a vast class of complex systems showing fractal geometric correlations. For models with long range interactions, such as Laplacian growth models, the identification by analytical methods of the transformation's basic elements is a very difficult task. Here we present a Monte Carlo renormalization approach which allows the direct numerical evaluation of the FST transfer matrix, overcoming the usual problems of analytical and numerical treatments. The scheme is explicitly applied to the diffusion limited aggregation case where a scale invariant regime is identified and the fractal dimension is computed. The Monte Carlo FST represents an alternative tool which can be easily generalized to other fractal growth models with nonlocal interactions.

Multicomponent turbulence, the spherical limit, and non-Kolmogorov spectra.

A set of models for homogeneous, isotropic turbulence is considered in which the Navier-Stokes equations for incompressible fluid flow are generalized to a set of N coupled equations in N velocity fields. It is argued that in order to be useful these models must embody a new group of symmetries, and a general formalism is laid out for their construction. The work is motivated by similar techniques that have had extraordinary success in improving the theoretical understanding of equilibrium phase transitions in condensed matter systems. We consider two classes of models: a simpler class (model I), which does not contain an exact Galilean symmetry, and a more complicated, extended class (model II), which does. The key result is that these models simplify when N is large. The so-called spherical limit N\ensuremath{\rightarrow}\ensuremath{\infty} can be solved exactly, yielding closed sets of nonlinear integral equations for the response and correlation functions. For model I, these equations, known as Kraichnan's direct interaction approximation equations, are solved fully in the scale-invariant turbulent regime. For model II, these equations are more complicated and their full solution is left for future work. Implications of these results for real turbulence (N=1) are discussed. In particular, it is argued that previously applied renormalization group techniques, based on an expansion in the exponent y that characterizes the driving spectrum, are incorrect and that the Kolmogorov exponent \ensuremath{\zeta} has a nontrivial dependence on N, with \ensuremath{\zeta}(N\ensuremath{\rightarrow}\ensuremath{\infty})=3/2 for both sets of models. This value is close to the experimental result \ensuremath{\zeta}\ensuremath{\simeq}5/3, which must therefore result from higher-order corrections in powers of 1/N. Prospects for calculating these corrections are briefly discussed: though daunting, such calculations might provide a controlled perturbation expansion for the Kolmogorov, and other, exponents. Our techniques may also be applied to other nonequilibrium dynamical problems, such as the Kardar-Parisi-Zhang equation for interface growth, and perhaps to turbulence in nonlinear wave systems.

Scale-invariant regime in Rayleigh-Taylor bubble-front dynamics.

A statistical model of Rayleigh-Taylor bubble fronts in two dimensions is introduced. Float and merger of bubbles lead to a scale-invariant regime, with a stable distribution of scaled bubble radii and a constant front acceleration. The model is solved for a simple merger law, showing that a family of such stable distributions exists. The basins of attraction of each of these are mapped. The properties of the scale-invariant distributions for various merger laws, including a merger law derived from the Sharp-Wheeler model, are analyzed. The results are in good agreement with computer simulations. Finally, it is shown that for some merger laws, a runaway bubble regime develops. A criterion for the appearance of runaway growth is presented.

Persistence of screening and self-criticality in the scale invariant dynamics of diffusion limited aggregation.

The origin of fractal properties in diffusion limited aggregation is related to the persistence of screening in the scale invariant growth regime. This effect is described by the effective noise reduction parameter S spontaneously generated by the scale invariant dynamics. The renormalization of this parameter under scale transformation shows the following: (i) The fixed point is attractive, implying the self-critical nature of the process. (ii) The fixed point value ${\mathit{S}}^{\mathrm{*}}$ is of the order of unity, showing that the small scale growth rules are already close to the scale invariant ones and that screening effects persisit in the asympotic regime.