Dynamic Scaling Hypothesis Research Articles

Non-equilibrium Dynamics of Vortices in Two-Dimensional Quantum Gases: Determining the Dynamical Scaling Region Using the Mahalanobis Distance

When a system undergoes a rapid quench from a disordered to an ordered phase, it does not order instantly but instead relaxes towards equilibrium over time. During this relaxation, the dynamical scaling hypothesis predicts that the length scale of ordered regions increases, with later patterns statistically similar to earlier ones except for a change in global scale. Quantum gases are one of many systems in which such out of equilibrium behaviour is predicted to occur. Here, we present a method for systematically testing when dynamical scaling is taking place, by quantifying the similarity of the rescaled two-point correlation function over time using the Mahalanobis distance. Data on the velocity field of a two-dimensional quantum fluid, generated from point vortex simulations, are used to illustrate the application of this method.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals.

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

Simulation of phase-ordering dynamics of 2d Ising model under influence of oscillatory shear

In this paper, we study phase-ordering dynamics of 2d Ising model under influence of oscillatory shear by computer simulation. We focus on the time evolution of the equal-time correlation functions to test the dynamical scaling hypothesis and to determine the governing growth laws. We find that at late times the system has two different behaviours which are a non-shear behaviour and a uniformly sheared behaviour. Initial phases and frequencies of oscillatory shear determine the behaviour of the system. On one hand, if the frequency is high, the system behaves as if it is not under shear. On the other hand, if the frequency is approached zero, the behaviour of the system will be determined by the initial phase of oscillatory shear. At the initial phase of 90 degree (or 270 degree), the system also behaves as if it is not under shear. Otherwise, the system behaves as if it is under uniform shear (Oscillatory shear is in the form of cosine function).

Quench dynamics of an ultracold two-dimensional Bose gas

We study the dynamics of a two-dimensional Bose gas after an instantaneous quench of an initially ultracold thermal atomic gas across the Berezinskii-Kosterlitz-Thouless phase transition, confirming via stochastic simulations that the system undergoes phase ordering kinetics and fulfills dynamical scaling hypothesis at late-time dynamics. Specifically, we find in that regime the vortex number decaying in time as $\langle N_v \propto t^{-1}\rangle$, consistent with a dynamical critical exponent $z \approx 2$ for both temperature and interaction quenches. Focusing on finite-size box-like geometries, we demonstrate that such an observation is within current experimental reach.

Violation of single-length-scaling dynamics via spin vortices in an isolated spin-1 Bose gas

We consider the phase ordering dynamics of an isolated quasi-two-dimensional spin-1 Bose gas quenched into an easy-plane ferromagnetic phase. Preparing the initial system in an unmagnetized anti-ferromagnetic state the subsequent ordering involves both polar core and Mermin-Ho spin vortices, with the ratio between the different vortices controllable by the quench parameter. Ferromagnetic domain growth occurs as these vortices annihilate. The distinct dynamics of the two types of vortices means that the domain growth law is determined by two macroscopic length scales, violating the standard dynamic scaling hypothesis. Nevertheless we find that universality of the ordering process manifests in the decay laws for the spin vortices.

Dynamical Critical Exponents in Driven-Dissipative Quantum Systems.

We study the phase ordering of parametrically and incoherently driven microcavity polaritons after an infinitely rapid quench across the critical region. We confirm that the system, despite its driven-dissipative nature, satisfies the dynamical scaling hypothesis for both driving schemes by exhibiting self-similar patterns for the two-point correlator at late times of the phase ordering. We show that polaritons are characterized by the dynamical critical exponent z≈2 with topological defects playing a fundamental role in the dynamics, giving logarithmic corrections both to the power-law decay of the number of vortices and to the associated growth of the characteristic length scale.

