Bounds on the decay of the autocorrelation in phase ordering dynamics

Kinetics of phase separation and aging dynamics of segregating fluid mixtures in the presence of quenched disorder

Quenched or frozen-in structural disorder is ubiquitous in real experimental systems. Much of the progress is achieved in understanding the phase separation of such systems using the diffusion-driven coarsening in an Ising model with quenched disorder. But there is a paucity of research on the phase-separation kinetics in fluids with quenched disorder. In this paper, we present results from a detailed molecular dynamics simulation, showing the effects of randomly placed localized impurities on the phase separation kinetics of binary fluid mixtures. Two different models are offered for representing the impurities. We observe a dramatic slowing down in the pattern formation with increasing impurity concentration. This sluggish domain growth kinetics follows a power-law with a disorder-dependent exponent. The correlation function and structure factor show a non-Porod behavior, indicating the roughening of the domain interfaces. We have also studied the effect of quenched disorder on the aging dynamics by calculating the two-time order parameter auto correlation function and find that the Fisher and Huse scaling law holds good in the presence of quenched disorder.

Growth and aging in a few phase-separating active matter systems

Via computer simulations we study evolution dynamics in systems of continuously moving active Brownian particles. The obtained results are discussed against those from the passive 2D Ising case. Following sudden quenches of random configurations to state points lying within the miscibility gaps and to the critical points, we investigate the far-from-steady-state dynamics by calculating quantities associated with structure and characteristic length scales. We also study aging for quenches into the miscibility gap and provide a quantitative picture for the scaling behavior of the two-time order-parameter correlation function. The overall structure and dynamics are consistent with expectations from the Ising model. This remains true for certain active lattice models as well, for which we present results for quenches to the critical points.

Influence of roughening transition on magnetic ordering

In the literature of magnetic phase transitions, in addition to a critical point, the existence of another special point has been discussed. This is related to the broadening of the interface between two different ordering phases and is referred to as the point of roughening transition. While the equilibrium properties associated with this transition are well understood, the influence of this on nonequilibrium dynamics still needs to be investigated. In this paper we present comprehensive results, from Monte Carlo simulations, on coarsening dynamics in a system, over a wide range of temperature, in space dimension d=3, for which there exists a roughening transition at a nonzero temperature T_{R}. An advanced analysis of the simulation data, on structure, growth, and aging, shows that the onset of unexpected glasslike slow dynamics in this system, that has received attention in recent times, for quenches to zero temperature, actually occurs at this transition point. This implies that the structure and aging depend upon the final temperature, when the latter lies between 0 and T_{R}. This is a very interesting exception to universality in coarsening dynamics. The results also demonstrate an important structure-dynamics connection in the phase-ordering dynamics. We compare the key results with those from d=2, for which there exists no nonzero roughening transition temperature. The absence of the above-mentioned anomalous features in the latter dimension places our conjecture on the role of the roughening transition on a firmer footing.

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.

Effect of annealed disorder on phase separation kinetics and aging phenomena in fluid mixtures

We use state-of-the-art molecular dynamics simulations to study the effects of annealed disorder on the phase-separating kinetics and aging phenomena of a segregating binary fluid mixture. In the presence of disorder, we observe a dramatic slowing down in the phase separation dynamics. The domain growth follows the power law with a disorder-dependent exponent. Due to the energetically favorable positions, the domain boundary roughens, which modifies the correlation function and structure factor to a non-Porod behavior. The correlation function and structure factor provide clear evidence that superuniversality does not hold in our system. The role of annealed disorder on the nonequilibrium aging dynamics is studied qualitatively by computing the two-time order-parameter autocorrelation function. The decay of the correlation function slows down significantly with the disorder. This quantity exhibits scaling laws with respect to the ratio of the domain length at the observation time and the age of the system. We find the scaling laws hold good for the disordered system and are therefore robust and generic to such segregating fluid mixtures.

Domain growth and aging in the random field XY model: A Monte Carlo study

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.

