Intermediate Power Law Research Articles

Quantum Quenches and Work Distributions in Ultralow-Density Systems

We present results on quantum quenches in lattice systems with a fixed number of particles in a much larger number of sites. Both local and global quenches in this limit generically have power-law work distributions ("edge singularities"). We show that this regime allows for large edge singularity exponents beyond that allowed by the constraints of the usual thermodynamic limit. This large-exponent singularity has observable consequences in the time evolution, leading to a distinct intermediate power-law regime in time. We demonstrate these results first using local quantum quenches in a low-density Kondo-like system, and additionally through global and local quenches in Bose-Hubbard, Aubry-Andre, and hard-core boson systems at low densities.

Engagement in the electoral processes: Scaling laws and the role of political positions

We report on a statistical analysis of the engagement in the electoral processes of all Brazilian cities by considering the number of party memberships and the number of candidates for mayor and councillor. By investigating the relationships between the number of party members and the population of voters, we have found that the functional forms of these relationships are well described by sublinear power laws (allometric scaling) surrounded by a multiplicative log-normal noise. We have observed that this pattern is quite similar to those we previously reported for the relationships between the number of candidates (mayor and councillor) and population of voters [Europhys. Lett. 96, 48001 (2011)], suggesting that similar universal laws may be ruling the engagement in the electoral processes. We also note that the power-law exponents display a clear hierarchy, where the more influential is the political position the smaller is the value of the exponent. We have also investigated the probability distributions of the number of candidates (mayor and councillor), party memberships, and voters. The results indicate that the most influential positions are characterized by distributions with very short tails, while less influential positions display an intermediate power-law decay before showing an exponential-like cutoff. We discuss the possibility that, in addition to the political power of the position, limitations in the number of available seats can also be connected with this changing of behavior. We further believe that our empirical findings point out to an under-representation effect, where the larger the city is, the larger are the obstacles for more individuals to become directly engaged in the electoral process.

Hierarchy of temporal responses of multivariate self-excited epidemic processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)\theta}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < \theta <1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.

Diffusion in time-dependent random environments: mass fluctuations and scaling properties

A mass-ejection model in a time-dependent random environment with both temporal and spatial correlations is introduced. When the environment has a finite correlation length, individual particle trajectories are found to diffuse at large times with a displacement distribution that approaches a Gaussian. The collective dynamics of diffusing particles reaches a statistically stationary state, which is characterized in terms of a fluctuating mass density field. The probability distribution of density is studied numerically for both smooth and non-smooth scale-invariant random environments. Competition between trapping in the regions where the ejection rate of the environment vanishes and mixing due to its temporal dependence leads to large fluctuations of mass. These mechanisms are found to result in the presence of intermediate power-law tails in the probability distribution of the mass density. For spatially differentiable environments, the exponent of the right tail is shown to be universal and equal to −3/2. However, at small values, it is found to depend on the environment. Finally, spatial scaling properties of the mass distribution are investigated. The distribution of the coarse-grained density is shown to possess some rescaling properties that depend on the scale, the amplitude of the ejection rate and the Hölder exponent of the environment.

The pressure induced insulator to metal transition in FeSb2

The evolution of the ground state properties of FeSb2 has been investigated via temperature (4.2–300 K), magnetic field (0–12 T) and pressure (0–8.8 GPa) dependent electrical resistivity studies. The temperature dependence of the resistivity follows activated behavior in the high temperature (HT) regime (T > 60 K), while variable range hopping (VRH) dictates the transport in the intermediate temperature (IT) regime (10 K > T > 45 K) and power law behavior is observed in the low temperature (LT) regime (T < 10 K). The pressure profoundly affects the resistivity in all the temperature regimes. The energy gap (Δ) extracted in the HT regime initially increases with pressure and then decreases, while the VRH parameter T0 deduced in the IT regime is seen to decrease monotonically and vanish beyond 5 GPa leading to an insulator to metal transition (MIT) on account of delocalization of the electronic states in the gap. The analysis of the logarithmic derivative of the conductivity indicates the MIT to occur at ∼6 GPa. The magnetoresistivity is found to be positive. The analysis of the resistivity behavior under pressure and magnetic field indicates that the former induces delocalization, while the latter tends to assist localization of the defect states inside the gap of FeSb2.