Erratum to: Domain Size Distribution in Segregating Binary Superfluids

Domain size distribution in phase separating binary Bose--Einstein condensates is studied theoretically by numerically solving the Gross--Pitaevskii equations at zero temperature. We show that the size distribution in the domain patterns arising from the dynamic instability obeys a power law in a scaling regime according to the dynamic scaling analysis based on the percolation theory. The scaling behavior is kept during the relaxation development until the characteristic domain size becomes comparable to the linear size of the system, consistent with the dynamic scaling hypothesis of the phase-ordering kinetics. Our numerical experiments indicate the existence of a different scaling regime in the size distribution function, which can be caused by the so-called coreless vortices.

Domain Size Distribution in Segregating Binary Superfluids

Domain size distribution in phase separating binary Bose–Einstein condensates is studied theoretically by numerically solving the Gross–Pitaevskii equations at zero temperature. We show that the size distribution in the domain patterns arising from the dynamic instability obeys a power law in a scaling regime according to the dynamic scaling analysis based on the percolation theory. The scaling behavior is kept during the relaxation dynamics until the characteristic domain size becomes comparable to the linear size of the system, consistent with the dynamic scaling hypothesis of the phase-ordering kinetics. Our numerical experiments indicate the existence of a different scaling regime in the size distribution function, which can be caused by the so-called coreless vortices.

How soon after a zero-temperature quench is the fate of the Ising model sealed?

We study the transient between a fully disordered initial condition and a percolating structure in the low-temperature non-conserved order parameter dynamics of the two-dimensional Ising model. We show that a stable structure of spanning clusters establishes at a time . Our numerical results yield for the square and kagome lattices, for the triangular and for the bowtie-a lattices. We generalise the dynamic scaling hypothesis to take into account this new time-scale. We discuss the implications of these results for other non-equilibrium processes.

Stability of earthquake clustering models: Criticality and branching ratios

We study the stability conditions of a class of branching processes prominent in the analysis and modeling of seismicity. This class includes the epidemic-type aftershock sequence (ETAS) model as a special case, but more generally comprises models in which the magnitude distribution of direct offspring depends on the magnitude of the progenitor, such as the branching aftershock sequence (BASS) model and another recently proposed branching model based on a dynamic scaling hypothesis. These stability conditions are closely related to the concepts of the criticality parameter and the branching ratio. The criticality parameter summarizes the asymptotic behavior of the population after sufficiently many generations, determined by the maximum eigenvalue of the transition equations. The branching ratio is defined by the proportion of triggered events in all the events. Based on the results for the generalized case, we show that the branching ratio of the ETAS model is identical to its criticality parameter because its magnitude density is separable from the full intensity. More generally, however, these two values differ and thus place separate conditions on model stability. As an illustration of the difference and of the importance of the stability conditions, we employ a version of the BASS model, reformulated to ensure the possibility of stationarity. In addition, we analyze the magnitude distributions of successive generations of the BASS model via analytical and numerical methods, and find that the compound density differs substantially from a Gutenberg-Richter distribution, unless the process is essentially subcritical (branching ratio less than 1) or the magnitude dependence between the parent event and the direct offspring is weak.

Molecular View of Phase Coexistence in Lipid Monolayers

We used computer simulations to study the effect of phase separation on the properties of lipid monolayers. This is important for understanding the lipid-lipid interactions underlying lateral heterogeneity (rafts) in biological membranes and the role of domains in the regulation of surface tension by lung surfactant. Molecular dynamics simulations with the coarse-grained MARTINI force field were employed to model large length (~80 nm in lateral dimension) and time (tens of microseconds) scales. Lipid mixtures containing saturated and unsaturated lipids and cholesterol were investigated under varying surface tension and temperature. We reproduced compositional lipid demixing and the coexistence of liquid-expanded and liquid-condensed phases as well as liquid-ordered and liquid-disordered phases. Formation of the more ordered phase was induced by lowering the surface tension or temperature. Phase transformations occurred via either nucleation or spinodal decomposition. In nucleation, multiple domains formed initially and subsequently merged. Using cluster analysis combined with Voronoi tessellation, we characterized the partial areas of the lipids in each phase, the phase composition, the boundary length, and the line tension under varying surface tension. We calculated the growth exponents for nucleation and spinodal decomposition using a dynamical scaling hypothesis. At low surface tensions, liquid-ordered domains manifest spontaneous curvature. Lateral diffusion of lipids is significantly slower in the more ordered phase, as expected. The presence of domains increased the monolayer surface viscosity, in particular as a result of domain reorganization under shear.