Kinetics of domain growth and aging in a two-dimensional off-lattice system

We have used molecular dynamics simulations for a comprehensive study of phase separation in a two-dimensional single-component off-lattice model where particles interact through the Lennard-Jones potential. Via state-of-the-art methods we have analyzed simulation data on structure, growth, and aging for nonequilibrium evolutions in the model. These data were obtained following quenches of well-equilibrated homogeneous configurations, with density close to the critical value, to various temperatures inside the miscibility gap, having vapor-"liquid" as well as vapor-"solid" coexistence. For the vapor-liquid phase separation we observe that ℓ, the average domain length, grows with time (t) as t^{1/2}, a behavior that has connection with hydrodynamics. At low-enough temperature, a sharp crossover of this time dependence to a much slower, temperature-dependent, growth is identified within the timescale of our simulations, implying "solid"-like final state of the high-density phase. This crossover is, interestingly, accompanied by strong differences in domain morphology and other structural aspects between the two situations. For aging, we have presented results for the order-parameter autocorrelation function. This quantity exhibits data collapse with respect to ℓ/ℓ_{w}, ℓ, and ℓ_{w} being the average domain lengths at times t and t_{w} (≤t), respectively, the latter being the age of a system. Corresponding scaling function follows a power-law decay: ∼(ℓ/ℓ_{w})^{-λ} for t≫t_{w}. The decay exponent λ, for the vapor-liquid case, is accurately estimated via the application of an advanced finite-size scaling method. The obtained value is observed to satisfy a bound.

Aging in the Long-Range Ising Model

The current understanding of aging phenomena is mainly confined to the study of systems with short-ranged interactions. Little is known about the aging of long-ranged systems. Here, the aging in the phase-ordering kinetics of the two-dimensional Ising model with power-law long-range interactions is studied via Monte Carlo simulations. The dynamical scaling of the two-time spin-spin autocorrelator is well described by simple aging for all interaction ranges studied. The autocorrelation exponents are consistent with λ=1.25 in the effectively short-range regime, while for stronger long-range interactions the data are consistent with λ=d/2=1. For very long-ranged interactions, strong finite-size effects are observed. We discuss whether such finite-size effects could be misinterpreted phenomenologically as subaging.

Quasideterministic dynamics, memory effects, and lack of self-averaging in the relaxation of quenched ferromagnets

We discuss the interplay between the degree of dynamical stochasticity, memory persistence, and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasideterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including the mean-field, and the short-range random-field Ising model.

Relaxation in a phase-separating two-dimensional active matter system with alignment interaction

Via computer simulations, we study kinetics of pattern formation in a two-dimensional active matter system. Self-propulsion in our model is incorporated via the Vicsek-like activity, i.e., particles have the tendency of aligning their velocities with the average directions of motion of their neighbors. In addition to this dynamic or active interaction, there exists passive inter-particle interaction in the model for which we have chosen the standard Lennard-Jones form. Following quenches of homogeneous configurations to a point deep inside the region of coexistence between high and low density phases, as the systems exhibit formation and evolution of particle-rich clusters, we investigate properties related to the morphology, growth, and aging. A focus of our study is on the understanding of the effects of structure on growth and aging. To quantify the latter, we use the two-time order-parameter autocorrelation function. This correlation, as well as the growth, is observed to follow power-law time dependence, qualitatively similar to the scaling behavior reported for passive systems. The values of the exponents have been estimated and discussed by comparing with the previously obtained numbers for other dimensions as well as with the new results for the passive limit of the considered model. We have also presented results on the effects of temperature on the activity mediated phase separation.

Aging exponents for nonequilibrium dynamics following quenches from critical points

Via Monte Carlo simulations we study nonequilibrium dynamics in the nearest-neighbor Ising model, following quenches to points inside the ordered region of the phase diagram. With the broad objective of quantifying the nonequilibrium universality classes corresponding to spatially correlated and uncorrelated initial configurations, in this paper we present results for the decay of the order-parameter autocorrelation function for quenches from the critical point. This autocorrelation is an important probe for the aging dynamics in far-from-equilibrium systems and typically exhibits power-law scaling. From the state-of-the-art analysis of the simulation results, we quantify the corresponding exponents (λ) for both conserved and nonconserved (order-parameter) dynamics of the model in space dimension d=3. Via structural analysis we demonstrate that the exponents satisfy a bound. We also revisit the d=2 case to obtain more accurate results. It appears that irrespective of the dimension, λ is approximately the same for both conserved and nonconserved dynamics.

