Inverse Power Law Behavior Research Articles

Anomalous quantum transport in fractal lattices

Fractal lattices are self-similar structures with repeated patterns on different scales. Quantum transport through such structures is subtle due to the possible co-existence of localized and extended states. Here, we study the dynamical properties of two fractal lattices, the Sierpiński gasket and the Sierpiński carpet. While the gasket exhibits sub-diffusive behavior, sub-ballistic transport occurs in the carpet. We show that the different dynamical behavior is in line with qualitative differences of the systems’ spectral properties. Specifically, in contrast to the Sierpiński carpet, the Sierpiński gasket exhibits an inverse power-law behavior of the level spacing distribution. As a possible technological application, we discuss a memory effect in the Sierpiński gasket which allows to read off the phase information of an initial state from the spatial distribution after long evolution times. We also show that interpolating between fractal and regular lattices allows for flexible tuning between different transport regimes.

Relationships between fracture toughness, Y2O3 fraction and phases content in modern dental Yttria-doped zirconias

The relationship between fracture toughness and Yttria content in modern zirconia ceramics was revised. For that purpose, we evaluated here 10 modern Y2O3-stabilized zirconia (YSZ) materials currently used in biomedical applications, namely prosthetic and implant dentistry. The most relevant range between 2-5 mol% Y2O3 was addressed by selecting from conventional opaque 3 mol% YSZ up to more translucent compositions (4−5 mol% YSZs). A technical 2YSZ was used to extend the range of our evaluation. The bulk mol% Y2O3 concentration was measured by X-Ray Fluorescence Spectroscopy. Phase quantification by Rietveld refinement considered two tetragonal phases or an additional cubic phase. A first-account of the fracture toughness (KIc) of the pre-sintered blocks is given, which amounted to 0.4 – 0.7 MPa√m. In the fully-densified state, an inverse power-law behavior was obtained between KIc and bulk mol% Y2O3 content, whether using only our measurements or including literature data, challenging some established relationships. A linear relationship between KIc and the fraction of the transformable t-phase was established within the range of 30–70 vol%.

Relating the curvature of De Sitter universe to open quantum Lamb shift spectroscopy

In this paper, we explore the connection between the curvature of the background De Sitter space-time with the spectroscopic study of entanglement of two atoms. Our set up is in the context of an Open Quantum System (OQS), where the two atoms, each having two energy levels and represented by Pauli spin tensor operators projected along any arbitrary direction. The system mimic the role of a pair of freely falling Unruh De-Witt detectors, which are allowed to non-adiabatically interact with a conformally coupled massless probe scalar field which has the role of background thermal bath. The effective dynamics of this combined system takes into account of the non-adiabatic interaction, which is commonly known as the Resonant Casimir Polder Interaction (RCPI) with the thermal bath. Our analysis revels that the RCPI of two stable entangled atoms in the quantum vacuum states in OQS depends on the de Sitter space-time curvature relevant to the temperature of the thermal bath felt by the static observer. We also find that, in OQS, RCPI produces a new significant contribution appearing in the effective Hamiltonian of the total system and thermal bath under consideration. We find that the Lamb shift is characterised by a decreasing inverse square power law behaviour, L^{-2}, when inter atomic Euclidean distance, L, is much larger than a characteristic length scale, k, which is the inverse surface gravity of the background De Sitter space. If the background space time would have been Minkowskian this shift decreases as, L^{-1}, and is independent of temperature. Thus, we establish a connection between the curvature of the De Sitter space-time with the Lamb shift spectroscopy.

Universality in spectral condensation

Self-organization is the spontaneous formation of spatial, temporal, or spatiotemporal patterns in complex systems far from equilibrium. During such self-organization, energy distributed in a broadband of frequencies gets condensed into a dominant mode, analogous to a condensation phenomenon. We call this phenomenon spectral condensation and study its occurrence in fluid mechanical, optical and electronic systems. We define a set of spectral measures to quantify this condensation spanning several dynamical systems. Further, we uncover an inverse power law behaviour of spectral measures with the power corresponding to the dominant peak in the power spectrum in all the aforementioned systems.

Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market

The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of the volatilities and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore, we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior of other stochastic models that are used as models for the finances. Furthermore, we have made an analysis by using Fokker–Planck equation independent on time with the aim of obtaining the cumulative probability distribution of volatilities , however, the probability density found does not exhibit the cubic inverse law.

Quantum decay law: critical times and the equivalence of approaches

Methods based on the use of Green’s functions or Jost functions and the Fock–Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock–Krylov amplitude in terms of the coefficients introduced in the Green’s functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective of understanding the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, , from the exponential to the inverse power law behaviour at large times. Evaluating for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.

The true quantum face of the “exponential” decay: Unstable systems in rest and in motion

Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P 0 (t ) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P 0 (t ) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P 0 (t ) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P 0 (t ) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum p→≠0. These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.

