Critical Coarsening Research Articles

Slow quench dynamics in classical systems: kinetic Ising model and zero-range process

While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.

Quenched dynamics of artificial colloidal spin ice

Artificial spin ices are ideal frustrated model systems in which to explore or design emergent phenomena with unprecedented characterization of the constituent degrees of freedom. In square spin ice, violations of the ice rule are topological excitations essential to the kinetics of the system, providing an ideal testbed for studying the dynamics of such defects under varied quench rates. In this work we describe the first test of the Kibble-Zurek mechanism and critical coarsening in colloidal square and colloidal hexagonal ice under quenches from a weakly interacting liquid state into a strongly interacting regime. As expected, for infinitely slow quenches, the system is defect free, while for increasing quench rate, an increasing number of defects remain in the sample. For square ice, we find regimes in which the defect population decreases as a power law with decreasing quench rate. A detailed scaling analysis shows that for a wide range of parameters, including quench rates that are accessible by experiments, the behavior is described by critical coarsening rather than by the Kibble-Zurek mechanism, since the defect-defect interactions are long ranged. For quenches closer to the critical point, however, there can be a competition between the two mechanisms.

Lane formation and critical coarsening in a model of bacterial competition.

We study competition of two nonmotile bacterial strains in a three-dimensional channel numerically and analyze how their configuration evolves in space and time. We construct a lattice model that takes into account self-replication, mutation, and killing of bacteria. When mutation is not significant, the two strains segregate and form stripe patterns along the channel. The formed lanes are gradually rearranged, with increasing length scales in the two-dimensional cross-sectional plane. We characterize it in terms of coarsening and phase ordering in statistical physics. In particular, for the simple model without mutation and killing, we find logarithmically slow coarsening, which is characteristic of the two-dimensional voter model. With mutation and killing, we find a phase transition from a monopolistic phase, in which lanes are formed and coarsened until the system is eventually dominated by one of the two strains, to an equally mixed and disordered phase without lane structure. Critical behavior at the transition point is also studied and compared with the generalized voter class and the Ising class. These results are accounted for by continuum equations, obtained by applying a mean-field approximation along the channel axis. Our findings indicate relevance of critical coarsening of two-dimensional systems in the problem of bacterial competition within anisotropic three-dimensional geometry.

Growth kinetics of the two-dimensional Ising model with finite cooling rates.

We investigate, via spin-flip kinetic Monte Carlo simulations, the entire growth kinetic process of the two-dimensional kinetic Ising model cooled to zero temperature with finite cooling rates. A new slow dynamics, suppressed, and unseen in the instantaneous quench emerges and eventually crosses over to the asymptotic standard coarsening behavior. We present a numerical observation of the excess defect number density that provides direct information on the generated defects due to the critical slowing down. We find that the excess defect density reveals the dynamics of defect generation, which is shown to be in good agreement with the scaling theory pioneered by Kibble and Zurek (KZ). The dynamic spin correlation function reveals that the impulse regime alluded by KZ is characterized by a unique critical coarsening process with the growth law dictated by the Kibble-Zurek mechanism. We determine a new dynamic exponent governing the growth kinetics at the onset times of the zero temperature. The proposed scaling scheme for the excess defect density leads to an analytic expression for this dynamic exponent involving the KZ exponents, indicating the extended influence of the KZ mechanism even down to the kinetics at the onset times of zero temperature. We also perform dynamic simulations of critical heating with finite rates from zero temperature where the power law relaxation (with respect to the inverse cooling rates) of the magnetization can be explained clearly by the KZ exponent.

Disorder Aggregation by Vortices in Non-Equilibrium Critical Annealing of Two-Dimensional XY-model

It is shown that the non-equilibrium vortex annealing of structural disorder in a two-dimensional XY-model leads to a non-equilibrium critical coarsening. Large clusters are separated into an individual subsystem, the relaxation dynamics of which is much slower than the relaxation dynamics of the cluster subsystem as a whole. Despite the fact that the main equilibrium critical properties, even the Berezinskii-Kosterlitz-Thouless phase transition temperature TBKT(p), of the two-dimensional XY-model with disorder thermalization do not change as compared with the system with quench disorder, the non-equilibrium critical relaxation shows significant differences.