Initial correlation dependence of aging in phase separating solid binary mixtures and ordering ferromagnets

Following quenches of initial configurations having long range spatial correlations, prepared at the demixing critical point, to points inside the miscibility gap, we study aging phenomena in solid binary mixtures. Results on the decay of the two-time order-parameter autocorrelation functions, obtained from Monte Carlo simulations of the two-dimensional Ising model, with Kawasaki exchange kinetics, are analyzed via state-of-the art methods. The outcome is compared with that obtained for the ordering in uniaxial ferromagnets. For the latter, we have performed Monte Carlo simulations of the same model using the Glauber mechanism. For both types of systems we provide comparative discussion of our results with reference to those concerning quenches with configurations having no spatial correlation. We also discuss the role of structure on the decay of these correlations.

Aging phenomena during phase separation in fluids: decay of autocorrelation for vapor-liquid transitions

We performed molecular dynamics simulations to study relaxation phenomena during vapor-liquid transitions in a single component Lennard-Jones system. Results from two different overall densities are presented: one in the neighborhood of the vapor branch of the coexistence curve and the other being close to the critical density. The nonequilibrium morphologies, growth mechanisms and growth laws in the two cases are vastly different. In the low density case growth occurs via diffusive coalescence of droplets in a disconnected morphology. On the other hand, the elongated structure in the higher density case grows via advective transport of particles inside the tube-like liquid domains. The objective in this work has been to identify how the decay of the order-parameter autocorrelation, an important quantity to understand aging dynamics, differs in the two cases. In the case of the disconnected morphology, we observe a very robust power-law decay, as a function of the ratio of the characteristic lengths at the observation time and at the age of the system, whereas the results for the percolating structure appear rather complex. To quantify the decay in the latter case, unlike the standard method followed in a previous study, here we have performed a finite-size scaling analysis. The outcome of this analysis shows the presence of a strong preasymptotic correction, while revealing that in this case also, albeit in the asymptotic limit, the decay follows a power-law. Even though the corresponding exponents in the two cases differ drastically, this study, combined with a few recent ones, suggests that power-law behavior of this correlation function is rather universal in coarsening dynamics.

Direct test of the Gaussian auxiliary field ansatz in nonconserved order parameter phase ordering dynamics

The assumption that the local order parameter is related to an underlying spatially smooth auxiliary field, u(r[over ⃗],t), is a common feature in theoretical approaches to non-conserved order parameter phase separation dynamics. In particular, the ansatz that u(r[over ⃗],t) is a Gaussian random field leads to predictions for the decay of the autocorrelation function which are consistent with observations, but distinct from predictions using alternative theoretical approaches. In this paper, the auxiliary field is obtained directly from simulations of the time-dependent Ginzburg-Landau equation in two and three dimensions. The results show that u(r[over ⃗],t) is equivalent to the distance to the nearest interface. In two dimensions, the probability distribution, P(u), is well approximated as Gaussian except for small values of u/L(t), where L(t) is the characteristic length-scale of the patterns. The behavior of P(u) in three dimensions is more complicated; the non-Gaussian region for small u/L(t) is much larger than that in two dimensions but the tails of P(u) begin to approach a Gaussian form at intermediate times. However, at later times, the tails of the probability distribution appear to decay faster than a Gaussian distribution.

Pattern, growth, and aging in aggregation kinetics of a Vicsek-like active matter model

Via molecular dynamics simulations, we study kinetics in a Vicsek-like phase-separating active matter model. Quantitative results, for isotropic bicontinuous pattern, are presented on the structure, growth, and aging. These are obtained via the two-point equal-time density-density correlation function, the average domain length, and the two-time density autocorrelation function. Both the correlation functions exhibit basic scaling properties, implying self-similarity in the pattern dynamics, for which the average domain size exhibits a power-law growth in time. The equal-time correlation has a short distance behavior that provides reasonable agreement between the corresponding structure factor tail and the Porod law. The autocorrelation decay is a power-law in the average domain size. Apart from these basic similarities, the overall quantitative behavior of the above-mentioned observables is found to be vastly different from those of the corresponding passive limit of the model which also undergoes phase separation. The functional forms of these have been quantified. An exceptionally rapid growth in the active system occurs due to fast coherent motion of the particles, mean-squared-displacements of which exhibit multiple scaling regimes, including a long time ballistic one.

Ballistic aggregation in systems of inelastic particles: Cluster growth, structure, and aging

We study far-from-equilibrium dynamics in models of freely cooling granular gas and ballistically aggregating compact clusters. For both the cases, from event-driven molecular dynamics simulations, we have presented detailed results on structure and dynamics in space dimensions d=1 and 2. Via appropriate analyses it has been confirmed that the ballistic aggregation mechanism applies in d=1 granular gases as well. Aging phenomena for this mechanism, in both the dimensions, have been studied via the two-time density autocorrelation function. This quantity is demonstrated to exhibit scaling property similar to that in the standard phase transition kinetics. The corresponding functional forms have been quantified and the outcomes have been discussed in connection with the structural properties. Our results on aging establish a more complete equivalence between the granular gas and the ballistic aggregation models in d=1.

