Ordering kinetics of the two-dimensional voter model with long-range interactions

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance r with probability P(r)∝r^{-α}. The model is characterized by different regimes, as α is varied. For α>4, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as L(t)∝sqrt[t], until consensus is reached in a time of the order of NlnN, with N being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as ρ(t)∝1/lnt. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For 0<α≤4, standard scaling is reinstated and the correlation length increases algebraically as L(t)∝t^{1/z}, with 1/z=2/α for 3<α<4 and 1/z=2/3 for 0<α<3. In addition, for α≤3, L(t) depends on N at any time t>0. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the N→∞ limit. In finite systems, consensus is reached in a time of the order of N for any α<4.

Phase ordering dynamics of the random-field long-range Ising model in one dimension

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.

Unveiling Charge-Transport Mechanisms in Electronic Devices Based on Defect-Engineered MoS2 Covalent Networks

Device performance of solution-processed 2D semiconductors in printed electronics has been limited so far by structural defects and high interflake junction resistance. Covalently interconnected networks of transition metal dichalcogenides potentially represent an efficient strategy to overcome both limitations simultaneously. Yet, the charge-transport properties in such systems have not been systematically researched. Here, the charge-transport mechanisms of printed devices based on covalent MoS₂ networks are unveiled via multiscale analysis, comparing the effects of aromatic versus aliphatic dithiolated linkers. Temperature-dependent electrical measurements reveal hopping as the dominant transport mechanism: aliphatic systems lead to 3D variable range hopping, unlike the nearest neighbor hopping observed for aromatic linkers. The novel analysis based on percolation theory attributes the superior performance of devices functionalized with π-conjugated molecules to the improved interflake electronic connectivity and formation of additional percolation paths, as further corroborated by density functional calculations. Valuable guidelines for harnessing the charge-transport properties in MoS₂ devices based on covalent networks are provided.

Asymptotic states of Ising ferromagnets with long-range interactions

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.

Maximal Diversity and Zipf's Law

Zipf's law describes the empirical size distribution of the components of many systems in natural and social sciences and humanities. We show, by solving a statistical model, that Zipf's law co-occurs with the maximization of the diversity of the component sizes. The law ruling the increase of such diversity with the total dimension of the system is derived and its relation with Heaps's law is discussed. As an example, we show that our analytical results compare very well with linguistics and population datasets.

Kinetics of the two-dimensional long-range Ising model at low temperatures

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_{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.

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.

The Interplay between Phase Separation and Gene-Enhancer Communication: A Theoretical Study

The phase separation occurring in a system of mutually interacting proteins that can bind on specific sites of a chromatin fiber is investigated here. This is achieved by means of extensive molecular dynamics simulations of a simple polymer model that includes regulatory proteins as interacting spherical particles. Our interest is particularly focused on the role played by phase separation in the formation of molecule aggregates that can join distant regulatory elements, such as gene promoters and enhancers, along the DNA. We find that the overall equilibrium state of the system resulting from the mutual interplay between binding molecules and chromatin can lead, under suitable conditions that depend on molecules concentration, molecule-molecule, and molecule-DNA interactions, to the formation of phase-separated molecular clusters, allowing robust contacts between regulatory sites. Vice versa, the presence of regulatory sites can promote the phase-separation process. Different dynamical regimes can generate the enhancer-promoter contact, either by cluster nucleation at binding sites or by bulk spontaneous formation of the mediating cluster to which binding sites are successively attracted. The possibility that such processes can explain experimental live-cell imaging data measuring distances between regulatory sites during time is also discussed.

Effects of frustration on fluctuation-dissipation relations

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)2470-004510.1103/PhysRevE.95.062136], 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.

Large Fluctuations and Dynamic Phase Transition in a System of Self-Propelled Particles

We study the statistics, in stationary conditions, of the work W_{τ} done by the active force in different systems of self-propelled particles in a time τ. We show the existence of a critical value W_{τ}^{†} such that fluctuations with W_{τ}>W_{τ}^{†} correspond to configurations where interaction between particles plays a minor role whereas those with W_{τ}<W_{τ}^{†} represent states with single particles dragged by clusters. This twofold behavior is fully mirrored by the probability distribution P(W_{τ}) of the work, which does not obey the large-deviation principle for W_{τ}<W_{τ}^{†}. This pattern of behavior can be interpreted as due to a phase transition occurring at the level of fluctuating quantities and an order parameter is correspondingly identified.