Critical dynamics of classical systems under slow quench

We study the slow quench dynamics of a one-dimensional nonequilibrium lattice gas model which exhibits a phase transition in the stationary state between a fluid phase with homogeneously distributed particles and a jammed phase with a macroscopic hole cluster. Our main result is that in the critical region (i.e., at the critical point and in its vicinity) where the dynamics are assumed to be frozen in the standard Kibble-Zurek argument, the defect density exhibits an algebraic decay in the inverse annealing rate with an exponent that can be understood using critical coarsening dynamics. However, in a part of the critical region in the fluid phase, the standard Kibble-Zurek scaling holds. We also find that when the slow quench occurs deep into the jammed phase, the defect density behavior is explained by the rapid quench dynamics in this phase.

Stabilisation of AMG solvers for saddle‐point stokes problems

SummaryWhen a block factorisation is used to precondition the saddle‐point equations of the discrete Stokes problem, the stability that this gives for the relaxation of residual errors may not be conserved in the coarse‐grid approximations (CGA) of algebraic multi‐grid (AMG) solvers. If the same first‐order interpolation is used in the inter‐grid transfer operators for the scalar and the vector fields, the conditioning degrades with each coarsening step until eventually a critical coarsening is reached beyond which residual errors are no longer damped and will become divergent with any further coarsening. It is shown that by introducing the same block pre‐conditioner as an integral part of the coarsening algorithm, stable smoothing can be maintained at all levels of the CGA. The pre‐conditioning need only be applied at preselected grid levels, one immediately before the critical threshold and others beyond that level if required. Excessive complexity in the CGA is thereby avoided. The method is purely algebraic and may be used for both classical AMG solvers and for smoothed‐aggregation AMG solvers. It should be applicable to other coupled vector and scalar fields in science and engineering that involve second‐order (block‐diagonal) and first‐order (block‐off‐diagonal) discrete difference operators. Copyright © 2015 John Wiley & Sons, Ltd.

Quantum quenches, dynamical transitions, and off-equilibrium quantum criticality

Several mean-field computations have revealed the existence of an out-of-equilibrium dynamical transition induced by quantum-quenching an isolated system starting from its symmetry-broken phase. In this work we focus on the quantum ${\ensuremath{\phi}}^{4}$ $N$-component field theory. By taking into account dynamical fluctuations at the Hartree-Fock level, corresponding to the leading order of the $1/N$ expansion, we derive the critical properties of the dynamical transition beyond mean-field theory (including at finite temperature). We find diverging time and length scales, dynamic scaling, and aging. Finally, we unveil a relationship with critical coarsening, an off-equilibrium regime that can be induced by quenching from the symmetric toward the symmetry-broken phase.

Stability of algebraic multigrid for Stokes problems

SUMMARYAn investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle‐point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed‐order and equal‐order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh‐dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed‐order interpolation to permit stable smoothing and therefore to deliver mesh‐independent convergence is explained. Two expedient options are suggested for obtaining mesh‐independent convergence for those AMG codes that are based on an equal‐order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.

Coarsening of a nonequilibrium kinetic Ising chain with absorbing transitions: spatialcorrelation of the order parameters and their dynamic scalings

A one-dimensional nonequilibrium kinetic Ising model possessing two symmetricabsorbing states exhibits simultaneous occurrence of two distinct phase transitions,namely the order–disorder transition and the active–absorbing transition. Westudy the coarsening (both critical and subcritical) process evolving towards thenonequilibrium steady state of this model via the spatial correlations of the orderparameters and their dynamic scaling features. For critical coarsening, the lengthscales associated with the two transitions exhibit the same superdiffusive growthbehavior. Critical dynamic scaling behaviors satisfied by the order parametercorrelations provide a general relationship between the defect relaxation and thegrowing length scale. Some of the standard scaling relations between the staticcritical exponents for the order–disorder transition are found to be modified for theabsorbing transition. For subcritical coarsening, the defect correlation exhibitsnon-monotonic spatial structure due to the interplay between branching and annihilationprocesses of defects. This structure reveals that, in addition to the usual diffusivelygrowing length scale, there exists a new length scale that grows subdiffusively intime.

