Ordinary Scaling Research Articles

Increasing Pump-Probe Signal toward Asymptotic Limits.

Optimization of pump-probe signal requires a complete understanding of how signal scales with experimental factors. In simple systems, signal scales quadratically with molar absorptivity, and linearly with fluence, concentration, and path length. In practice, scaling factors weaken beyond certain thresholds (e.g., OD > 0.1) due to asymptotic limits related to optical density, fluence and path length. While computational models can accurately account for subdued scaling, quantitative explanations often appear quite technical in the literature. This Perspective aims to present a simpler understanding of the subject with concise formulas for estimating absolute magnitudes of signal under both ordinary and asymptotic scaling conditions. This formulation may be more appealing for spectroscopists seeking rough estimates of signal or relative comparisons. We identify scaling dependencies of signal with respect to experimental parameters and discuss applications for improving signal under broad conditions. We also review other signal enhancement methods, such as local-oscillator attenuation and plasmonic enhancement, and discuss respective benefits and challenges regarding asymptotic limits that signal cannot exceed.

Dimensional transmutation and nonconventional scaling behavior in a model of self-organized criticality

This paper addresses two unusual scaling regimes (types of critical behavior) predicted by the field-theoretic renormalization group analysis for a self-organized critical system with turbulent motion of the environment. The system is modeled by the anisotropic stochastic equation for a “running sandpile” introduced by Hwa and Kardar in [Phys. Rev. Lett. 62, 1813 (1989)]. The turbulent motion is described by the isotropic Kazantsev–Kraichnan “rapid-change” velocity ensemble for an incompressible fluid. The original Hwa–Kardar equation allows for independent scaling of the spatial coordinates [Formula: see text] (the coordinate along the preferred dimension) and [Formula: see text] (the coordinates in the orthogonal subspace to the preferred direction) that becomes impossible once the isotropic velocity ensemble is coupled to the equation. However, it is found that one of the regimes of the system’s critical behavior (the one where the isotropic turbulent motion is irrelevant) recovers the anisotropic scaling through “dimensional transmutation.” The latter manifests as a dimensionless ratio acquiring nontrivial canonical dimension. The critical regime where both the velocity ensemble and the nonlinearity of the Hwa–Kardar equation are relevant simultaneously is also characterized by “atypical” scaling. While the ordinary scaling with fixed IR irrelevant parameters is impossible in this regime, the “restricted” scaling where the times, the coordinates, and the dimensionless ratio are scaled becomes possible. This result brings to mind scaling hypotheses modifications (Stell’s weak scaling or Fisher’s generalized scaling) for systems with significantly different characteristic scales.

Random-field critical scaling in a model of dipolar glass and relaxor ferroelectric behavior.

Heat-bath Monte Carlo simulations have been used to study a 12-state discretized Heisenberg model with a type of random field on simple cubic lattices of size 128×128×128. The 12 states correspond to the [110] directions of a cube. The model has the standard nonrandom two-spin exchange term with coupling energy J and a random field which consists of adding an energy h_{R} to two of the 12 spin states, chosen randomly and independently at each site. We report on the case h_{R}/J=3, which has a sharp phase transition at about T_{c}/J=1.40625. Below T_{c}, the model has long-range ferroelectric order oriented along one of the eight [111] directions. At T_{c}, the behavior of the peak in the structure factor, S(k), at small |k| is a straight line on a log-log plot, which gives the result η[over ¯]=1.214±0.014. The onset of orientational order below T_{c} is very rapid for this value of h_{R}. There are peaks in the specific heat and longitudinal susceptibility at T_{c}. Below T_{c} there is a strong correction to ordinary scaling, which is probably caused by the cubic anisotropy, which is a dangerous irrelevant variable.

Biophotons and Emergence of Quantum Coherence-A Diffusion Entropy Analysis.

We study the emission of photons from germinating seeds using an experimental technique designed to detect light of extremely small intensity. We analyze the dark count signal without germinating seeds as well as the photon emission during the germination process. The technique of analysis adopted here, called diffusion entropy analysis (DEA) and originally designed to measure the temporal complexity of astrophysical, sociological and physiological processes, rests on Kolmogorov complexity. The updated version of DEA used in this paper is designed to determine if the signal complexity is generated either by non-ergodic crucial events with a non-stationary correlation function or by the infinite memory of a stationary but non-integrable correlation function or by a mixture of both processes. We find that dark count yields the ordinary scaling, thereby showing that no complexity of either kinds may occur without any seeds in the chamber. In the presence of seeds in the chamber anomalous scaling emerges, reminiscent of that found in neuro-physiological processes. However, this is a mixture of both processes and with the progress of germination the non-ergodic component tends to vanish and complexity becomes dominated by the stationary infinite memory. We illustrate some conjectures ranging from stress induced annihilation of crucial events to the emergence of quantum coherence.