Persistence of unvisited sites in quantum walks on a line

We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk there is no connection between the behaviour of persistence and the scaling of variance. In particular, we find that for a two-state quantum walks persistence follows an inverse power-law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk the scaling of persistence is given by two contributions. The first is the inverse power-law. The second contribution to the asymptotic behaviour of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the two-state walks both the exponent of the inverse power-law and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse power-law behaviour.

Optical conductivity of nodal metals

Fermi liquid theory is remarkably successful in describing the transport and optical properties of metals; at frequencies higher than the scattering rate, the optical conductivity adopts the well-known power law behavior σ1(ω) ∝ ω−2. We have observed an unusual non-Fermi liquid response σ1(ω) ∝ ω−1±0.2 in the ground states of several cuprate and iron-based materials which undergo electronic or magnetic phase transitions resulting in dramatically reduced or nodal Fermi surfaces. The identification of an inverse (or fractional) power-law behavior in the residual optical conductivity now permits the removal of this contribution, revealing the direct transitions across the gap and allowing the nature of the electron-boson coupling to be probed. The non-Fermi liquid behavior in these systems may be the result of a common Fermi surface topology of Dirac cone-like features in the electronic dispersion.

Numerical validation of the generalized Harvey–Shack surface scatter theory

The generalized Harvey–Shack (GHS) surface scatter theory is numerically compared to the classical small perturbation method, the Kirchhoff approximation method, and the rigorous method of moments for one-dimensional ideally conducting surfaces whose surface power spectral density function is Gaussian or exhibits an inverse power law (fractal) behavior. In spite of its simple analytic form, our numerical comparison shows that the new GHS theory is valid (with reasonable accuracy) over a broader range of surface parameter space than either of the two classical surface scatter theories.

Influence of the Dielectric Environment on the Photoluminescence Intermittency of CdSe Quantum Dots

We show experimentally that the photoluminescence intermittency (blinking) of single CdSe quantum dots (QDs) is influenced by the dielectric properties of the embedding environment (matrix), the type of ligands and the capping shell. For the on-times, we observe (and tentatively explain) a strong deviation from the commonly reported inverse power law behaviour, which can be taken into account by an exponential cut-off at long times. We assign this component to the photoejection of the electron, while the power law behaviour is a combination of hole- and electron-trapping processes. The cut-off times and their distributions depend strongly on the polarity of the environment. Also, the off-times show, though on a much longer timescale, deviations from the inverse power laws. We suggest a model including surface states and self-trapped states, which quantitatively explains the experimental observations.

Relaxation time distributions for an anomalously diffusing particle

As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the dynamics, successively considering the memory kernel, the particle’s mean velocity, and the scattering function. All these quantities are studied from a unique angle, namely, the discussion of the possible existence of a distribution of relaxation times characterizing their time decay. Although a very popular concept, a relaxation time distribution cannot be associated with any time-decreasing quantity (from a mathematical point of view, the decay has to be described by a completely monotonic function). Technically, we use a memory kernel decaying as a Mittag-Leffler function (the Mittag-Leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times). We show that, in the case of a subdiffusive motion, relaxation time distributions can be defined for the memory kernel and for the scattering function, but not for the particle’s mean velocity. The situation is opposite in the superdiffusive case.

Time-dependent models for a decade of SN 1993J

A classical and a relativistic law of motion for a supernova remnant (SNR) are deduced assuming an inverse power law behavior for the density of the interstellar medium and applying the thin layer approximation. A third equation of motion is found in the framework of relativistic hydrodynamics with pressure, applying momentum conservation. These new formulas are calibrated against a decade of observations of SN 1993J. The existing knowledge of the diffusive processes of ultrarelativistic electrons is reviewed in order to explain the behavior of the ‘U’ shaped profile of intensity versus distance from the center of SN 1993J.

Chipping away at memory

Inverse power-law behavior is known to be characteristic of adaptation, learning, and memory. Herein, we propose a phenomenological model of forgetting based on renewal theory that introduces a new psychophysical concept, chipping; discrete events that chip away at chunks of memory and thereby produce forgetting. The neural mechanism producing these chips is the 1/f-noise that is generically produced in complex neuronal networks.

Complexity and synchronization

We study a fully connected network (cluster) of interacting two-state units as a model of cooperative decision making. Each unit in isolation generates a Poisson process with rate g . We show that when the number of nodes is finite, the decision-making process becomes intermittent. The decision-time distribution density is characterized by inverse power-law behavior with index mu=1.5 and is exponentially truncated. We find that the condition of perfect consensus is recovered by means of a fat tail that becomes more and more extended with increasing number of nodes N . The intermittent dynamics of the global variable are described by the motion of a particle in a double well potential. The particle spends a portion of the total time tau(S) at the top of the potential barrier. Using theoretical and numerical arguments it is proved that tau(S) is proportional to (1/g)ln(const x N) . The second portion of its time, tau(K), is spent by the particle at the bottom of the potential well and it is given by tau(K)=(1/g)exp(const x N) . We show that the time tau(K) is responsible for the Kramers fat tail. This generates a stronger ergodicity breakdown than that generated by the inverse power law without truncation. We establish that the condition of partial consensus can be transmitted from one cluster to another provided that both networks are in a cooperative condition. No significant information transmission is possible if one of the two networks is not yet self-organized. We find that partitioning a large network into a set of smaller interacting clusters has the effect of converting the fat Kramers tail into an inverse power law with mu=1.5 .