Equilibrium structure and off-equilibrium kinetics of a magnet with tunable frustration

We study numerically a two-dimensional random-bond Ising model where frustration can be tuned by varying the fraction a of antiferromagnetic coupling constants. At low temperatures the model exhibits a phase with ferromagnetic order for sufficiently small values of a, a<a_{f}. In an intermediate range, a_{f}<a<a_{a}, the system is paramagnetic, with spin-glass order expected right at zero temperature. For even larger values, a>a_{a}, an antiferromagnetic phase exists. After a deep quench from high temperatures, slow evolution is observed for any value of a. We show that different amounts of frustration, tuned by a, affect the dynamical properties in a highly nontrivial way. In particular, the kinetics is logarithmically slow in phases with ferromagnetic or antiferromagnetic order, whereas evolution is faster, i.e., algebraic, when spin-glass order is prevailing. An interpretation is given in terms of the different nature of phase space.

Development and regression of a large fluctuation

We study the evolution leading to (or regressing from) a large fluctuation in a statistical mechanical system. We introduce and study analytically a simple model of many identically and independently distributed microscopic variables n_{m} (m=1,M) evolving by means of a master equation. We show that the process producing a nontypical fluctuation with a value of N=∑_{m=1}^{M}n_{m} well above the average 〈N〉 is slow. Such process is characterized by the power-law growth of the largest possible observable value of N at a given time t. We find similar features also for the reverse process of the regression from a rare state with N≫〈N〉 to a typical one with N≃〈N〉.

Coarsening and percolation in a disordered ferromagnet

By studying numerically the phase-ordering kinetics of a two-dimensional ferromagnetic Ising model with quenched disorder (either random bonds or random fields) we show that a critical percolation structure forms at an early stage. This structure is then rendered more and more compact by the ensuing coarsening process. Our results are compared to the nondisordered case, where a similar phenomenon is observed, and they are interpreted within a dynamical scaling framework.

Role of initial state and final quench temperature on aging properties in phase-ordering kinetics

We study numerically the two-dimensional Ising model with nonconserved dynamics quenched from an initial equilibrium state at the temperature T_{i}≥T_{c} to a final temperature T_{f} below the critical one. By considering processes initiating both from a disordered state at infinite temperature T_{i}=∞ and from the critical configurations at T_{i}=T_{c} and spanning the range of final temperatures T_{f}∈[0,T_{c}[ we elucidate the role played by T_{i} and T_{f} on the aging properties and, in particular, on the behavior of the autocorrelation C and of the integrated response function χ. Our results show that for any choice of T_{f}, while the autocorrelation function exponent λ_{C} takes a markedly different value for T_{i}=∞ [λ_{C}(T_{i}=∞)≃5/4] or T_{i}=T_{c} [λ_{C}(T_{i}=T_{c})≃1/8] the response function exponents are unchanged. Supported by the outcome of the analytical solution of the solvable spherical model we interpret this fact as due to the different contributions provided to autocorrelation and response by the large-scale properties of the system. As changing T_{f} is considered, although this is expected to play no role in the large-scale and long-time properties of the system, we show important effects on the quantitative behavior of χ. In particular, data for quenches to T_{f}=0 are consistent with a value of the response function exponent λ_{χ}=1/2λ_{C}(T_{i}=∞)=5/8 different from the one [λ_{χ}∈(0.5-0.56)] found in a wealth of previous numerical determinations in quenches to finite final temperatures. This is interpreted as due to important preasymptotic corrections associated to T_{f}>0.

Phase ordering in disordered and inhomogeneous systems

We study numerically the coarsening dynamics of the Ising model on a regular lattice with random bonds and on deterministic fractal substrates. We propose a unifying interpretation of the phase-ordering processes based on two classes of dynamical behaviors characterized by different growth laws of the ordered domain size, namely logarithmic or power law, respectively. It is conjectured that the interplay between these dynamical classes is regulated by the same topological feature that governs the presence or the absence of a finite-temperature phase transition.

Condensation of fluctuations in and out of equilibrium

Condensation of fluctuations is an interesting phenomenon conceptually distinct from condensation on average. One striking feature is that, contrary to what happens on average, condensation of fluctuations may occur even in the absence of interaction. The explanation emerges from the duality between large deviation events in the given system and typical events in a new and appropriately biased system. This phenomenon is investigated in the context of the Gaussian model, chosen as a paradigmatical noninteracting system, before and after an instantaneous temperature quench. It is shown that the bias induces a mean-field-like effective interaction responsible for the condensation on average. Phase diagrams, covering both the equilibrium and the off-equilibrium regimes, are derived for observables representative of generic behaviors.

