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Dynamics of pattern formation in self-organizing systems subjected to spontaneous suppression of diffusion through a recently proposed source-sink-boundary templating technique is presented based on three-dimensional numerical simulations. This template disorder is introduced to divide the system into three different subregions, i.e., a source (outer subregion) at higher potential and a sink (inner subregion) region at a lower potential with the help of a boundary which is at the highest potential. The boundary allows only one-way diffusion from source to sink so diffusion stops completely whenever the sink is filled to its capacity. This technique was shown to be able to spontaneously form stable and complex patterns in the source and the sink parts. While the technique is applicable to any self-organizing or phase-separating system whenever certain conditions are met, a physically realizable system of liquid films supported on chemically heterogeneous substrates which self-organizes through morphological phase separation is chosen to study the dynamics. Free-energy functional of this system shows an asymmetric double-well potential and the Maxwell's double tangent construction on this free energy gives rise to two distinct and finite equilibrium phases. Disorder is introduced in the form of chemical heterogeneity to divide the whole system into different subregions each consisting of two equilibrium phases, thus enabling more finite equilibrium points in the same system. The dynamics uncovers two different pathways for stable localized structures, viz., the defect and the direct pathway. Thinner films are likely to favor the defect pathway whereby the equilibrium pattern forms after coarsening, whereas thicker films will predominantly choose the direct pathway of pattern formation by phase separation. The final localized structure is same irrespective of the chosen pathway. The dynamics also shows that the patterns formed in the inner subregion part can be seen as a result of "localization" and those formed in the outer subregion are due to arrested diffusion under geometric or curvature constraints. Simple measures of localization in terms of free energy and its constituents are also presented. The identification of localization in the above systems opens up the possibility of a systematic approach to create controlled structures at nano- and submicrometer scales with desired shape and size. These structures constitute new building blocks of advanced and tailored materials. [ABSTRACT FROM AUTHOR]

We study numerically the aging properties of the two-dimensional Ising model with quenched disorder considered in our recent paper [Phys. Rev. E 95, 062136 (2017)], where frustration can be tuned by varying the fraction of antiferromagnetic interactions. Specifically, we focus on the scaling properties of the autocorrelation and linear response functions after a quench of the model to a low temperature. We find that the interplay between equilibrium and aging occurs differently in the various regions of the phase diagram of the model. When the quench is made into the ferromagnetic phase the two-time quantities are made by the sum of an equilibrium and an aging part, whereas in the paramagnetic phase these parts combine in a multiplicative way. Scaling forms are shown to be obeyed with good accuracy, and the corresponding exponents and scaling functions are determined and discussed in the framework of what is known in clean and disordered systems. [ABSTRACT FROM AUTHOR]

We present the first systematic observation of scaling of thermal hysteresis with the temperature scanning rate around an abrupt thermodynamic transition in correlated electron systems. We show that the depth of supercooling and superheating in vanadium sesquioxide (V2O3) shifts with the temperature quench rates. The dynamic scaling exponent is close to the mean field prediction of 2/3. These observations, combined with the purely dissipative continuous ordering seen in "quench-and-hold" experiments, indicate departures from classical nucleation theory toward a barrier-free phase ordering associated with critical dynamics. Observation of critical-like features and scaling in a thermally induced abrupt phase transition suggests that the presence of a spinodal-like instability is not just an artifact of the mean field theories but can also exist in the transformation kinetics of real systems, surviving fluctuations. [ABSTRACT FROM AUTHOR]

We present comprehensive numerical results from a study of model H, which describes phase separation kinetics in binary fluid mixtures. We study the pattern dynamics of both density and velocity fields in d = 2, 3. The density length scales show three distinct regimes, in accordance with analytical arguments. The velocity length scale shows a diffusive behavior. We also study the scaling behavior of the morphologies for density and velocity fields and observe dynamical scaling in the relevant correlation functions and structure factors. Finally, we study the effect of quenched random field disorder on spinodal decomposition in model H., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

We present results for phase-ordering kinetics in the Coulomb glass (CG) model, which describes electrons on a lattice with unscreened Coulombic repulsion. The filling factor is denoted by K∈[0,1]. For a square lattice with K=0.5 (symmetric CG), the ground state is a checkerboard with alternating electrons and holes. In this paper, we focus on the asymmetric CG where K≲0.5, i.e., the ground state is checkerboard-like with excess holes distributed uniformly. There is no explicit quenched disorder in our system, though the Coulombic interaction gives rise to frustration. We find that the evolution morphology is in the same dynamical universality class as the ordering ferromagnet. Further, the domain growth law is slightly slower than the Lifshitz-Cahn-Allen law, L(t)∼t^{1/2}, i.e., the growth exponent is underestimated. We speculate that this could be a signature of logarithmic growth in the asymptotic regime.

