Power-law Regime Research Articles

Stress-regime-dependence of inelastic anisotropy in forged age-hardening aluminium alloys at elevated temperature: Constitutive modeling, identification and validation

Many structural materials exhibit stress-regime-dependent anisotropy of inelastic responses. Examples are age-hardening aluminium alloys forgings. Experimental creep curves indicate that inelastic strain rates depend significantly on the loading direction within the power law regime, while in the power law breakdown range the anisotropy is weak and can be neglected. The aim of this paper is to analyze anisotropic behavior of forged AA2014 alloy and to develop a constitutive model to describe inelastic response under multi-axial stress state. Microstructural observations suggest that the anisotropy is primarily caused by elongated grains and coarse particles on grain boundaries. To account for inhomogeneous inelastic deformation and stress redistribution in different microstructural zones a composite model is developed. Constitutive equations for inelastic-hard and inelastic-soft constituents and anisotropic rules for mechanical interactions between them are elaborated. Hardening/recovery and overageing processes in grain interiors are characterized by internal state variables and kinetic equations. The model is calibrated against families of creep curves for two loading directions in a wide stress and temperature ranges. For the validation, creep tests under the loading with the angle of 30∘ relative to the longitudinal grain axis as well as tensile tests are simulated by the model and results are compared with experimental data.

Dispersion modeling in pore networks: A comparison of common pore-scale models and alternative approaches

Mass transfer in porous media resulting from dispersion occurs in a wide variety of applications such as water treatment, flow batteries, flow in aquifers, enhanced oil recovery, and packed-bed reactors. The underlying mechanisms of dispersion are the molecular diffusion superimposed on the advective transport induced by the fluid flow. Modeling dispersion in pore networks can be performed at a much lower computational cost compared to that in direct numerical simulations (DNS) such as finite element or the lattice Boltzmann methods, so it can be regarded as a suitable alternative provided its accuracy is sufficient. The most common approach to model dispersion in network models is based on the first-order upwind scheme, despite its known limitations in terms of accuracy for certain flow and transport regimes. In this study, three alternative pore-scale models for dispersion, which are more accurate than the existing ones, were derived and tested in pore network simulations. These models were adopted from the CFD literature and are based on a spatial discretization of the advection-diffusion equation using the hybrid and power-law finite difference schemes and the exact solution of the one-dimensional advection-diffusion equation. Finally, considering dispersion problems over arbitrary porous structures, consisting of stick-and-ball geometries, and different flow and mass transfer arrangements, the developed models were validated. Validation was carried-out through comparisons between results obtained with DNS, using a finite element solver, and those from pore network simulations. It is shown that under a wide range of dispersion regimes (up to the onset of the dispersion power-law regime), the relative error (with respect to DNS results) introduced by the power-law and exact solution-based models is consistently below 1%, whereas the use of the upwind scheme leads to >10% of relative error, depending on the dispersion regime. All the dispersion models developed in this study were implemented as part of the open-source network modeling package, OpenPNM.

Tension propagation during DNA hairpin folding

Characterization of DNA/RNA folding dynamics is an interesting problem involving the complex force balance between the relaxation of the growing duplex and the stretched single-stranded segments. In a previous paper, we reported that the helicity plays an important role in determining the folding time for a chain of size N, where is the Flory exponent. Here we analyze this process in further detail by using molecular dynamics, with particular emphasis on tension propagation along a single strand on the unfolded segment. We directly observe that the single-strand segments are always stretched during the folding process by the tension induced by base pair formation, propagating ahead of the y -junction. Our molecular dynamics simulations verify the existence of a stem-flower structure in the unfolded segments, with a power-law dependence of the stem length Ns on the duplex length n. For the longest hairpin structures considered, we also find that the power-law regime terminates before the first half of the folding is completed. We demonstrate that the structure’s helicity is an integral aspect of the folding dynamics by comparing our findings with those obtained from a hypothetical non-helical model.