Dynamic heterogeneities in critical coarsening: exact results for correlation and responsefluctuations in finite-sized spherical models

We study dynamic heterogeneities in the out-of-equilibrium coarsening dynamicsof the spherical ferromagnet after a quench from infinite temperature to itscritical point. A standard way of probing such heterogeneities is by monitoringthe fluctuations of correlation and susceptibility, coarse-grained over mesoscopicregions. We discuss how to define such fluctuating coarse-grained correlationsand susceptibilities in models where no quenched disorder is present. Ourfocus for the spherical model is on coarse-graining over the whole volume ofN spins, which requires accounting for non-Gaussian fluctuations of the spin variables. The latter are treated as a perturbationabout the leading order Gaussian statistics. We obtain exact results for these quantities,which enable us to characterize the joint distribution of correlation and susceptibilityfluctuations. We find that this distribution is qualitatively different, even for equilibriumabove criticality, from the spin-glass scenario where correlation and susceptibilityfluctuations are linked in a manner akin to the fluctuation-dissipation relation between theaverage correlation and susceptibility. Our results show that coarsening at criticality isclearly heterogeneous above the upper critical dimension and suggest that, as inother glassy systems, there is a well-defined timescale on which fluctuations acrossthermal histories are largest. Surprisingly, however, neither this timescale nor theamplitude of the heterogeneities increases with the age of the system, as would beexpected from the growing correlation length. Below the upper critical dimension,the strength of correlation and susceptibility fluctuations varies on a timescaleproportional to the age of the system; the corresponding amplitude also grows withage, but does not scale with the correlation volume as might have been expectednaively.

Dynamic behavior of the voter model on fractals: logarithmic-periodic oscillations as a signature of time discrete scale invariance

The understanding of the dynamic behavior of the voter model, in low-dimensional media, is a very interesting open topic. In fact, due to the absence of the interfacial tension, only the interfacial noise becomes relevant during the coarsening processes, bringing the possibility of studing a new physical process. In this way, it is known that below the upper critical dimension (d < 2) and starting from a disordered configuration, a critical coarsening process takes place, and the density of interfaces, ρ(t), decays as a power-law function of time. Recently published numerical studies performed on low-dimensional fractal substrates (dF < 2) [Physica A 362, 338 (2006)] show the existence of logarithmic-periodic oscillations superimposed on the standard ρ(t) power-law behavior, but the origin of those oscillations remains unclear. In this work, we provide an explanation of these oscillations in terms of the interplay between the dynamics of the voter model and the discrete scale invariance of the underlying fractal substrate. Our arguments are verified by means of extensive numerical simulations carried out on different fractal substrates.

Fluctuation–dissipation relations in critical coarsening: crossover from unmagnetized to magnetized initial states

We study the non-equilibrium dynamics of the spherical ferromagnet quenched to its critical temperature, as a function of the magnetization of the initial state. The two limits of unmagnetized and fully magnetized initial conditions can be understood as corresponding to times that are respectively much shorter and much longer than a magnetization timescale, as in a recent field theoretical analysis of the n-vector model. We calculate exactly the crossover functions interpolating between these two limits, for the magnetization correlator and response and the resulting fluctuation–dissipation ratio (FDR). For d > 4 our results match those obtained recently from a Gaussian field theory. For d < 4, non-Gaussian fluctuations arising from the spherical constraint need to be accounted for. We extend our framework from the fully magnetized case to achieve this, providing an exact solution for the relevant integral kernel. The resulting crossover behaviour is very rich, with the asymptotic FDR X∞ depending non-monotonically on the scaled age of the system. This is traced back to non-monotonicities of the two-time correlator, themselves the consequence of large magnetization fluctuations on the crossover timescale. We correct a trivial error in our earlier calculation for fully magnetized initial states; the corrected FDR is consistent with renormalization group expansions to first order in 4 − d for the longitudinal fluctuations of the O(n) model in the limit n → ∞.