Background: Anterior Cruciate Ligament (ACL) injuries are common in the active population of the Armed Forces. Symptomatic instability prompts individuals to seek a cure or a sheltered appointment. Despite the increasing numbers of ACL reconstructions performed, the outcomes have not been so spectacular with only a meager percentage of our patients returning to preinjury levels of activity. With the premise that an all-inside ACL reconstruction is likely to result in better functional outcomes, the aim of this study was to compare the short-term functional outcomes of a large consecutive series of patients undergoing ACL reconstruction using the translateral all-inside ACL reconstruction technique (AI) and standard anteromedial portal technique (AM) with a minimum follow-up of one year., Methods: A total of 240 patients with isolated ACL tear underwent ACL reconstruction via the AI or AM technique. Their preoperative and postoperative scores were compared to look for any significant differences in functional outcomes., Results: The two groups were matched for age, BMI, mechanism of injury, and interval from injury to surgery. There was no difference in their preoperative scores. Postoperatively, although there were significant improvements across both groups, there was no significant difference between the groups at any point of time., Conclusion: The AI technique has garnered interest in recent literature in addressing ACL injuries. This study found no discernible benefit of the AI technique when compared to the AM technique in terms of functionality following an ACL reconstruction at any point of time up to 1 year following surgery., (© 2022 Director General, Armed Forces Medical Services. Published by Elsevier, a division of RELX India Pvt. Ltd.)

We present the phase separation dynamics of a binary (AB), simple fluid (SF), and amphiphilic polymer (AP) mixture using dissipative particle dynamics simulation at d = 3. We study the effect of different AP topologies, including block copolymers, ring block copolymers (RCP), and miktoarm star polymers, on the evolution morphologies, dynamic scaling functions, and length scale of the AB mixture. Our results demonstrate that the presence of APs leads to significantly different evolution morphologies in SF. However, the deviation from dynamical scaling is prominent, mainly for RCP. Typically, the characteristic length scale for SF follows the power law R(t) ∼ tϕ, where ϕ is the growth exponent. In the presence of high AP, we observe diffusive growth (ϕ → 1/3) at early times, followed by saturation in length scale (ϕ → 0) at late times. The extent of saturation varies with constraints imposed on the APs, such as topology, composition ratio, chain length, and stiffness. At lower composition ratios, the system exhibits inertial hydrodynamic growth (ϕ → 2/3) at asymptotic times without clearly exhibiting the viscous hydrodynamic regime (ϕ → 1) at earlier times in our simulations. Our results firmly establish the existence of hydrodynamic growth regimes in low surfactant-influenced phase separation kinetics of binary fluids and settle the related ambiguity in d = 3 systems., (© 2023 Author(s). Published under an exclusive license by AIP Publishing.)

We investigate the influence of long-range (LR) interactions on the phase ordering dynamics of the one-dimensional random-field Ising model (RFIM). Unlike the usual RFIM, a spin interacts with all other spins through a ferromagnetic coupling that decays as r^{-(1+σ)}, where r is the distance between two spins. In the absence of LR interactions, the size of coarsening domains R(t) exhibits a crossover from pure system behavior R(t)∼t^{1/2} to an asymptotic regime characterized by logarithmic growth: R(t)∼(lnt)^{2}. The LR interactions affect the preasymptotic regime, which now exhibits ballistic growth R(t)∼t, followed by σ-dependent growth R(t)∼t^{1/(1+σ)}. Additionally, the LR interactions also affect the asymptotic logarithmic growth, which becomes R(t)∼(lnt)^{α(σ)} with α(σ)<2. Thus, LR interactions lead to faster growth than for the nearest-neighbor system at short times. Unexpectedly, this driving force causes a slowing down of the dynamics (α<2) in the asymptotic logarithmic regime. This is explained in terms of a nontrivial competition between the pinning force caused by the random field and the driving force introduced by LR interactions. We also study the spatial correlation function and the autocorrelation function of the magnetization field. The former exhibits superuniversality for all σ, i.e., a scaling function that is independent of the disorder strength. The same holds for the autocorrelation function when σ<1, whereas a signature of the violation of superuniversality is seen for σ>1.