Nonadiabatic dynamics of electron scattering from adsorbates in surface bands

We present a comparative study of nonadiabatic dynamics of electron scattering in quasi-two-dimensional surface band which is induced by the long-range component of the interactions with a random array of adsorbates. Using three complementary model descriptions of intraband spatiotemporal propagation of quasiparticles that go beyond the single-adsorbate scattering approach we are able to identify distinct subsequent regimes of evolution of an electron following its promotion into an unoccupied band state: i early quadratic or ballistic decay of the initial-state survival probability within the Heisenberg uncertainty window, ii preasymptotic exponential decay governed by the self-consistent Fermi golden rule scattering rate, and iii asymptotic decay described by a combined inverse power-law and logarithmic behavior. The developed models are applied to discuss the dynamics of intraband adsorbate-induced scattering of hot electrons excited into the n = 1 image-potential band on Cu100 surface during the first stage of a two-photon photoemission process. Estimates of crossovers between the distinct evolution regimes enable assessments of the lifespan of a standard quasiparticle behavior and thereby of the range of applicability of the widely used Fermi golden rule and optical Bloch equations approach for description of adsorbate-induced quasiparticle decay and dephasing in ultrafast experiments.

Diagnostic cellular organization features extracted from autofluorescence images

Depth-resolved NADH autofluorescence images are shown to differentiate between normal and precancerous engineered tissues. An inverse power law behavior of the power spectral density (PSD) of these images is observed, indicating a self-affine organization of mitochondrial NADH at length scales 1-10 microm. Power exponents of the PSD functions vary significantly with tissue depth and precancerous state, giving insight into the morphological changes associated with precancerous lesions and providing substantial potential for noninvasive clinical diagnosis of squamous epithelial lesions and tumors.

Diffusive escape and reentry through a channel in the cavity wall

We show that the kinetics of diffusive escape from a cavity through a narrow channel in the cavity wall and successive reentry can be described by a formal kinetic scheme for reversible dissociation of a spherical binding site with appropriately defined effective association and dissociation rate constants. Initially the population of the cavity decays exponentially, with the rate constant determined by the cavity volume and the channel length and radius. A crossover to the universal inverse power law behavior, which does not depend on the channel geometry, occurs at long times. These simple predictions are in excellent agreement with the results of Brownian dynamics simulations.

Escape and reentry of a Brownian particle through a hole in a cavity

We show that the kinetics of diffusive escape from a cavity through a small hole can be described, to a sufficient degree of accuracy, by a formal kinetic scheme for reversible dissociation of a spherical binding site with appropriately defined effective association and dissociation rate constants. At short times, the population of the cavity decays exponentially, while at long times a crossover to the inverse power law behavior is observed. These simple predictions are in excellent agreement with the results of Brownian dynamics simulations.

Modeling distributed kinetics in isolated semiconductor quantum dots

A detailed modeling of recently observed nonexponential fluorescence intermittency in colloidal semiconductor quantum dots (QDs) is presented. In particular, experiments have shown that both ``on''-time and ``off''-time probability densities generated from single-QD fluorescence trajectories follow an inverse power law, $P({\ensuremath{\tau}}_{\mathrm{o}\mathrm{n}/\mathrm{o}\mathrm{f}\mathrm{f}})\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{o}\mathrm{n}/\mathrm{o}\mathrm{f}\mathrm{f}}^{1+\ensuremath{\alpha}},$ over multiple decades in time, where the exponent $1+\ensuremath{\alpha}$ can, in general, differ for ``on'' versus ``off'' episodes. Several models are considered and tested against their ability to predict inverse power law behavior in both $P({\ensuremath{\tau}}_{\mathrm{on}})$ and $P({\ensuremath{\tau}}_{\mathrm{off}}).$ A physical picture involving electron tunneling to, and return from, traps located several nanometers away from the QD is found to be consistent with the observed $P({\ensuremath{\tau}}_{\mathrm{off}})$ but does not yield the inverse power-law behavior seen in $P({\ensuremath{\tau}}_{\mathrm{on}}).$ However, a simple phenomenological model based on exponentially distributed and randomly switched on and off decay rates is analyzed in detail and shown to yield an inverse power-law behavior in both $P({\ensuremath{\tau}}_{\mathrm{on}})$ and $P({\ensuremath{\tau}}_{\mathrm{off}}).$ Monte Carlo calculations are used to simulate the resulting blinking behavior, and are subsequently compared with experimental observations. Most relevantly, these comparisons indicate that the experimental $\mathrm{o}\stackrel{\ensuremath{\rightarrow}}{n}\mathrm{off}$ blinking kinetics are independent of excitation intensity, in contradiction with previous multiphoton models of on/off intermittency based on an Auger-assisted ionization of the QD by recombination of a second electron-hole pair.