Caputo Fractional Derivative and Quantum-Like Coherence.

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

We present three examples of chemical reaction networks whose ordinary differential equation scaling limit are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these reaction networks display the full range of behaviors: one is stable (positive recurrent), one is unstable (transient) and one is marginally stable (null recurrent). We study these differences and characterize the invariant measures by Lyapunov function techniques. In particular, we design a natural set of such functions which scale homogeneously to infinity, taking advantage of the same scaling behavior of the reaction rates.

Daily measurement of slow slip from low-frequency earthquakes is consistent with ordinary earthquake scaling.

Slow slip transients on faults can last from seconds to months and stitch together the earthquake cycle. However, no single geophysical instrument is able to observe the full range of slow slip because of bandwidth limitations. Here, we connect seismic and geodetic data from the Mexican subduction zone to explore an instrumental blind spot. We establish a calibration of the daily median amplitude of the seismically recorded low-frequency earthquakes to the daily geodetically recorded moment rate of previously established slow slip events. This calibration allows us to use the precise evolution of low-frequency earthquake activity to quantitatively measure the moment of smaller, subdaily slip events that are unresolvable by geodesy alone. The resulting inferred slow slip moments scale with duration and inter-event time like ordinary earthquakes. These new quantifications help connect slow and fast events in a broad spectrum of transient slip and suggest that slow slip events behave much like ordinary earthquakes.

A novel refocusing distance measurement method using super‐resolved light field images

AbstractThis paper proposes a novel distance measurement method using superresolved light‐field images. We first superresolve the light field using the hybrid cross‐resolution input based on PatchMatch and convolutional neural network method, which combines and takes advantage of these two methods. In this way, we can explore the similarity between images and deal with challenging scenes effectively. The scaling factor is up to eight times, which is much larger than the ordinary superresolution scaling factor, and our superresolution method can also maintain a satisfactory accuracy. With the traditional idea of focus ranging, the light‐field 3D measurement method is established. A proper definition evaluation function suitable for light‐field data is selected to evaluate the imaging quality, and the triangle barycenter interpolation method is proposed to measure 3D scenes. Experimental results demonstrate that the proposed method not only improves the quality of the reconstructed high‐resolution light field but also has the ability of the distance measurement. It indicates that the light‐field imaging is a promising 3D measurement technique.

Insights from inside the spinodal: Bridging thermalization time scales with smoothed particle hydrodynamics.

We report the influence of the strength of heat bath coupling on the demixing behavior in spinodal decomposing one component liquid-vapor systems. The smoothed particle hydrodynamics (SPH) method with a van der Waals equation of state is used for the simulation. A thermostat for SPH is introduced that is based on the Berendsen thermostat. It controls the strength of heat bath coupling and allows for quenches with exponential temperature decay at a certain thermalization time scale. The present method allows us to bridge several orders of magnitude in the thermalization time scale. The early stage is highly affected by the choice of time scale. A transition from exponential growth to a 1/2 ordinary power law scaling in the characteristic lengths is observed. At high initial temperatures the growth is logarithmic. The comparison with pure thermal simulations reveals latent heat to raise the mean system temperature. Large thermalization time scales and thermal conductivity are figured out to affect a stagnation of heating, which is explained with convective processes. Furthermore, large thermalization time scales are responsible for a stagnation of growth of domains, which is temporally embedded between early and late stage of phase separation. Therefore, it is considered as an intermediate stage. We present an aspect concerning this stage, namely that choosing larger thermalization time scales increases the duration. Moreover, it is observed that diffuse interfaces are formed during this stage, provided that the stage is apparent. We show that the differences in the evolution between pure thermal simulations and simulations with an instantaneously scaled mean temperature can be explained by the thermalization process, since a variation of the time scale allows for the bridging between these cases of limit.