This paper investigates the phase separation kinetics of ternary fluid mixtures composed of a polymeric component (C) and two simple fluids (A and B) using dissipative particle dynamics simulations with a system dimensionality of d = 3. We model the affinities between the components to enable the settling of the polymeric component at the interface of fluids A and B. Thus, the system evolves to form polymer coated morphologies, enabling alteration of the fluids' interfacial properties. This manipulation can be utilized across various disciplines, such as the stabilization of emulsions and foams, rheological control, biomimetic design, and surface modification. We probe the effects of various parameters, such as the polymeric concentration, chain stiffness, and length, on the phase separation kinetics of the system. The simulation results show that changes in the concentration of flexible polymers exhibit perfect dynamic scaling for coated morphologies. The growth rate decreases as the polymeric composition is increased due to reduced surface tension and restricted connectivity between A- and B-rich clusters. Variations in the polymer chain rigidity at fixed composition ratios and degrees of polymerization slow the evolution kinetics of AB fluids marginally, although the effect is more pronounced for perfectly rigid chains. Whereas flexible polymer chain lengths at fixed composition ratios slow down the segregation kinetics of AB fluids slightly, varying the chain lengths of perfectly rigid polymers leads to a significant deviation in the length scale and dynamic scaling for the evolved coated morphologies. The characteristic length scale follows a power-law growth with a growth exponent ϕ that shows a crossover from the viscous to the inertial hydrodynamic regime, where the values of ϕ depend on the constraints imposed on the system.

An amalgam of nematic liquid crystals and active matter, referred to as living liquid crystals, is a promising self-healing material with futuristic applications for targeted delivery of information and microcargo. We provide a phenomenological model to study the symbiotic pattern dynamics in this contemporary system using the Toner-Tu model for active matter (AM), the Landau-de Gennes free energy for liquid crystals (LCs), and an experimentally motivated coupling term that favours coalignment of the active and nematic components. Our extensive theoretical studies unfold two novel steady states, chimeras and solitons, with sharp regions of distinct orientational order that sweep through the coupled system in synchrony. The induced dynamics in the passive nematic is unprecedented. We show that the symbiotic dynamics of the AM and LC components can be exploited to induce and manipulate order in an otherwise disordered system.

During coarsening, small structures disappear, leaving behind only large ones. Here we study the spectral energy transfers in Model A, where the order parameter ϕ evolves via nonconserved dynamics. We show that the nonlinear interactions dissipate fluctuations and facilitate energy transfers among the Fourier modes so that only ϕ(k=0), where k is the wave number, survives at the end and approaches the asymptotic value +1 or -1. We contrast the coarsening evolution for the initial conditions with 〈ϕ(x,t=0)〉=0 and with uniformly positive or negative ϕ(x,t=0).

We present simulation results from a comprehensive molecular dynamics (MD) study of surface-directed spinodal decomposition (SDSD) in unstable symmetric binary mixtures at wetting surfaces. We consider long-ranged and short-ranged surface fields to investigate the early stage wetting kinetics. The attractive part of the long-ranged potential is of the form V(z)∼z^{-n}, where z is the distance from the surface and n is the power-law exponent. We find that the wetting-layer thickness R&#95;{1}(t) at very early times exhibits a power-law growth with an exponent α=1/(n+2). It then crosses over to a universal fast-mode regime with α=3/2. In contrast, for the short-ranged surface potential, a logarithmic behavior in R&#95;{1}(t) is observed at initial times. Remarkably, similar rapid growth is seen in this case too. We provide phenomenological arguments to understand these growth laws. Our MD results firmly establish the existence of universal fast-mode kinetics and settle the related controversy.