Room Temperature Ionic Liquid−Lithium Salt Mixtures: Optical Kerr Effect Dynamical Measurements

The addition of lithium salts to ionic liquids causes an increase in viscosity and a decrease in ionic mobility that hinders their possible application as an alternative solvent in lithium ion batteries. Optically heterodyne-detected optical Kerr effect spectroscopy was used to study the change in dynamics, principally orientational relaxation, caused by the addition of lithium bis(trifluoromethylsulfonyl)imide to the ionic liquid 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide. Over the time scales studied (1 ps-16 ns) for the pure ionic liquid, two temperature-independent power laws were observed: the intermediate power law (1 ps to approximately 1 ns), followed by the von Schweidler power law. The von Schweidler power law is followed by the final complete exponential relaxation, which is highly sensitive to temperature. The lithium salt concentration, however, was found to affect both power laws, and a discontinuity could be found in the trend observed for the intermediate power law when the concentration (mole fraction) of lithium salt is close to chi(LiTf(2)N) = 0.2. A mode coupling theory (MCT) schematic model was also used to fit the data for both the pure ionic liquid and the different salt concentration mixtures. It was found that dynamics in both types of liquids are described very well by MCT.

Flow mechanisms and rheology of dense aqueous suspensions of titanium(IV) oxide

Rheograms obtained from testing concentrated suspensions of titanium(IV) oxide in water indicated that, depending on temperature and concentration, Newtonian, shear thickening, shear thinning or mixed behaviour could occur. Tests at constant shear rate and increasing temperature indicated that different flow mechanisms were dominant depending on temperature, shear rate and composition. Phenomenologically, the temperature ramp results could be interpreted in terms of three flow types, namely, fluid (Newtonian) flow, plastic flow and an intermediate power law flow, a description which in many respects is similar to that used for deformation in crystalline materials. Power law and plastic flow could be combined into a single hybrid equation which, together with an expression for Newtonian flow, gave a reasonably good description of behaviour both from a qualitative and quantitative point of view. Structure parameters appearing in the model depended only on solids concentration and not on the externally controlled variables of temperature and strain rate, suggesting that perhaps much of the apparent thinning or thickening behaviour is an artefact of changes in flow mechanism.

Spectral and particle dispersion properties of steady two-dimensional multiscale flows

The spectral and particle dispersion characteristics of steady multiscale laminar thin-layer flows are investigated through numerical simulations of a two-dimensional layer-averaged model. The model assumes a semiparabolic velocity profile and is solved using a semi-Lagrangian spline method. The main features of the flows are turbulentlike and consistent with previous experimental studies. The Eulerian wavenumber spectra and the Lagrangian frequency spectra oscillate around power laws that reflect the self-similarity of the forcing. In the weak forcing regime, the exponents of these power laws can be related to the multiscale geometry and the intensity scaling of the forcing. The Lagrangian spectra also show low-frequency plateaus, which arise from the slow motions far away from the applied forces. The absolute dispersion of tracer particles in these steady planar flows presents a ballistic stage followed by a diffusive regime, which results from the decorrelated motions of particles lying on streamlines of different periods. Relative dispersion shows an additional intermediate stage consisting of several separation bursts, which originate from the intense strain regions imposed by the different forcing scales. While these bursts can cause locally superquadratic mean square separation, the trapping by steady recirculation regions rules out an intermediate relative dispersion power law regardless of the number of scales in the flow.

Poincaré recurrences and transient chaos in systems with leaks

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Intra‐meander hyporheic flow in alluvial rivers

Several geomorphological fluvial features are able to induce hyporheic exchange between the rivers and the alluvial sediments. However, while the small‐scale exchange induced by bed forms has been thoroughly investigated, the role of the larger features remains poorly understood. Here, we focus on the hyporheic flows driven by the channel sinuosity in the intrameander zone. A physically based model is adopted to simulate the morphodynamic evolution of three different meandering rivers, from the first stages of the meander evolution until the incipient meander cutoff. For each stage, the sinuosity‐driven intrameander hyporheic flow is computed. In this way, the hyporheic flow field, the fluxes exchanged with the river, and the residence times are described during the whole meander evolution. The main result concerns the existence of a remarkable zonation induced by the flow field in the intrameander zone. The more the meander evolves, the more the zonation becomes pronounced, and the probability distribution of the residence times shows a bimodal shape with an intermediate power law behavior. Some general rules governing the typical timescales of the intrameander hyporheic flow and the mean exchanged flux are also deduced.