Microstructural evolution of commercial pure iron during directional annealing

Microstructural evolution during the directional annealing of the cold-rolled pure iron has been investigated. Columnar grains with the largest aspect ratio of 23.6 were produced in the specimen cold-rolled for 70% of thickness reduction at the hot zone temperature (HZT) of 850 °C and a temperature gradient (TG) of 200 °C/cm with an optimum drawing velocity of 5 μm/s. It was found that the columnar grain structure was developed due to abnormal grain growth. There is a critical coarsening temperature, below which the secondary recrystallization does not occur during the annealing; consequently no columnar grain structure can be produced. No obvious evidence shows that the grain size after primary recrystallization has an influence on the microstructural evolution. Due to the strong crystallographic texture of the small grains ahead of the hot zone, the columnar grains have small aspect ratios. Most of the boundaries of the columnar grain are high angle coincidence site lattice (CSL) grain boundaries. The grains with twin or low angle grain boundaries are apt to form island grains in the columnar structure.

CRITICAL COARSENING DYNAMICS OF SPIN S =1/2, 3/2 ANTIFERROMAGNETIC ISING MODEL ON TRIANGULAR LATTICE

The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.

Spin, bond and global fluctuation–dissipation relations in the non-equilibrium spherical ferromagnet

We study the out-of-equilibrium dynamics of the spherical ferromagnet after a quench to its critical temperature. We calculate correlation and response functions for spin observables which probe lengthscales much larger than the lattice spacing but smaller than the system size, and find that the asymptotic fluctuation-dissipation ratio (FDR) X^inf is the same as for local observables. This is consistent with our earlier results for the Ising model in dimension d=1 and d=2. We also check that bond observables, both local and long-range, give the same asymptotic FDR. In the second part of the paper the analysis is extended to global observables, which probe correlations among all N spins. Here non-Gaussian fluctuations arising from the spherical constraint need to be accounted for, and we develop a systematic expansion in 1/sqrt{N} to do this. Applying this to the global bond observable, i.e. the energy, we find that non-Gaussian corrections change its FDR to a nontrivial value which we calculate exactly for all dimensions d>2. Finally, we consider quenches from magnetized initial states. Here even the FDR for the global spin observable, i.e. the magnetization, is nontrivial. It differs from the one for unmagnetized states even in d>4, signalling the appearance of a distinct dynamical universality class of magnetized critical coarsening. For lower d the FDR is irrational even to first order in 4-d and d-2, the latter in contrast to recent results for the n-vector model.

Universality of fluctuation-dissipation ratios: The ferromagnetic model

We calculate analytically the fluctuation-dissipation ratio (FDR) for Ising ferromagnets quenched to criticality, both for the long-range model and its short-range analog in the limit of large dimension. Our exact solution shows that, for both models, if the system is unmagnetized while if the initial magnetization is nonzero. This indicates that two different classes of critical coarsening dynamics need to be distinguished depending on the initial conditions, each with its own nontrivial FDR. We also analyze the dependence of the FDR on whether local and global observables are used. These results clarify how a proper local FDR (and the corresponding effective temperature) should be defined in long-range models in order to avoid spurious inconsistencies and maintain the expected correspondence between local and global results; global observables turn out to be far more robust tools for detecting nonequilibrium FDRs.

Critical coarsening without surface tension: the universality class of the voter model.

We show that the two-dimensional voter model, usually considered to be only a marginal coarsening system, represents a broad class of models for which phase ordering takes place without surface tension. We argue that voter-like growth is generically observed at order-disorder nonequilibrium transitions solely driven by interfacial noise between dynamically symmetric absorbing states.