The biaxial phase in nematic liquid crystals has been elusive for several decades after its prediction in the 1970s. A recent experimental breakthrough was achieved by Liu et al. [Proc. Natl. Acad. Sci. USA 113, 10479 (2016)0027-842410.1073/pnas.1601235113] in a liquid-crystalline medium with magnetic nanoparticles. They exploited the different length scales of dipolar and magnetonematic interactions to obtain an equilibrium state where the magnetic moments are at an angle to the nematic director. This tilt introduces a second distinguished direction for orientational ordering or biaxiality in the two-component system. Using coarse-grained Ginzburg-Landau free-energy models for the nematic and magnetic fields, we provide a theoretical framework which allows for the manipulation of morphologies and quantitative estimates of biaxial order.

It is known that, after a quench to zero temperature (T=0), two-dimensional (d=2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d=2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T=0 and T≠0. For T=0 and the finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T≠0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.

The anisotropic shape of calamitic liquid crystal (LC) particles results in distinct values of energy when the nematogens are placed side by side or end to end. This anisotropy in energy which is governed by a parameter κ^{'} has deep consequences on equilibrium and nonequilibrium properties. Using the Gay-Berne (GB) model, which exhibits the nematic (Nm) as well as the low-temperature smectic (Sm) order, we undertake large-scale Monte Carlo and molecular dynamics simulations to probe the effect of κ^{'} on the equilibrium phase diagram and the nonequilibrium domain growth following a quench in the temperature T or coarsening. There are two transitions in the GB model: (i) isotropic to Nm at T&#95;{c}^{1} and (ii) Nm to Sm at T&#95;{c}^{2}T&#95;{c}^{1}→T

We study the breaking of time-reversal invariance (TRI) by the application of a magnetic field in the quantum kicked rotor (QKR), using Izrailev's finite-dimensional model. There is a continuous crossover from TRI to time-reversal noninvariance (TRNI) in the spectral and eigenvector fluctuations of the QKR. We show that the properties of this TRI to TRNI transition depend on α^{2}/N, where α is the chaos parameter of the QKR and N is the dimensionality of the evolution operator matrix. For α^{2}/N≳N, the transition coincides with that in random matrix theory. For α^{2}/N

We use large-scale Monte Carlo simulations to obtain comprehensive results for domain growth and aging in the random field XY model in dimensions d=2,3. After a deep quench from the paramagnetic phase, the system orders locally via annihilation of topological defects, i.e., vortices and antivortices. The evolution morphology of the system is characterized by the correlation function and the structure factor of the magnetization field. We find that these quantities obey dynamical scaling, and their scaling function is independent of the disorder strength Δ. However, the scaling form of the autocorrelation function is found to be dependent on Δ, i.e., superuniversality is violated. The large-t behavior of the autocorrelation function is explored by studying aging and autocorrelation exponents. We also investigate the characteristic growth law L(t,Δ) in d=2,3, which shows an asymptotic logarithmic behavior: L(t,Δ)∼Δ^{-φ}(lnt)^{1/ψ}, with exponents φ,ψ>0.

The behavior of many magnetic and dielectric solids, and the more contemporary magnetic superlattices, is governed by dipolar interactions. They are anisotropic and long ranged, having varied consequences ranging from ground states with complicated magnetic order to the presence of glassy dynamics characterized by a plethora of relaxation times. These systems are well captured by the dipolar Ising model (DIM) with nearest-neighbor exchange interactions (J) and long-range dipolar interactions (D). Depending on the relative interaction strength Γ=J/D, there are four phases of distinct magnetic order and symmetry. Using Monte Carlo simulations, we perform deep quenches to study domain growth or coarsening in the d=3 DIM. This important nonequilibrium phenomenon has not been addressed as dipolar interactions are notoriously difficult to handle theoretically. Our study reveals that, in spite of the anisotropy in interactions and diversity in ground state configurations, we observe universality in the ordering dynamics of all phases.

We undertake a detailed numerical study of the Active Model B proposed by Wittkowski et al., [Nature Commun. 5, 4351 (2014)]2041-172310.1038/ncomms5351. We find that the introduction of activity has a drastic effect on the ordering kinetics. First, the domain growth law shows a crossover from the usual Lifshitz-Slyozov growth law for phase separation (L∼t^{1/3}, where t is the time) to a novel growth law (L∼t^{1/4}) at late times. Second, the equal-time correlation function of the density field exhibits dynamical scaling for a given activity strength λ, but the scaling function depends on λ.