Scaling in the aging dynamics of the site-diluted Ising model

We study numerically the phase-ordering kinetics of the two-dimensional site-diluted Ising model. The data can be interpreted in a framework motivated by renormalization-group concepts. Apart from the usual fixed point of the nondiluted system, there exist two disorder fixed points, characterized by logarithmic and power-law growth of the ordered domains. This structure gives rise to a rich scaling behavior, with an interesting crossover due to the competition between fixed points, and violation of superuniversality.

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.

Crossover in growth law and violation of superuniversality in the random-field Ising model

We study the nonconserved phase-ordering dynamics of the d=2,3 random-field Ising model, quenched to below the critical temperature. Motivated by the puzzling results of previous work in two and three dimensions, reporting a crossover from power-law to logarithmic growth, together with superuniversal behavior of the correlation function, we have undertaken a careful investigation of both the domain growth law and the autocorrelation function. Our main results are as follows: We confirm the crossover to asymptotic logarithmic behavior in the growth law, but, at variance with previous findings, we find the exponent in the preasymptotic power law to be disorder dependent, rather than being that of the pure system. Furthermore, we find that the autocorrelation function does not display superuniversal behavior. This restores consistency with previous results for the d=1 system, and fits nicely into the unifying scaling scheme we have recently proposed in the study of the random-bond Ising model.

Fluctuation-dissipation relations and field-free algorithms for the computation of response functions

We discuss the relation between the fluctuation-dissipation relation derived by Chatelain and Ricci-Tersenghi [C. Chatelain, J. Phys. A 36, 10739 (2003); F. Ricci-Tersenghi, Phys. Rev. E 68, 065104(R) (2003)] and that by Lippiello-Corberi-Zannetti [E. Lippiello, F. Corberi, and M. Zannetti, Phys. Rev. E 71, 036104 (2005)]. In order to do that, we rederive the fluctuation-dissipation relation for systems of discrete variables evolving in discrete time via a stochastic nonequilibrium Markov process. The calculation is carried out in a general formalism comprising the Chatelain, Ricci-Tersenghi, result and that by Lippiello-Corberi-Zannetti as special cases. The applicability, generality, and experimental feasibility of the two approaches are thoroughly discussed. Extending the analytical calculation to the variance of the response function, we show the advantage of field-free numerical methods with respect to the standard method, where the perturbation is applied. We also show that the signal-to-noise ratio is better (by a factor square root of 2) in the algorithm of Lippiello-Corberi-Zannetti with respect to that of Chatelain-Ricci Tersenghi.

Complex phase ordering of the one-dimensional Heisenberg model with conserved order parameter

We study the phase-ordering kinetics of the one-dimensional Heisenberg model with conserved order parameter by means of scaling arguments and numerical simulations. We find a rich dynamical pattern with a regime characterized by two distinct growing lengths. Spins are found to be coplanar over regions of a typical size LV(t), while inside these regions smooth rotations associated to a smaller length LC(t) are observed. Two different and coexisting ordering mechanisms are associated to these lengths, leading to different growth laws LV(t) approximately t1/3 and LC(t) approximately t1/4 violating dynamical scaling.

Nonlinear response and fluctuation-dissipation relations

A unified derivation of the off-equilibrium fluctuation dissipation relations (FDRs) is given for Ising and continuous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDRs allows one to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second application, the important problem of the definition of an off-equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.

Influence of thermal fluctuations on the geometry of interfaces of the quenched Ising model

We study the role of the quench temperature Tf in the phase-ordering kinetics of the Ising model with single spin flip in d=2,3 . Equilibrium interfaces are flat at Tf=0 , whereas at Tf>0 they are curved and rough (above the roughening temperature in d=3 ). We show, by means of scaling arguments and numerical simulations, that this geometrical difference is important for the phase-ordering kinetics as well. In particular, while the growth exponent z=2 of the size of domains L(t) approximately t 1/z is unaffected by Tf, other exponents related to the interface geometry take different values at Tf=0 or Tf>0 . For Tf>0 a crossover phenomenon is observed from an early stage where interfaces are still flat and the system behaves as at Tf=0 , to the asymptotic regime with curved interfaces characteristic of Tf>0 . Furthermore, it is shown that the roughening length, although subdominant with respect to L(t) , produces appreciable correction to scaling up to very long times in d=2 .