Dimensionality dependence of aging in kinetics of diffusive phase separation: Behavior of order-parameter autocorrelation

Behavior of two-time autocorrelation during the phase separation in solid binary mixtures is studied via numerical solutions of the Cahn-Hilliard equation as well as Monte Carlo simulations of the Ising model. Results are analyzed via state-of-the-art methods, including the finite-size scaling technique. Full forms of the autocorrelation in space dimensions 2 and 3 are obtained empirically. The long-time behavior is found to be power law, with exponents unexpectedly higher than the ones for the ferromagnetic ordering. Both Cahn-Hilliard and Ising models provide consistent results.

Aging in ferromagnetic ordering: full decay and finite-size scaling of autocorrelation

Nonequilibrium dynamics in Ising and Ginzburg-Landau models were studied for a nonconserved order parameter that mimics ordering in ferromagnets. The focus was on the understanding of the decay of the two time (t, t(w); t > tw) order-parameter correlation function. For this quantity, a full form has been obtained empirically which, for t ≫ t(w), provides a power-law ∼ (ℓ/ℓ(w))(-λ), ℓ and ℓ(w) being the characteristic lengths at t and tw, respectively. This empirical form was used for a finite-size scaling analysis to obtain the exponent λ in space dimensions d = 2 and 3. Our estimates of λ and understanding of the finite-size effects, for the models considered, provide useful information on the relevance of thermal noise. The values of λ obtained are in good agreement with the predictions of a theory based on Gaussian auxiliary field ansatz.

Effects of density conservation and hydrodynamics on aging in nonequilibrium processes

Aging in kinetics of three different phase transitions, viz., magnetic, a binary solid, and a single component fluid, are studied via Monte Carlo and molecular dynamics simulations in three space dimensions with the objective of identifying the effects of order-parameter conservation and hydrodynamics. We observe that the relevant autocorrelations exhibit power-law decay in a ferromagnet and binary solid but with different exponents. At early time the fluid autocorrelation function nicely follows that of the binary solid, the order parameter being conserved for both of them, as opposed to a ferromagnet. At a late time the fluid data crosses over to an exponential decay which we identify as a hydrodynamic effect and we provide analytical justification for this behavior.

Aging and crossovers in phase-separating fluid mixtures

We use state-of-the-art molecular dynamics simulations to study hydrodynamic effects on aging during kinetics of phase separation in a fluid mixture. The domain growth law shows a crossover from a diffusive regime to a viscous hydrodynamic regime. There is a corresponding crossover in the autocorrelation function from a power-law behavior to an exponential decay. While the former is consistent with theories for diffusive domain growth, the latter results as a consequence of faster advective transport in fluids for which an analytical justification has been provided.

Correction to scaling in the response function of the two-dimensional kinetic Ising model

The aging part Rag(t,s) of the impulsive response function of the two-dimensional ferromagnetic Ising model, quenched below the critical point, is studied numerically employing an algorithm without the imposition of the external field. We find that the simple scaling form Rag(t,s)=s-(1+a)f(t/s), which is usually believed to hold in the aging regime, is not obeyed. We analyze the data assuming the existence of a correction to scaling. We find a=0.273+/-0.006, in agreement with previous numerical results obtained from the zero field cooled magnetization. We investigate in detail also the scaling function f(t/s) and we compare the results with the predictions of analytical theories. We make an ansatz for the correction to scaling, deriving an analytical expression for Rag(t,s). This gives a satisfactory qualitative agreement with the numerical data for Rag(t,s) and for the integrated response functions. With the analytical model we explore the overall behavior, extrapolating beyond the time regime accessible with the simulations. We explain why the data for the zero field cooled susceptibility are not too sensitive to the existence of the correction to scaling in Rag(t,s), making this quantity the most convenient for the study of the asymptotic scaling properties.