Ferronematics (FNs) are suspensions of magnetic nanoparticles in nematic liquid crystals (NLCs). They have attracted much experimental attention, and are of great interest both scientifically and technologically. There are very few theoretical studies of FNs, even in equilibrium. In this paper, we study the non-equilibrium phenomenon of domain growth after a thermal quench (or coarsening) in this coupled system. Our modeling is based on coupled time-dependent Ginzburg-Landau (TDGL) equations for two order parameters: the LC tensor order parameter Q, and the magnetization M. We consider both shallow and deep quenches from a high-temperature disordered phase. The system coarsens by the collision and annihilation of topological defects. We focus on slaved coarsening, where a disordered Q (or M) field is driven to coarsen by an ordered M (or Q) field. We present detailed results for the morphologies and growth laws, which exhibit unusual features purely due to the magneto-nematic coupling. To the best of our knowledge, this is the first study of non-equilibrium phenomena in FNs.

Using a dissipative particle dynamics (DPD) simulation method, we study the phase separation dynamics in block copolymer (BCP) melts in d = 3, subjected to external stimuli such as light. An initial homogeneous BCP melt is rapidly quenched to a temperature T < T c , where T c is the critical temperature. We then allow the system to undergo alternate light "on" and "off" cycles. An on-cycle breaks the stimuli-sensitive bonds connecting both the blocks A and B in the BCP melt, and during the off-cycle, the broken bonds recombine. By simulating the effect of light, we isolate scenarios where phase separation begins with the light off (set 1); the cooperative interactions within the system allow it to undergo microphase separation. When the phase separation starts with the light on (set 2), the system undergoes macrophase separation due to bond breaking. Here, we report the role of alternate cycles on domain morphology by varying the bond-breaking probability for both set 1 and set 2, respectively. We observe that the scaling functions depend upon the conditions mentioned above that change the time scale of the evolving morphologies in various cycles. However, in all the cases, the average domain size respects the power-law growth: R(t) ∼t φ at late times, where φ is the dynamic growth exponent. After a short-lived diffusive growth (φ∼ 1/3) at early times, φ illustrates a crossover from the viscous hydrodynamic (φ∼ 1) to the inertial hydrodynamic (φ∼ 2/3) regimes at late times.

We study the low-temperature domain growth kinetics of the two-dimensional Ising model with long-range coupling J(r)∼r^{-(d+σ)}, where d=2 is the dimensionality. According to the Bray-Rutenberg predictions, the exponent σ controls the algebraic growth in time of the characteristic domain size L(t), L(t)∼t^{1/z}, with growth exponent z=1+σ for σ<1 and z=2 for σ>1. These results hold for quenches to a nonzero temperature T>0 below the critical temperature T&#95;{c}. We show that, in the case of quenches to T=0, due to the long-range interactions, the interfaces experience a drift which makes the dynamics of the system peculiar. More precisely, we find that in this case the growth exponent takes the value z=4/3, independently of σ, showing that it is a universal quantity. We support our claim by means of extended Monte Carlo simulations and analytical arguments for single domains.

We study the dynamics of an intruder moving through a fluidized granular medium in three dimensions (d=3). The intruder and grains have both translational and rotational degrees of freedom. The energy-dissipation mechanism is solid friction between all pairs of particles. We keep the granular system fluidized even at rather high densities by randomly perturbing the linear and angular velocities of the grains. We apply a constant external force of magnitude F to the intruder and obtain its steady-state velocity V&#95;{s} in the center-of-mass frame of the grains. The F-V&#95;{s} relation is of great interest in the industrial processing of granular matter and has been the subject of most experiments on this problem. We also obtain the mobility, which is proportional to the inverse viscosity, as a function of the volume fraction ϕ. This is shown to diverge at the jamming volume fraction. For ϕ below the jamming fraction, we find that V&#95;{s}∼F for small F and V&#95;{s}∼F^{1/2} for large F. The intruder shows diffusive motion in the plane perpendicular to the direction of the external force.