Mott quantum criticality in the one-band Hubbard model: Dynamical mean-field theory, power-law spectra, and scaling

Recent studies of electrical transport, both theoretical and experimental, near the bandwidth-tuned Mott metal-insulator transition have uncovered apparent quantum critical scaling of the electrical resistivity at elevated temperatures, despite the fact that the actual low-temperature phase transition is of first order. This raises the question whether there is a hidden Mott quantum critical point. Here we argue that the dynamical mean-field theory of the Hubbard model admits, in the low-temperature limit, asymptotically scale-invariant (i.e. power-law) solutions, corresponding to the metastable insulator at the boundary of metal-insulator coexistence region, which can be linked to the physics of the pseudogap Anderson model. While our state-of-the-art numerical renormalization group calculations reveal that this asymptotic regime is restricted to very small energies and temperatures and hence difficult to access numerically, we uncover the existence of a wide crossover regime where the single-particle spectrum displays a \emph{different} power law. We show that it is this power-law regime, corresponding to approximate local quantum criticality, which is continuously connected to and responsible for the apparent quantum critical scaling above the classical critical end point. We connect our findings to experiments on tunable Mott materials.

Compound redistribution due to droplet evaporation on a thin polymeric film: Theory

A thin polymeric film in contact with a fluid body may leach low-molecular-weight compounds into the fluid. If this fluid is a small droplet, the compound concentration within the liquid increases due to continuous leaching in addition to the evaporation of the droplet. This may eventually lead to an inversion of the transport process and a redistribution of the compounds within the thin film. In order to gain an understanding of the compound redistribution, we apply a macroscopic model for the evaporation of a droplet and combine that with a diffusion model for the compound transport. In the model, material deposition and the resulting contact line pinning are associated with the precipitation of a fraction of the dissolved material. We find three power law regimes for the size of the deposit area as a function of the initial droplet size, dictated by the competition between evaporation, diffusion, and the initial compound concentrations in the droplet and the thin film. The strength of the contact line pinning determines the deposition profile of the precipitate, characterized by a pronounced edge and a linearly decaying profile toward the center of the stain. Our predictions for the concentration profile within the solid substrate resemble patterns found experimentally.

Shear Rheology of Unentangled and Marginally Entangled Ring Polymer Melts from Large-Scale Nonequilibrium Molecular Dynamics Simulations.

We present results for the steady state shear rheology of non-concatenated, unentangled and marginally entangled ring poly(ethylene oxide) (PEO) melts from detailed, atomistic nonequilibrium molecular dynamics (NEMD) simulations, and compare them to the behavior of the corresponding linear melts. The applied flow field spans a wide range of shear rates, from the linear (Newtonian) to the highly non-linear (described by a power law) regime. For all melts studied, rings are found to exhibit shear thinning but to a lesser degree compared to linear counterparts, mostly due to their reduced deformability and stronger resistance to alignment in the direction of flow. These features are attributed to the more compact structure of ring molecules compared to linear chains; the latter are capable of adopting wider and more open conformations even under shear due to the freedom provided by the free ends. Similar to linear melts, rings also exhibit a first and a second normal stress coefficient; the latter is negative. The ratio of the magnitude of the two coefficients remains practically constant with shear rate and is systematically higher than the corresponding one for linear melts. Emphasis was also given to the statistics of terminal (re-orientational) relaxation times which we computed by analyzing all chains in the simulated systems one by one; it was demonstrated that long time dynamics are strongly heterogeneous both for rings and (especially) linears. Repeating the analysis under flow conditions, and as expected, we found that the applied flow field significantly suppresses dynamic heterogeneity, especially for high shear rates well beyond the Newtonian plateau. Finally, a detailed geometrical analysis revealed that the average population of ring–ring threading events in the longest melt studied here (the PEO-5k ring) remains practically unaffected by the imposed flow rate even at strong shear rates, except for multi-threadings which disappear. To further analyze this peculiar and rather unexpected effect, we computed the corresponding survival times and penetration lengths, and found that the overwhelming majority of threadings under shear are extremely weak constraints, as they are characterized by very small penetration lengths, thus also by short survival times. They are expected therefore to play only a minor (if any) role on chain dynamics.

Fabrication of Guar-Only Electrospun Nanofibers by Exploiting a High- and Low-Molecular Weight Blend.