Depolarized light scattering versus optical Kerr effect. II. Insight into the dynamic susceptibility of molecular liquids

We have previously discussed [J. Chem. Phys. 125, 114502 (2006)] that optical Kerr effect (OKE) and depolarized light scattering (DLS) data of molecular liquids reveal, each in their native domain, the same characteristic signatures of the glass transition dynamics; in particular, the intermediate power law of OKE is equivalent with the excess wing of the frequency-domain data, long since known in dielectric spectroscopy. We now extend the discussion to show that the excess wing is an equally common feature in DLS. We further discuss the time-temperature superposition property of OKE data in relation to our DLS and literature dielectric-spectroscopic results, and the merits of their mode coupling theory analyses. Spectroscopic signatures of a liquid-crystal-forming system (nematogen) are discussed in the same frame.

Evolution of dynamic susceptibility in molecular glass formers—a critical assessment

Dielectric, depolarized light scattering (LS) and optical Kerr effect (OKE) data are criticallydiscussed in an attempt to achieve a common interpretation of the evolution of dynamicsusceptibility in molecular glass formers at temperatures down to the glass transitionTg. The so-called intermediate power-law, observed in OKE data below a certain temperatureTx, is identified with the excess wing, long since known from dielectricspectroscopy, with a temperature-independent exponent. This is in contrastwith several recent analyses that concluded a considerable temperaturedependence of spectral shapes. We introduce a new approach to disentangleα-peak and excess wing contributions in the dielectric spectra,which allows for frequency–temperature superposition (FTS) of theα-process at alltemperatures above Tg. From the LS spectra we conclude, in particular, that FTS holds even at temperatures wellabove the melting point, i.e. in normal equilibrium liquids. Attempting to correlate thefragility and stretching, our conclusions are opposite to those made previously. Specifically,we observe that a high fragility is associated with a less stretched relaxation function.

Theory of earthquake recurrence times

The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions and when averaged over multiple regions. These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan's argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori‐Utsu law of earthquake triggering. We then develop the theory of the statistics of interevent times in the framework of the Epidemic‐Type Aftershock Sequence (ETAS) model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression well fits the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by the Omori‐Utsu law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension d ≈ 1.8, the average clustering ratio n ≈ 0.9, and the productivity exponent α ≈ 0.9 times the b value of the Gutenberg‐Richter law.

Dynamics of a discotic liquid crystal in the isotropic phase

Optically heterodyne-detected optical Kerr effect (OHD-OKE) experiments are conducted to study the orientational dynamics of a discotic liquid crystal 2,3,6,7,10,11-hexakis(pentyloxy)triphenylene (HPT) in the isotropic phase near the columnar-isotropic (C-I) phase transition. The OHD-OKE signal of HPT is characterized by an intermediate power law t(-0.76+/-0.02) at short times (a few picoseconds), a von Schweidler power law t(-0.26+/-0.01) at intermediate times (hundreds of picoseconds), and an exponential decay at long times (tens of nanoseconds). The exponential decay has Arrhenius temperature dependence. The functional form of the total time dependent decay is identical to the one observed previously for a large number of molecular supercooled liquids. The mode coupling theory schematic model based on the Sjogren [Phys. Rev. A 33, 1254 (1986)] model is able to reproduce the HPT data over a wide range of times from <1 ps to tens of nanoseconds. The studies indicate that the HPT C-I phase transition is a strong first order transition, and the dynamics in the isotropic phase display a complex time dependent profile that is common to other molecular liquids that lack mesoscopic structure.