Kinetics of phase separation and coarsening in dilute surfactant pentaethylene glycol monododecyl ether solutions

We investigated the phase separation phenomena in dilute surfactant pentaethylene glycol monodedecyl ether (C(12)E(5)) solutions focusing on the growth law of separated domains. The solutions confined between two glass plates were found to exhibit the phase inversion, characteristic of the viscoelastic phase separation; the majority phase (water-rich phase) nucleated as droplets and the minority phase (micelle-rich phase) formed a network temporarily, then they collapsed into an usual sea-island pattern where minority phase formed islands. We found from the real-space microscopic imaging that the dynamic scaling hypothesis did not hold throughout the coarsening process. The power law growth of the domains with the exponent close to 1/3 was observed even though the coarsening was induced mainly by hydrodynamic flow, which was explained by Darcy's law of laminar flow.

Comparison of branching models for seismicity and likelihood maximization through simulated annealing

[1] Stochastic branching models provide a good description of some aspects of the temporal organization in seismicity. Generally, they assume that magnitudes are independent of history, as in the widely used Epidemic Type Aftershock Sequence (ETAS) model. Here, we consider a recent epidemic‐like model where time‐magnitude and magnitude‐magnitude correlations are introduced via a dynamical scaling (DS) hypothesis, namely, the magnitude difference between earthquakes fixes the time scale for correlations. We also consider a variation of the ETAS model where the c parameter of the Omori law is not fixed but depends on the parent magnitude. We develop a novel procedure to maximize the log likelihood of the different models. This method is based on a Monte Carlo sampling in the parameter space with a variable step size during the evolution to converge to the given accuracy. The log likelihood indicates that the DS model provides the best fit for the major California sequences and the whole catalog, setting as lower magnitude thresholds Mc ≥ 3.5. For Mc = 3, the best fit is obtained by a model with c depending on both the parent magnitude and goes into Mc as in the generalized Omori law. The better performance of the DS model with respect to the ETAS model is attributed to correlations in magnitudes.

Evolution of resist roughness during development: stochastic simulation and dynamic scaling analysis

The study of surface-roughness evolution of resist films during development may elucidate the material origins of line-edge roughness. We use a stochastic simulator of resist development and analyze the resulting surface roughness evolution with dynamic scaling theory in polymers with homogeneous and inhomogeneous local dissolution behavior. In all cases, a power-law increase of root mean square (rms) roughness and correlation length is found. In homogenous polymers, a slow rms roughness increase is reported and the scaling exponents are shown to obey the dynamic scaling hypothesis of Family-Viscek. Dissolution inhomogeneity is inserted on a monomer scale (using copolymers), on a chain scale (using mixture of copolymers), or on a film scale by the activation of photo acid generator (PAG) molecules and the concomitant acid diffusion. Our simulator predicts that any kind of inhomogeneity may cause much larger rms roughness than homogeneous solubility. Furthermore, PAG-induced inhomogeneity results in a faster increase and larger values of roughness compared to chain-level inhomogeneity and this, in turn, leads to larger values compared to polymers with monomer-scale inhomogeneity. The differences in roughness are abruptly magnified when one reduces the resist solubility to levels in the range of the clearing dose. A comparison with experimental results shows good agreement with the simulation predictions.

Morphological evolution in ballistic deposition

Ballistic deposition model has been employed to simulate the morphological evolution of thin-film growth under non-normal incident flux that occurs frequently in realistic physical vapor deposition. We show that the growth front clearly deviates from the Kardar, Parisi, and Zhang universality class and the growth front exhibits a characteristic length scale that gives rise to a wavelength selection. The implication is that under realistic physical vapor deposition conditions, the growth front morphology is expected not to obey the well-established dynamic scaling hypothesis.