We present a facile approach to electrospin nanofibers of guar galactomannan by blending high- and low-molecular weights (MWs) of guar. We discover that while neither native high MW guar nor hydrolyzed low MW guar is electrospinnable on its own, their combination leads to synergism in producing defect-free nanofibers. Such an approach of fabricating nanofibers from blending multiple MWs of the same polymer may provide an easy route to produce nanofibers of biopolymers which are typically hard to electrospin. Rheological studies reveal that a limiting amount of native guar is needed for electrospinnability, and for those systems that have the proportionate amount of native guar, there is a critical total concentration above which fibers form. Interestingly, a plot of blend viscosity versus guar concentration reveals two power-law regimes with an inflection point, above which fiber formation can be achieved akin to the behavior observed for pure (i.e., nonblend) polymers.

Effects of inhibitory and excitatory neurons on the dynamics and control of avalanching neural networks.

The statistical analysis of the collective neural activity known as avalanches provides insight into the proper behavior of brains across many species. We consider a neural network model based on the work of Lombardi, Herrmann, De Arcangelis etal. that captures the relevant dynamics of neural avalanches, and we show how tuning the fraction of inhibitory neurons in this model alters the connectivity of the network over time, removes exponential cut-offs present in the distributions of avalanche size and duration, and transitions the power spectral density of the network into an "epileptic" regime. We propose that the brain operates away from this power-law regime of low inhibitory fraction to protect itself from the dominating avalanches present in these extended distributions. We present control strategies that curtail these power-law distributions through either random or, more effectively, targeted disabling of excitatory neurons.

Induced Gravitational Collapse, Binary-Driven Hypernovae, Long Gramma-ray Bursts and Their Connection with Short Gamma-ray Bursts

There is increasing observational evidence that short and long Gamma-ray bursts (GRBs) originate in different subclasses, each one with specific energy release, spectra, duration, etc, and all of them with binary progenitors. The binary components involve carbon-oxygen cores (CO core ), neutron stars (NSs), black holes (BHs), and white dwarfs (WDs). We review here the salient features of the specific class of binary-driven hypernovae (BdHNe) within the induced gravitational collapse (IGC) scenario for the explanation of the long GRBs. The progenitor is a CO core -NS binary. The supernova (SN) explosion of the CO core , producing at its center a new NS ( ν NS), triggers onto the NS companion a hypercritical, i.e., highly super-Eddington accretion process, accompanied by a copious emission of neutrinos. By accretion the NS can become either a more massive NS or reach the critical mass for gravitational collapse with consequent formation of a BH. We summarize the results on this topic from the first analytic estimates in 2012 all the way up to the most recent three-dimensional (3D) smoothed-particle-hydrodynamics (SPH) numerical simulations in 2018. Thanks to these results it is by now clear that long GRBs are richer and more complex systems than thought before. The SN explosion and its hypercritical accretion onto the NS explain the X-ray precursor. The feedback of the NS accretion, the NS collapse and the BH formation produce asymmetries in the SN ejecta, implying the necessity of a 3D analysis for GRBs. The newborn BH, the surrounding matter and the magnetic field inherited from the NS, comprises the inner engine from which the GRB electron-positron ( e + e − ) plasma and the high-energy emission are initiated. The impact of the e + e − on the asymmetric ejecta transforms the SN into a hypernova (HN). The dynamics of the plasma in the asymmetric ejecta leads to signatures depending on the viewing angle. This explains the ultrarelativistic prompt emission in the MeV domain and the mildly-relativistic flares in the early afterglow in the X-ray domain. The feedback of the ν NS pulsar-like emission on the HN explains the X-ray late afterglow and its power-law regime. All of the above is in contrast with a simple GRB model attempting to explain the entire GRB with the kinetic energy of an ultrarelativistic jet extending through all of the above GRB phases, as traditionally proposed in the “collapsar-fireball” model. In addition, BdHNe in their different flavors lead to ν NS-NS or ν NS-BH binaries. The gravitational wave emission drives these binaries to merge producing short GRBs. It is thus established a previously unthought interconnection between long and short GRBs and their occurrence rates. This needs to be accounted for in the cosmological evolution of binaries within population synthesis models for the formation of compact-object binaries.

Crack propagation in elastomers subjected to cyclic loading

This work tackles crack propagation mechanisms of styrene butadiene and natural rubbers, both subject to cyclic loading under a positive load ratio. The experimental evidence is based on pure shear loading of notched specimens supported by SEM analysis of fracture surfaces. Relationship between crack growth rate and tearing energy is revisited. The experimental results show that the fatigue behaviour of studied vulcanisates can be clearly separated in the power-law regime. Furthermore, the reported results from the literature extend this statement to any load condition. Fracture surface observations reveal also differences in energy dissipation mechanisms inferred to structural mutations in the case of natural rubber. Alternation of rough and smooth fracture surfaces associated with change in crack growth rate is more likely to occur for styrene butadiene rubber. All observations suggest that crack branching is a secondary ranked dissipation mechanism for the studied materials compared to crack deviation.