Structural relaxation in complex liquids: Non-Markovian dynamics in a bistable potential

The time correlation function C(t) identical with <x(0)x(t)> of the distance fluctuations of a particle moving in a bistable potential under the action of fractional Gaussian noise (fGn) is calculated from a Smoluchowski-type equation derived from a generalized Langevin equation (GLE). The time derivative of this function, dC(t)dt, is compared with data from optical Kerr effect measurements of liquid crystal dynamics in the vicinity of the isotropic-to-nematic transition, which are related to the time derivative of an orientational correlation function. A number of characteristic features of the experimental decay curves, including short and intermediate time power law behavior and long time exponential relaxation, are qualitatively reproduced by the analytical calculations, even though the latter do not explicitly treat orientational degrees of freedom. The GLE formalism with fGn was, in fact, originally proposed as a model of protein conformational fluctuations, so the present results suggest that it may also serve more generally as a model of structural relaxation in complex condensed phase media.

Depolarized light scattering versus optical Kerr effect spectroscopy of supercooled liquids: Comparative analysis

Recently, heterodyne-detected optical Kerr effect (HD-OKE) spectroscopy was used to study dynamics of supercooled molecular liquids. The studies revealed an apparently new physical phenomenon that had not been reported before from the related depolarized light scattering (DLS), namely, an intermediate power law (nearly logarithmic decay) of the response functions [H. Cang et al., J. Chem. Phys. 118, 2800 (2003)]. Conceptually, HD-OKE and DLS data reflect optical anisotropy fluctuations mainly due to molecular reorientation dynamics in time and frequency domains, respectively. The above-mentioned effects are revealed in the mesoscopic range less, similar1 GHz ( greater, similar100 ps), where no direct comparison of the techniques was reported. In this Communication, we attempt such a comparison of exemplifying HD-OKE literature data of the glass-forming salol (phenyl salicylate), benzophenone, and liquid-crystal forming 4-cyano-4(')-pentylbiphenyl with DLS data of the same systems that we measured down to ca. 200 MHz by a combined tandem Fabry-Perot interferometer plus tandem-grating-monochromator technique. Generally, we find a satisfactory agreement, albeit in some cases with subtle differences at frequencies greater, similar10 GHz. We conclude that, in the mesoscopic dynamic range, HD-OKE and DLS studies provide consistent and comparable information, and therefore their conclusions must agree. We argue that the intermediate power law of HD-OKE is in essence a manifestation of the excess wing of the corresponding frequency-domain data, known long since from broadband dielectric spectroscopy and anticipated from DLS studies of supercooled liquids.

Dynamics in Supercooled Ionic Organic Liquids and Mode Coupling Theory Analysis

Optically heterodyne-detected optical Kerr effect experiments are applied to study the orientational dynamics of the supercooled ionic organic liquids N-propyl-3-methylpyridinium bis(trifluoromethylsulfonyl)imide (PMPIm) and 1-ethyl-3-methylimidazolium tosylate (EMImTOS). The orientational dynamics are complex with relaxation involving several power law decays followed by a final exponential decay. A mode coupling theory (MCT) schematic model, the Sjögren model, was able to reproduce the PMPIm data very successfully over a wide range of times from 1 ps to hundreds of ns for all temperatures studied. Over the temperature range from room temperature down to the critical temperature Tc of 231 K, the OHD-OKE signal of PMPIm is characterized by the intermediate power law t(-1.00+/-0.04) at short times, a von Schweidler power law t(-0.51+/-0.03) at intermediate times, and a highly temperature-dependent exponential (alpha relaxation) at long times. This form of the decay is identical to the form observed previously for a large number of organic van der Waals liquids. MCT analysis indicates that the theory can explain the experimental data very well for a range of temperatures above Tc, but as might be expected, there are some deviations from the theoretical modeling at temperatures close to Tc. For EMImTOS, the orientational dynamics were studied on the ps time scale in the deeply supercooled region near its glass transition temperature. The orientational relaxation of EMImTOS clearly displays the feature associated with the boson peak at approximately 2 ps, which is the first time domain evidence of the boson peak in ionic organic liquids. Overall, all the dynamical features observed earlier for organic van der Waals liquids using the same experimental technique are also observed for organic ionic liquids.