Short-time dynamics of finite-size mean-field systems

We study the short-time dynamics of a mean-field model with non-conserved orderparameter (Curie–Weiss with Glauber dynamics) by solving the associated Fokker–Planckequation. We obtain closed-form expressions for the first moments of the order parameter,near to both the critical and spinodal points, starting from different initial conditions. Thisallows us to confirm the validity of the short-time dynamical scaling hypothesis in bothcases. Although the procedure is illustrated for a particular mean-field model, ourresults can be straightforwardly extended to generic models with a single orderparameter.

Dynamic scaling study of vapor deposition polymerization: A Monte Carlo approach

The morphological scaling properties of linear polymer films grown by vapor deposition polymerization are studied by 1+1D Monte Carlo simulations. The model implements the basic processes of random angle ballistic deposition (F) , free-monomer diffusion (D) and monomer adsorption along with the dynamical processes of polymer chain initiation, extension, and merger. The ratio G=D/F is found to have a strong influence on the polymer film morphology. Spatial and temporal behavior of kinetic roughening has been extensively studied using finite-length scaling and height-height correlations H(r,t). The scaling analysis has been performed within the no-overhang approximation and the scaling behaviors at local and global length scales were found to be very different. The global and local scaling exponents for morphological evolution have been evaluated for varying free-monomer diffusion by growing the films at G=10 , 10(2), 10(3), and 10(4) and fixing the deposition flux F. With an increase in G from 10 to 10(4), the average growth exponent beta approximately 0.50 was found to be invariant, whereas the global roughness exponent alpha(g) decreased from 0.87 (1) to 0.73 (1) along with a corresponding decrease in the global dynamic exponent z(g) from 1.71(1) to 1.38(2). The global scaling exponents were observed to follow the dynamic scaling hypothesis, z(g)=alpha(g)/beta. With a similar increase in G however, the average local roughness exponent alpha(l) remained close to 0.46 and the anomalous growth exponent beta(*) decreased from 0.23(4) to 0.18(8). The interfaces display anomalous scaling and multiscaling in the relevant height-height correlations. The variation in H(r,t) with deposition time t indicates nonstationary growth. A comparison has been made between the simulational findings and the experiments wherever applicable.

Non-Equilibrium Relaxation Analysis on Two-Dimensional Melting

The melting transition in the hard-disk system is considered. Non-equilibrium relaxation analysis of the six-fold bond-orientational order parameter has been carried out. The critical point between the hexatic and the fluid phase is determined on the basis of the dynamic scaling hypothesis. The value of the critical exponent η is determined from the fluctuation of the order parameter at the criticality as η= 0.25(2) which is consistent with the prediction by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Statistical properties and universality in earthquake and solar flare occurrence

Earthquakes are phenomena of great complexity, however some simple general laws govern the statistics of their occurrence. Some of these most important laws exhibit scale invariance, as the Gutenberg-Richter law and the Omori law. The origin of these scaling behaviours is not yet fully understood and a natural fondamental question concerns the existence of these features also in other complex phenomena. A direct inspection of experimental catalogues has shown that the stochastic processes underlying solar flare and earthquake occurrence have universal properties. Another intensively debated question is the existence of correlations between magnitudes of subsequent earthquakes. Our recent analysis of the Southern California Catalogue has shown that non-zero magnitude correlations exist. A branching model based on a dynamical scaling hypothesis, relating magnitude to time, reproduces the hierarchical organization in time and magnitude of events and the observed magnitude correlations.

Dynamical scaling and generalized Omori law

The power law decay of the aftershocks rate is observed only after a characteristic time scale c. The dependence of c on the mainshock magnitude MM and on the lower cut‐off magnitude MI is well established. By considering ten sequences recorded in the California Catalog we show that the aftershock number distribution becomes independent of both MM and MI if time is rescaled by an appropriate time scale fixed by the difference MM − MI. This result is interpreted within a more general dynamical scaling hypothesis recently formulated, relating time differences to magnitude differences. The above hypothesis gives predictions in good agreement with the recent findings by Peng et al. (2007).