Power Law Size Distributions in Geoscience Revisited

The size or energy of diverse structures or phenomena in geoscience appears to follow power law distributions. A rigorous statistical analysis of such observations is tricky, though. Observables can span several orders of magnitude, but the range for which the power law may be valid is typically truncated, usually because the smallest events are too tiny to be detected and the largest ones are limited by the system size. We revisit several examples of proposed power law distributions dealing with potentially damaging natural phenomena. Adequate fits of the distributions of sizes are especially important in these cases, given that they may be used to assess long‐term hazard. After reviewing the theoretical background for power law distributions, we improve an objective statistical fitting method and apply it to diverse data sets. The method is described in full detail, and it is easy to implement. Our analysis elucidates the range of validity of the power law fit and the corresponding exponent and whether a power law tail is improved by a truncated lognormal. We confirm that impact fireballs and Californian earthquakes show untruncated power law behavior, whereas global earthquakes follow a double power law. Rain precipitation over space and time and tropical cyclones show a truncated power law regime. Karst sinkholes and wildfires, in contrast, are better described by truncated lognormals, although wildfires also may show power law regimes. Our conclusions only apply to the analyzed data sets but show the potential of applying this robust statistical technique in the future.

High-temperature deformation mechanisms and processing maps of equiatomic CoCrFeMnNi high-entropy alloy

The hot compressive deformation mechanism and processing maps of the equiatomic FCC CoCrFeMnNi high-entropy alloy (HEA) were studied in the temperature range between 1023 and 1323 K and in the strain rate range between 10−3 and 10 s−1. At high strain rates above 1 s-1, strain hardening was dominant even at the very high temperature of 0.84Tm, which may be attributed to the sluggish diffusion coefficient and low stacking fault energy of the CoCrFeMnNi HEA, leading to suppression of dynamic recovery. According to the processing maps, the best condition for hot working was near 10−3 s−1 at 1323 K. Power-law breakdown and unstable flow occurred at low temperatures and high strain rates where the strain hardening was pronounced. The activation energy for plastic flow measured in the power-law creep regime when considering the dependence of elastic modulus on temperature was 312.2 kJ/mol; this value is close to the activation energy for the weighted diffusion coefficient calculated by weighting the contribution of each element in the CoCrFeMnNi HEA (284 kJ/mol). The size and fraction of the dynamically recrystallized grains increased as the strain rate decreased and the temperature increased, as in conventional metals. Both discontinuous dynamic recrystallization and continuous dynamic recrystallization (CDRX) occurred. CDRX became more distinct as the temperature increased. The deformation mechanism and behavior of the CoCrFeMnNi HEA were very similar to those of FCC pure metals in terms of the stress exponent and the effect of the stacking fault energy and diffusivity on the creep rates.

Receptor-Ligand Rebinding Kinetics in Confinement

Rebinding kinetics of molecular ligands plays a key role in the operation of biomachinery, from regulatory networks to protein transcription, and is also a key factor in design of drugs and high-precision biosensors. In this study, we investigate initial release and rebinding of ligands to their binding sites grafted on a planar surface, a situation commonly observed in single-molecule experiments and that occurs in vivo, e.g., during exocytosis. Via scaling arguments and molecular dynamic simulations, we analyze the dependence of nonequilibrium rebinding kinetics on two intrinsic length scales: the average separation distance between the binding sites and the total diffusible volume (i.e., height of the experimental reservoir in which diffusion takes place or average distance between receptor-bearing surfaces). We obtain time-dependent scaling laws for on rates and for the cumulative number of rebinding events. For diffusion-limited binding, the (rebinding) on rate decreases with time via multiple power-law regimes before the terminal steady-state (constant on-rate) regime. At intermediate times, when particle density has not yet become uniform throughout the diffusible volume, the cumulative number of rebindings exhibits a novel, to our knowledge, plateau behavior because of the three-dimensional escape process of ligands from binding sites. The duration of the plateau regime depends on the average separation distance between binding sites. After the three-dimensional diffusive escape process, a one-dimensional diffusive regime describes on rates. In the reaction-limited scenario, ligands with higher affinity to their binding sites (e.g., longer residence times) delay entry to the power-law regimes. Our results will be useful for extracting hidden timescales in experiments such as kinetic rate measurements for ligand-receptor interactions in microchannels, as well as for cell signaling via diffusing molecules.