Phase-ordering kinetics on graphs

We study numerically the phase-ordering kinetics following a temperature quench of the Ising model with single spin-flip dynamics on a class of graphs, including geometrical fractals and random fractals, such as the percolation cluster. For each structure we discuss the scaling properties and compute the dynamical exponents. We show that the exponent a_{chi} for the integrated response function, at variance with all the other exponents, is independent of temperature and of the presence of pinning. This universal character suggests a strict relation between a_{chi} and the topological properties of the networks, in analogy to what is observed on regular lattices.

Scaling and universality in the aging kinetics of the two-dimensional clock model

We study numerically the aging dynamics of the two-dimensional p -state clock model after a quench from an infinite temperature to the ferromagnetic phase or to the Kosterlitz-Thouless phase. The system exhibits the general scaling behavior characteristic of nondisordered coarsening systems. For quenches to the ferromagnetic phase, the value of the dynamical exponents suggests that the model belongs to the Ising-type universality class. Specifically, for the integrated response function chi(t,s) approximately or = s(-a)chif(t/s), we find a(chi) consistent with the value a(chi)=0.28 found in the two-dimensional Ising model.

Test of local scale invariance from the direct measurement of the response function in the Ising model quenched to and below T(C)

In order to check on a recent suggestion that local scale invariance [M. Henkel, Phys. Rev. Lett. 87, 265701 (2001)] might hold when the dynamics is of Gaussian nature, we have carried out the measurement of the response function in the kinetic Ising model with Glauber dynamics quenched to T(C) in d=4, where Gaussian behavior is expected to apply, and in the two other cases of the d=2 model quenched to T(C) and to below T(C), where instead deviations from Gaussian behavior are expected to appear. We find that in the d=4 case there is an excellent agreement between the numerical data, the local scale invariance prediction and the analytical Gaussian approximation. No logarithmic corrections are numerically detected. Conversely, in the d=2 cases, both in the quench to T(C) and to below T(C), sizable deviations of the local scale invariance behavior from the numerical data are observed. These results do support the idea that local scale invariance might miss to capture the non-Gaussian features of the dynamics. The considerable precision needed for the comparison has been achieved through the use of a fast new algorithm for the measurement of the response function without applying the external field. From these high quality data we obtain a=0.27+/-0.002 for the scaling exponent of the response function in the d=2 Ising model quenched to below T(C), in agreement with previous results.

Crossover between Ising and XY -like behavior in the off-equilibrium kinetics of the one-dimensional clock model

We study the phase-ordering kinetics following a quench to a final temperature Tf of the one-dimensional p-state clock model. We show the existence of a critical value pc=4, where the properties of the dynamics change. At Tf=0, for p<or=pc the dynamics is analogous to that of the kinetic Ising model, characterized by Brownian motion and annihilation of interfaces. Dynamical scaling is obeyed with the same dynamical exponents and scaling functions of the Ising model. For p>pc, instead, the dynamics is dominated by a texture mechanism analogous to the one-dimensional XY model and dynamical scaling is violated. During the phase-ordering process at Tf>0, before equilibration occurs, a crossover between an early XY-like regime and a late Ising-like dynamics is observed for p>pc.

Aging dynamics and the topology of inhomogenous networks

We study phase ordering on networks and we establish a relation between the exponent a(x) of the aging part of the integrated auto-response function and the topology of the underlying structures. We show that a(x) > 0 in full generality on networks which are above the lower critical dimension d(L), i.e., where the corresponding statistical model has a phase transition at finite temperature. For discrete symmetry models on finite ramified structures with T(c) = 0, which are at the lower critical dimension d(L), we show that a(x) is expected to vanish. We provide numerical results for the physically interesting case of the 2 - d percolation cluster at or above the percolation threshold, i.e., at or above d(L), and for other networks, showing that the value of a(x) changes according to our hypothesis. For O(N) models we find that the same picture holds in the large-N limit and that a(x) only depends on the spectral dimension of the network.