Phase-ordering simulation of one-dimensional conserved kink system

The kink-antikink kinetics of one-dimensional phase ordering under conserved order parameter dynamics is studied numerically. The average domain size is found to grow logarithmically, and the distribution of domain size and order parameter correlation function are shown to satisfy a scaling relation. The two-time autocorrelation function follows a power law of A (t(0))(t) approximately t(-lambda) , where lambda depends on the start time of the calculation t(0) . If t(0) is in the scaling regime, lambda takes a constant value of 3.0. Thus the scaling functions are sensitive to the initial configuration of domains. When the initial kink positions are given by uniform random numbers, the scaling functions agree with those obtained by cell dynamical system simulation.

Autocorrelation functions for phase separation in ternary mixtures

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.

Autocorrelation exponent of conserved spin systems in the scaling regime following a critical quench

We study the autocorrelation function of a conserved spin system following a quench at the critical temperature. Defining the correlation length L(t) approximately t(1/z), we find that for times t' and t satisfying L(t')<<L(t)<<L(t')(phi) well inside the scaling regime, the spin autocorrelation function behaves like s(t)s(t') approximately L(t')(-(d-2+eta))[L(t')/L(t)](lambda(')(c)). For the O(n) model in the n-->infinity limit, we show that lambda(')(c)=d+2 and phi=z/2. We give a heuristic argument suggesting that this result is, in fact, valid for any dimension d and spin vector dimension n. We present numerical simulations for the conserved Ising model in d=1 and d=2, which are fully consistent with the present theory.

Comment on "Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models"

Recently Phys. Rev. E 68, 016116 (2003)]] proposed an alternative way to compute numerically the fluctuation-dissipation ratios in nonequilibrium critical systems. Using well-known facts of nonequilibrium critical dynamics I show that the leading contributions of the quantities they consider are in fact one-time quantities which are independent of the waiting time. The ratio of these one-time quantities determines the slope of the straight lines observed in the fluctuation-dissipation plots of Mayer et al.

Scaling of the linear response in simple aging systems without disorder

The time-dependent scaling of the thermoremanent and zero-field-cooled susceptibilities in ferromagnetic spin systems undergoing aging after a quench to a temperature at or below criticality is studied. A recent debate on their interpretation is resolved by showing that for systems with a short-ranged equilibrium spin-spin correlator and above their roughening temperature, the field-cooled susceptibility chi(FC)(t)-chi(0) approximately t(-A), where chi(0) is related to the equilibrium magnetization and the exponent A is related to the time-dependent scaling of the interface width between ordered domains. The same effect also dominates the scaling of the zero-field-cooled susceptibility chi(ZFC)(t,s), but does not enter into the thermoremanent susceptibility rho(TRM)(t,s). However, there may be large finite-time corrections to the scaling of rho(TRM)(t,s) which are explicitly derived and may be needed in order to extract reliable aging exponents. Consistency with the predictions of local scale invariance is confirmed in the Glauber-Ising and spherical models.

Experimental study of spinodal decomposition in a 1D conserved order parameter system

We study the zigzag instability coarsening of splay-bend walls formed in a nematic liquid crystal under external fields. The vertexes of zigzag can be considered as kinks in a one-dimensional order parameter system and the geometrical constraints associated with the necessary equal length sum of zig and zag segments impose a conserved quantity in this Cahn-Hilliard-type problem. In the late stage of coarsening, the characteristic length of the system L(t) shows a logarithmic increase in time and the dynamical scaling law holds. We then try to extract the nontrivial asymptotic scaling exponent lambda of the two-time correlation function, defined by lim(<phi(0,t)phi(0,t('))> approximately [L(t)/L(t('))](-lambda). The scaling exponents with respective time references, t(')=32 and 64 s, after quench are found to be lambda approximately 2 which is larger than the value with respective time reference t(')=0, predicted by numerical simulation.

Limits and regulations for mycotoxins in food and feed

Mycotoxins are natural contaminants whose presence in food- and feedstuffs cannot be completely avoided. Since several mycotoxins have been associated with and implicated in human and animal diseases there is a need to establish maximum levels, guidelines or action levels for them in some kinds of commodities. International and government authorities in many countries have been investing in mycotoxins research and initiating administrative actions for elaboration of legislation and implementing regulatory measures for the control of mycotoxins. Codex Alimentarius Commission is established international legislation on food and feed. In European Union specific limits and regulations for mycotoxins and other contaminants are constructed under the general Codex standards and based on proposal from European Commission. The legal basis for European Commission became available with the framework Council Regulation (EEC) No 315/93. In this paper, legislation regarding maximum levels for certain mycotoxins in food- and feedstuffs in European Community and other countries were reviewed and discussed.