SST Dynamics at Different Scales: Evaluating the Oceanographic Model Resolution Skill to Represent SST Processes in the Southern Ocean

AbstractIn this study we demonstrate the many strengths of scale analysis: we use it to evaluate the Nucleus for European Modelling of the Ocean model skill in representing sea surface temperature (SST) in the Southern Ocean by comparing three model resolutions: 1/12°, 1/4°, and 1°. We show that while 4–5 times resolution scale is sufficient for each model resolution to reproduce the magnitude of satellite Earth Observation (EO) SST spatial variability to within ±10%, the representation of ∼100‐km SST variability patterns is substantially (e.g., ∼50% at 750 km) improved by increasing model resolution from 1° to 1/12°. We also analyzed the dominant scales of the SST model input drivers (short‐wave radiation, air‐sea heat fluxes, wind stress components, wind stress curl, and bathymetry) variability with the purpose of determining the optimal SST model input driver resolution. The SST magnitude of variability is shown to scale with two power law regimes separated by a scaling break at ∼200‐km scale. The analysis of the spatial and temporal scales of dominant SST driver impact helps to interpret this scaling break as a separation between two different dynamical regimes: the (relatively) fast SST dynamics below ∼200 km governed by eddies, fronts, Ekman upwelling, and air‐sea heat exchange, while above ∼200 km the SST variability is dominated by long‐term (seasonal and supraseasonal) modes and the SST geography.

Convolution Lagrangian perturbation theory for biased tracers beyond general relativity

We compare analytic predictions for real and Fourier space two-point statistics for biased tracers from a variety of Lagrangian Perturbation Theory approaches against those from state of the art N-body simulations in $f(R)$ Hu-Sawicki and the nDGP braneworld modified gravity theories. We show that the novel physics of gravitational collapse in scalar tensor theories with the chameleon or the Vainshtein screening mechanism can be effectively factored in with bias parameters analytically predicted using the Peak-Background Split formalism when updated to include the environmental sensitivity of modified gravity theories as well as changes to the halo mass function. We demonstrate that Convolution Lagrangian Perturbation Theory (CLPT) and Standard Perturbation Theory (SPT) approaches provide accurate analytic methods to predict the correlation function and power spectra, respectively, for biased tracers in modified gravity models and are able to characterize both the BAO, power-law and small scale regimes needed for upcoming galaxy surveys such as DESI, Euclid, LSST and WFIRST.

Dynamics of the cosmological models with perfect fluid in Einstein\u2013Gauss\u2013Bonnet gravity: low-dimensional case

In this paper we performed investigation of the spatially-flat cosmological models whose spatial section is product of three- (“our Universe”) and extra-dimensional parts. The matter source chosen to be the perfect fluid which exists in the entire space. We described all physically sensible cases for the entire range of possible initial conditions and parameters as well as brought the connections with vacuum and Lambda -term regimes described earlier. In the present paper we limit ourselves with D=1, 2 (number of extra dimensions). The results suggest that in D=1 there are no realistic compactification regimes while in D=2 there is if alpha > 0 (the Gauss–Bonnet coupling) and the equation of state omega < 1/3; the measure of the initial conditions leading to this regime is increasing with growth of omega and reaches its maximum at omega rightarrow 1/3 - 0. We also describe some pecularities of the model, distinct to the vacuum and Lambda -term cases – existence of the isotropic power-law regime, different role of the constant-volume solution and the presence of the maximal density for D = 2, alpha < 0 subcase and associated features.