Anisotropic Turbulent Advection of a Passive Vector Field: Effects of the Finite Correlation Time

The turbulent passive advection under the environment (velocity) field with finite correlation time is studied. Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is investigated by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and prescribed pair correlation function. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to nontrivial fixed points of the RG equations and depend on the relation between the exponents in the energy energy spectrum e ∝ k ⊥ 1-ξ and the dispersion law ω ∝ k ⊥ 2-η . The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field itself. In contrast to the well-known isotropic Kraichnan model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: instead of power-like corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L . Due to the presence of the anisotropy in the model, all multiloop diagrams are equal to zero, thus this result is exact.

Passive advection of a vector field: Anisotropy, finite correlation time, exact solution, and logarithmic corrections to ordinary scaling.

In this work we study the generalization of the problem considered in [Phys. Rev. E 91, 013002 (2015)] to the case of finite correlation time of the environment (velocity) field. The model describes a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow. Inertial-range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, with finite correlation time and preassigned pair correlation function. Due to the presence of distinguished direction n, all the multiloop diagrams in this model vanish, so that the results obtained are exact. The inertial-range behavior of the model is described by two regimes (the limits of vanishing or infinite correlation time) that correspond to the two nontrivial fixed points of the RG equations. Their stability depends on the relation between the exponents in the energy spectrum E∝k(⊥)(1-ξ) and the dispersion law ω∝k(⊥)(2-η). In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale L.

Logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field.

Inertial-range asymptotic behavior of a vector (e.g., magnetic) field, passively advected by a strongly anisotropic turbulent flow, is studied by means of the field-theoretic renormalization group and the operator product expansion. The advecting velocity field is Gaussian, not correlated in time, with the pair correlation function of the form ∝δ(t-t')/k(⊥)(d-1+ξ), where k(⊥)=|k(⊥)| and k(⊥) is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow")--the d-dimensional generalization of the ensemble introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)]. The stochastic advection-diffusion equation for the transverse (divergence-free) vector field includes, as special cases, the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized Navier-Stokes equation. In contrast to the well-known isotropic Kraichnan's model, where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents, here the dependence on the integral turbulence scale L has a logarithmic behavior: Instead of powerlike corrections to ordinary scaling, determined by naive (canonical) dimensions, the anomalies manifest themselves as polynomials of logarithms of L. The key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized. The detailed proof of this fact is given for the correlation functions of arbitrary order.

Self-organised criticality in stochastic sandpiles: Connection to directed percolation

We introduce a stochastic sandpile model where finite drive and dissipation are coupled to the activity field. The absorbing phase transition here, as expected, belongs to the directed percolation (DP) universality class. We focus on the small drive and dissipation limit, i.e. the so-called self-organised critical (SOC) regime and show that the system exhibits a crossover from ordinary DP scaling to a dissipation-controlled scaling which is independent of the underlying dynamics or spatial dimension. The new scaling regime continues all the way to the zero bulk drive limit suggesting that the corresponding SOC behaviour is only DP, modified by the dissipation-controlled scaling. We demonstrate this for the continuous and discrete Manna model driven by noise and bulk dissipation.

Absence of the ordinary and extraordinary Hall effects scaling in granular ferromagnets at metal-insulator transition

Universality of the extraordinary Hall effect scaling was tested in granular three-dimensional Ni-SiO2 films across the metal-insulator transition. Three types of magnetotransport behavior have been identified: metallic, weakly insulating and strongly insulating. Scaling between both the ordinary and extraordinary Hall effects and material resistivity is absent in the weakly insulating range characterized by logarithmic temperature dependence of conductivity. The results provide compelling experimental confirmation to recent models of granular metals predicting transition from logarithmic to exponential conductivity temperature dependence when inter-granular conductance drops below the quantum conductance value and loss of Hall effect scaling when inter-granular conductance is higher than the quantum one. The effect was found at high temperatures and reflects the granular structure of material rather than low temperature quantum corrections.

Growth models on the Bethe lattice

I report on an extensive numerical investigation of various discrete growth models describing equilibrium and nonequilibrium interfaces on a substrate of a finite Bethe lattice. An unusual logarithmic scaling behavior is observed for the nonequilibrium models describing the scaling structure of the infinite-dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This gives rise to the classification of different growing processes on the Bethe lattice in terms of logarithmic scaling exponents which depend on both the model and the coordination number of the underlying lattice. The equilibrium growth model also exhibits a logarithmic temporal scaling but with an ordinary power law scaling behavior with respect to the appropriately defined lattice size. The results may imply that no finite upper critical dimension exists for the KPZ equation.