Exact multilocal renormalization of the effective action: application to the random sine Gordon model statics and nonequilibrium dynamics

We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.

Universality of the off-equilibrium response function in the kinetic Ising chain

The off-equilibrium response function chi(t,t(w)) and autocorrelation function C(t,t(w)) of an Ising chain with spin-exchange dynamics are studied numerically and compared with the same quantities in the case of spin-flip dynamics. It is found that, even though these quantities are different in the two cases, the parametric plot of chi(t,t(w)) versus C(t,t(w)) is the same. While this result could be expected in higher dimensionality, where chi(C) is related to the equilibrium state, it is far from trivial in the one-dimensional case where this relation does not hold. The origin of the universality of chi(C) is traced back to the optimization of domains position with respect to the perturbing external field. This mechanism is investigated resorting to models with a single domain moving in a random environment.

Kinetics of phase separation in ternary mixtures

We present detailed results from Monte Carlo simulations of the kinetics of phase separation in ternary mixtures. We focus on the case of ABV mixtures (where V denotes a vacancy) and investigate segregation kinetics resulting from V-mediated dynamics. We provide heuristic arguments for the existence of different morphologies in various parameter regimes. Furthermore, we present comprehensive numerical results for various characteristic features of the domain growth process, e.g., real-space correlation functions, domain-size distribution functions, and growth laws.

Spinodal decomposition and the tomita sum rule

The scaling properties of a phase-ordering system with a conserved order parameter are studied. The theory developed leads to scaling functions satisfying certain general properties including the Tomita sum rule. The theory also gives good agreement with numerical results for the order parameter scaling function in three dimensions. The values of the associated nonequilibrium decay exponents are given by the known lower bounds.

Ordering dynamics of heisenberg spins with torque: crossover, spin waves, and defects

We study the effect of a torque induced by the local molecular field on the phase ordering dynamics of the Heisenberg model when the total magnetization is conserved. The torque drives the zero-temperature ordering dynamics to a new fixed point, characterized by exponents z=2 and lambda approximately 5. This "torque-driven" fixed point is approached at times such that t>>g(2), where g is the strength of the torque. All physical quantities, like the domain size L(t) and the equal and unequal time correlation functions, obey a crossover scaling form over the entire range of g. An attempt to understand this crossover behavior from the approximate Gaussian closure scheme fails completely, implying that the dynamics at late times cannot be understood from the dynamics of defects alone. We provide convincing arguments that the spin configurations can be decomposed in terms of defects and spin waves which interact with each other even at late times. In the absence of the torque term, the spin waves decay faster, but even so we find that the Gaussian closure scheme is inconsistent. In the latter case the inconsistency may be remedied by including corrections to a simple Gaussian distribution. For completeness we include a discussion of the ordering dynamics at T(c), where the torque is shown to be relevant, with exponents z=4-varepsilon/2 and lambda=d (where varepsilon=6-d). We show to all orders in perturbation theory that lambda=d as a consequence of the conservation law.

Evolution of speckle during spinodal decomposition

Time-dependent properties of the speckled intensity patterns created by scattering coherent radiation from materials undergoing spinodal decomposition are investigated by numerical integration of the Cahn-Hilliard-Cook equation. For binary systems which obey a local conservation law, the characteristic domain size is known to grow in time tau as R=[Btau](n) with n=1/3, where B is a constant. The intensities of individual speckles are found to be nonstationary, persistent time series. The two-time intensity covariance at wave vector k can be collapsed onto a scaling function Cov(deltat,t), where deltat=k(1/n)B(tau(2)-tau(1)) and t=k(1/n)B(tau(1)+tau(2))/2. Both analytically and numerically, the covariance is found to depend on deltat only through deltat/t in the small-t limit and deltat/t (1-n) in the large-t limit, consistent with a simple theory of moving interfaces that applies to any universality class described by a scalar order parameter. The speckle-intensity covariance is numerically demonstrated to be equal to the square of the two-time structure factor of the scattering material, for which an analytic scaling function is obtained for large t. In addition, the two-time, two-point order-parameter correlation function is found to scale as C(r/(B(n)sqaureroot[tau1(2n)+tau2(2n)]),tau1/tau2), even for quite large distances r. The asymptotic power-law exponent for the autocorrelation function is found to be lambda approximately 4.47, violating an upper bound conjectured by Fisher and Huse.