Gaussian and critical scalings in the magnetoconductivity fluctuations of Y3Ba5Cu8O18 superconductor

We have studied the superconducting transition and the magnetoconductivity fluctuations in the polycrystalline Y3Ba5Cu8O[Formula: see text] (Y358) superconductor under magnetic fields upto 1 T. A two-step superconducting transition could be observed as a consequence of the granular structure of the sample, which is strongly affected by the applied magnetic field. Gaussian and genuine critical 3D-XY-E fluctuation regimes were identified. A critical scaling regime beyond 3D-XY was identified for magnetic fields upto 0.25 T, corresponding to the averaged exponent 0.19 and suggesting the occurrence of the weak first-order character of the superconducting transition. In the approximation to the zero resistance a power law regime could be observed, corresponding to the averaged exponent 2.37, which are smaller than previously reported for the Y358 system. Our results are discussed in terms of the Y358 and YBa2Cu3O[Formula: see text](Y123) results in the literature.

Theory for IMPS on Rough and Finite Fractal Dye Sensitized Solar Cell

A generalized theoretical model of intensity modulated photocurrent spectroscopy (IMPS) for the random morphology in a dye sensitized solar cell (DSSC) under uniform illumination is developed. The generalized IMPS expression for the disordered semiconducting/conducting glass interface of a DSSC is obtained in term of power spectral density of roughness. Influence of surface roughness on the dynamic response of DSSC originate due to the coupling of characteristic phenomenological and morphological length scales. A detailed analysis of IMPS response is performed over finite self- affine fractals to highlight roughness induced anomalies and cause of photocurrent enhancement. The IMPS of a rough DSSC exhibit three characteristic frequency regimes: lifetime of charge carrier dependent low frequency regime, surface irregularity dependent intermediate power-law regime and diffusion controlled high frequency regime. Finally, our theory facilitates the understanding of dynamics and kinetics of charge carriers under the influence of ubiquitous surface disorder.

A model for the stress tensor in dilute suspensions of rigid spheroids in a generalized Newtonian fluid

When non-spherical particles, like rods or discs, are added to a liquid the increase in viscosity of the mixture is often described by the Lipscomb model. One important result of Lipscomb's model is that it predicts a strong increase in the viscosity of the suspension with increasing aspect ratio of the filler particles. Despite the fact that this model was originally proposed for a Newtonian matrix fluid it is also applied to polymer melts filled with non-spherical particles. Such an approach completely decouples the influence of the particle shape from the nonlinear properties of the suspending fluid. Yet, since polymer melts often exhibit strong non-Newtonian behavior, e.g. shear thinning, it is to be expected that such a superposition will give a wrong prediction of the suspension viscosity.To investigate this problem we performed a numerical study of a suspension based on a non-Newtonian matrix fluid and rigid spheroidal particles. In particular, we simulated different flows of a Carreau fluid around spheroidal particles and used numerical homogenization to obtain the intrinsic viscosity of the suspension as function of applied rate of deformation, thinning exponent and aspect ratio. In the Newtonian regime we also compare with results from literature. In the transition region from Newtonian to non-Newtonian behavior we obtained lower values of the intrinsic viscosity. In the power-law regime of the Carreau model, i.e. at high deformation rates, we found that the intrinsic viscosity of the suspension is independent of the applied rate of deformation. Further we obtained from the simulations that the intrinsic viscosity at high deformation rates strongly depends not only on the aspect ratio of the particles but also one the thinning exponent in the Carreau model, implying that the superposition approach in fact leads to a wrong prediction of the suspension viscosity at high deformation rates.

Stochastic modeling of nonstationary earthquake time series with long-term clustering effects

Earthquake time series are widely used to characterize the main features of regional seismicity and to provide useful insights into earthquake dynamics. Properties such as intermittency and nonstationary clustering are common in earthquake time series, highlighting the complex nature of the earthquake generation process. In the present work we introduce a stochastic model with memory effects that reproduces the temporal scaling behavior observed in regional seismicity. For nonstationary earthquake activity, where the average seismic rate fluctuates, the solution of the stochastic model is the $q$-generalized gamma function that presents two power-law regimes for short and long waiting times, respectively, while for stationary activity it reduces to the standard gamma function. To validate the derived model, we study nonstationary earthquake time series from Southern California and Japan. The analysis shows that for various threshold magnitudes and spatial areas and after rescaling with the mean waiting time, the normalized probability density functions fall onto a unique curve, which is characterized by two power-law regimes for short and long waiting times, respectively, a scaling behavior that can exactly be recovered by the derived $q$-generalized gamma function. The results show the validity of the stochastic model and the derived scaling function, further signifying both short- and long-term clustering effects and memory in the evolution of seismicity.