Fast and efficient multidimensional scaling algorithm for mobile positioning

Mobile station (MS) localisation that plays an important role in the process of target continuous localisation has received considerable attention. In this study, a new framework based on subspace approach for positioning an MS at minimum localisation system with the use of time-of-arrival measurements is introduced. Unlike ordinary multidimensional scaling algorithm using eigendcomposition or inverse computation to estimate the MS position, a computationally simple weighting estimator is proposed by introducing Lagrange multiplier and mean-square error weighting matrix. Computer simulations are included to corroborate the theoretical development and to contrast the estimator performance with several conventional algorithms as well as the Cramer–Rao lower bound (CRLB). It is shown that the new method with low computational complexity attains the CRLB for zero-mean white Gaussian range error at moderate noise level.

Density-matrix renormalization study of frustrated fermions on a triangular lattice

We show that the two-dimensional density-matrix renormalization analysis is useful to detect the symmetry breaking in the fermionic model on a triangular lattice. Under the cylindrical boundary conditions with chemical potentials on edge sites, we find that the open edges work as perturbation to select the strongest correlations {\it only in the presence of a long range order}. We also demonstrate that the ordinary size scaling analysis on the charge gap as well as that of the local charge density under this boundary condition could determine the metal-insulator phase boundary, which scales almost perfectly with the density of states and the exact solutions in the weak and strong coupling region, respectively.

Perturbation-induced emergence of Poisson-like behavior in non-Poisson systems

The response of a system with ON–OFF intermittency to an external harmonicperturbation is discussed. ON–OFF intermittency is described by means of a sequence ofrandom events, i.e., the transitions from the ON to the OFF state and vice versa. Theunperturbed waiting times (WTs) between two events are assumed to satisfy arenewal condition, i.e., the WTs are statistically independent random variables.The response of a renewal model with non-Poisson ON–OFF intermittency, associated withnon-exponential WT distribution, is analyzed by looking at the changes induced in the WTstatistical distribution by the harmonic perturbation. The scaling properties are alsostudied by means of diffusion entropy analysis.It is found that, in the range of fast and relatively strong perturbation, the non-Poissonsystem displays a Poisson-like behavior in both WT distribution and scaling.In particular, the histogram of perturbed WTs becomes a sequence of equallyspaced peaks, with intensity decaying exponentially in time. Further, the diffusionentropy detects an ordinary scaling (related to normal diffusion) instead of theexpected unperturbed anomalous scaling related to the inverse power-law decay.Thus, an analysis based on the WT histogram and/or on scaling methods has to beconsidered with some care when dealing with perturbed intermittent systems.

Pressure-dependent scaling scenarios in experiments of spontaneous imbibition

The scaling properties of the rough liquid-air interface formed in the spontaneous imbibition of a viscous liquid by a model porous medium are found to be very sensitive to the magnitude of the pressure difference applied at the liquid inlet. Interface fluctuations change from obeying intrinsic anomalous scaling at large negative pressure differences, to being super-rough with the same dynamic exponent z approximately =3 at less negative pressure differences, to finally obeying ordinary Family-Vicsek scaling with z approximately =2 at large positive pressure differences. This rich scenario reflects the relative importance on different length scales of capillary and permeability disorder, and the role of surface tension and viscous pressure in damping interface fluctuations.

Beyond Scaling and Locality in Turbulence

An analytic perturbation theory is suggested in order to find finite-size corrections to the scaling power laws. In the frame of this theory it is shown that the first order finite-size correction to the scaling power laws has following form $$S(r) \cong cr^{\alpha_0}[\ln(r/\eta)]^{\alpha_1}$$ , where η is a finite-size scale (in particular for turbulence, it can be the Kolmogorov dissipation scale). Using data of laboratory experiments and numerical simulations it is shown shown that a degenerate case with α 0=0 can describe turbulence statistics in the near-dissipation range r > η, where the ordinary (power-law) scaling does not apply. For moderate Reynolds numbers the degenerate scaling range covers almost the entire range of scales of velocity structure functions (the log-corrections apply to finite Reynolds number). Interplay between local and non-local regimes has been considered as a possible hydrodynamic mechanism providing the basis for the degenerate scaling of structure functions and extended self-similarity. These results have been also expanded on passive scalar mixing in turbulence. Overlapping phenomenon between local and non-local regimes and a relation between position of maximum of the generalized energy input rate and the actual crossover scale between these regimes are briefly discussed.