Three homeotropically aligned nematic liquid crystals: Comparison of ultrafast to slow time-scale dynamics

The dynamics of two nematic liquid crystals, 4-(trans-4(')-n-octylcyclohexyl)isothiocyanatobenzene and 4-(4-pentyl-cyclohexyl)-benzonitrile, are investigated as a function of temperature both in the homeotropically aligned nematic phase and in the isotropic phase using optical heterodyne-detected optical Kerr effect experiments, which measures the time derivative of the polarizability-polarizability-correlation function (orientational relaxation). Data are presented over a time range of 500 fs-70 micros for the nematic phase and 500 fs to a few hundred nanoseconds for the isotropic phase. The nematic dynamics are compared with a previously studied liquid crystal in the nematic phase. All three liquid crystals have very similar dynamics in the nematic phase that are very different from the isotropic phase. On the slowest time scale (20 ns-70 micros), a temperature-independent power law, the final power law, t(-f) with f approximately 0.5, is observed. On short time scales (approximately 3 ps to approximately 1 ns), a temperature-dependent intermediate power law is observed with an exponent that displays a linear dependence on the nematic order parameter. Between the intermediate power law and the final power law, there is a crossover region that has an inflection point. For times that are short compared to the intermediate power law (approximately <2 ps), the data decay much faster, and can be described as a third power law, although this functional form is not definitive. The isotopic phase data have the same features as found in previous studies of nematogens in the isotropic phase, i.e., the temperature-independent intermediate power law and von Schweidler power law at short to intermediate times, and a highly temperature-dependent long time exponential decay that is well described by the Landau-de Gennes theory. The results show that liquid-crystal dynamics in the nematic phase exhibit universal behavior.

Ultrafast to Slow Orientational Dynamics of a Homeotropically Aligned Nematic Liquid Crystal

The orientational dynamics of a homeotropically aligned nematic liquid crystal, 4'-pentyl-4-biphenylcarbonitrile (5-CB), is studied over more than six decades of time (500 fs to 2 mus) using optical heterodyne detected optical Kerr effect experiments. In contrast to the dynamics of nematogens in the isotropic phase, the data do not decay as a highly temperature-dependent exponential on the longest time scale, but rather, a temperature-independent power law spanning more than two decades of time, the final power law, is observed. On short time scales (approximately 3 ps to approximately 1 ns) another power law, the intermediate power law, is observed that is temperature dependent. The power law exponent of the correlation function associated with the intermediate power law displays a linear dependence on the change in the nematic order parameter with temperature. Between the intermediate power law and the final power law, there is a crossover region that displays an inflection point. The temperature-dependent orientational dynamics in the nematic phase are shown to be very different than those observed in the isotropic phase.

Free‐Free Spectral Energy Distributions of Hierarchically Clumped HiiRegions

In an effort to understand unusual power-law spectral slopes observed in some hypercompact HII regions, we consider the radio continuum energy distribution from an ensemble of spherical clumps. An analytic expression for the free-free emission from a single spherical clump is derived. The radio continuum slope (with F_\nu \nu^\alpha) is governed by the population of clump optical depths N(tau), such that (a) at frequencies where all clumps are thick, a continuum slope of +2 is found, (b) at frequencies where all clumps are optically thin, a flattened slope of -0.11 is found, and (c) at intermediate frequencies, a power-law segment of significant bandwidth with slopes between these two limiting values can result. For the ensemble distribution, we adopt a power-law distribution N(tau) tau^{-\gamma}, and find that significant power-law segments in the SED with slopes from +2 to -0.11 result only for a relatively restricted range of $\gamma$ values of 1 to 2. Further, a greater range of clump optical depths for this distribution leads to a wider bandwidth over which the intermediate power-law segment exists. The model is applied to the source W49N-B2 with an observed slope of \alphab +0.9, but that may be turning over to become optically thin around 2 mm. An adequate fit is found in which most clumps are optically thin and there is little shadowing of rearward clumps by foreground clumps (i.e., the geometrical covering factor C<<1). The primary insight gained from our study is that in the Rayleigh-Jeans limit for the Planck function that applies for the radio band, it is the distribution in optical depth of the clump population that is solely responsible for setting the continuum shape, with variations in the size and temperature of clumps serving to modulate the level of free-free emission.