This paper is the second in a two-part exposition on surface-directed spinodal decomposition (SDSD), i.e., the interplay of kinetics of wetting and phase separation at a surface which is wetted by one of the components of a binary mixture. In our first paper [P. Das, P. K. Jaiswal, and S. Puri, Phys. Rev. E 102, 012803 (2020)2470-004510.1103/PhysRevE.102.012803], we studied SDSD on chemically heterogeneous and physically flat substrates. In this paper, we study SDSD on a chemically homogeneous but morphologically patterned substrate. Such substrates arise in a vast variety of technological applications. Our goal is to provide a theoretical understanding of SDSD in this context. We present detailed numerical results for domain growth both inside and above the grooves in the substrate. The morphological evolution can be understood in terms of the interference of SDSD waves originating from the different surfaces comprising the substrate.

Dyson has shown an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. In this paper, we introduce finite-range Coulomb gas models as a generalization of the Dyson models with a finite range of eigenvalue interactions. As the range of interaction increases, there is a transition from Poisson statistics to classical random matrix statistics. These models yield distinct universality classes of random matrix ensembles. They also provide a theoretical framework to study banded random matrices, and dynamical systems the matrix representation of which can be written in the form of banded matrices.

In part I of this two-stage exposition [Pandey, Kumar, and Puri, preceding paper, Phys. Rev. E 101, 022217 (2020)10.1103/PhysRevE.101.022217], we introduced finite-range Coulomb gas (FRCG) models, and developed an integral-equation framework for their study. We obtained exact analytical results for d=0,1,2, where d denotes the range of eigenvalue interaction. We found that the integral-equation framework was not analytically tractable for higher values of d. In this paper, we develop a Monte Carlo (MC) technique to study FRCG models. Our MC simulations provide a solution of FRCG models for arbitrary d. We show that, as d increases, there is a transition from Poisson to Wigner-Dyson classical random matrix statistics. Thus FRCG models provide a route for transition from Poisson to Wigner-Dyson statistics. The analytical formulation obtained in part I, and MC techniques developed in this paper, are used to study banded random matrices (BRMs) and quantum kicked rotors (QKRs). We demonstrate that, for a BRM of bandwidth b and a QKR of chaos parameter α, the appropriate FRCG model has range d=b^{2}/N=α^{2}/N, for N→∞. Here, N is the dimensionality of the matrix in the BRM, and the evolution operator matrix in the QKR.

We present the first systematic observation of scaling of thermal hysteresis with the temperature scanning rate around an abrupt thermodynamic transition in correlated electron systems. We show that the depth of supercooling and superheating in vanadium sesquioxide (V&#95;{2}O&#95;{3}) shifts with the temperature quench rates. The dynamic scaling exponent is close to the mean field prediction of 2/3. These observations, combined with the purely dissipative continuous ordering seen in "quench-and-hold" experiments, indicate departures from classical nucleation theory toward a barrier-free phase ordering associated with critical dynamics. Observation of critical-like features and scaling in a thermally induced abrupt phase transition suggests that the presence of a spinodal-like instability is not just an artifact of the mean field theories but can also exist in the transformation kinetics of real systems, surviving fluctuations.

We present results from Monte Carlo simulations of hysteresis in the zero-temperature ( T=0 ) dynamics of the Sherrington-Kirkpatrick spin glass model. We study the statistics of magnetization-jumps (denoted as Deltam ) in response to a time-dependent magnetic field H (t) , which increases or decreases with constant increments Delta as H(t)-->H(t)+/-Delta . In particular, we focus on the field dependence of the Deltam -distribution function P(Deltam,H) . We formulate arguments to understand the variation of P ( Deltam,H ) along the hysteresis loop in the weak-disorder limit.

We present numerical and analytical results for the autocorrelation functions which characterize domain growth in ternary mixtures. The numerical results are obtained from Monte Carlo simulations of the spin-1 Blume-Emery-Griffiths model with spin-exchange kinetics. Further, we model the autocorrelation functions using an approach based on the continuous-time random walk formalism. The aging property of these functions is related to the time dependence of the domain-size distribution. Our analytical results are found to be in good agreement with the numerical data.