Fluctuation-dissipation relation in an Ising model without detailed balance

We consider the modified Ising model introduced by de Oliveira, Mendes, and Santos [J. Phys. A 26, 2317 (1993)], where the temperature depends locally on the spin configuration and detailed balance and local equilibrium are not obeyed. We derive a relation between the linear response function and correlation functions that generalizes the fluctuation-dissipation theorem. In the stationary states of the model, which are the counterparts of the Ising equilibrium states, the fluctuation-dissipation theorem breaks down due to the lack of time reversal invariance. In the nonstationary phase-ordering kinetics, the parametric plot of the integrated response function chi(t,t(w)) vs the autocorrelation function is different from that of the kinetic Ising model. However, splitting chi(t,t(w)) into a stationary and an aging term chi(t,t(w)) = chi(st)(t-t(w)) + chi(ag)(t,t(w)), we find chi(ag)(t,t(w)) approximately t(w)(-a(chi)) f(t/t(w)), and a numerical value of a(chi) consistent with a(chi)= 1/4, as in the kinetic Ising model.

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.

Comment on "Scaling of the linear response in simple aging systems without disorder"

We have repeated the simulations of Henkel, Paessens, and Pleimling (HPP) [Phys. Rev. E 69, 056109 (2004)] for the field-cooled susceptibility chi(FC)(t) - chi0 approximately t(-A) in the quench of ferromagnetic systems to and below T(C). We show that, contrary to the statement made by HPP, the exponent A coincides with the exponent a of the linear response function R(t,s) approximately s(-(1+a))f(R)(t/s). We point out what are the assumptions in the argument of HPP that lead them to the conclusion A < a.

Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function

We derive for Ising spins an off-equilibrium generalization of the fluctuation dissipation theorem, which is formally identical to the one previously obtained for soft spins with Langevin dynamics [L.F. Cugliandolo, J. Kurchan, and G. Parisi, J. Phys. I 4, 1641 (1994)]. The result is quite general and holds both for dynamics with conserved and nonconserved order parameters. On the basis of this fluctuation dissipation relation, we construct an efficient numerical algorithm for the computation of the linear response function without imposing the perturbing field, which is alternative to those of Chatelain [J. Phys. A 36, 10 739 (2003)] and Ricci-Tersenghi [Phys. Rev. E 68, 065104(R) (2003)]. As applications of the new algorithm, we present very accurate data for the linear response function of the Ising chain, with conserved and nonconserved order parameter dynamics, finding that in both cases the structure is the same with a very simple physical interpretation. We also compute the integrated response function of the two-dimensional Ising model, confirming that it obeys scaling chi (t, t(w)) approximately equal to t(-a)(w) f (t/t(w)) , with a =0.26+/-0.01 , as previously found with a different method.

Generic features of the fluctuation dissipation relation in coarsening systems

The integrated response function in phase-ordering systems with scalar, vector, conserved, and nonconserved order parameter is studied at various space dimensionalities. Assuming scaling of the aging contribution chi(ag) (t, t(w) )= tw (- a(chi) ) chi; (t/ t(w) ) we obtain, by numerical simulations and analytical arguments, the phenomenological formula describing the dimensionality dependence of a(chi) in all cases considered. The primary result is that a(chi) vanishes continuously as d approaches the lower critical dimensionality d(L). This implies that (i) the existence of a nontrivial fluctuation dissipation relation and (ii) the failure of the connection between statics and dynamics are generic features of phase ordering at d(L).

Scaling of the linear response function from zero-field-cooled and thermoremanent magnetization in phase-ordering kinetics

In this paper we investigate the relation between the scaling properties of the linear response function R(t,s), of the thermoremanent magnetization (TRM) and of the zero-field-cooled (ZFC) magnetization in the context of phase-ordering kinetics. We explain why the retrieval of the scaling properties of R(t,s) from those of TRM and ZFC magnetization is not trivial. Preasymptotic contributions generate a long crossover in TRM, while ZFC magnetization is affected by a dangerous irrelevant variable. Lack of understanding of both these points has generated some confusion in the literature. The full picture relating the exponents of all the quantities involved is explicitly illustrated in the framework of the large-N model. Following this scheme, an assessment of the present status of numerical simulations for the Ising model can be made. We reach the conclusion that on the basis of the data available up to now, statements on the scaling properties of R(t,s) can be made from ZFC magnetization but not from TRM. From ZFC data for the Ising model with d=2,3,4 we confirm the previously found linear dependence on dimensionality of the exponent a entering R(t,s) approximately s(-(1+a))f(t/s). We also find evidence that a recently derived form of the scaling function f(x), using local scale invariance arguments [M. Henkel, M. Pleimling, C. Godrèche, and J. M. Luck, Phys. Rev. Lett. 87, 265701 (2001)], does not hold for the Ising model.

Ordering of the lamellar phase under a shear flow

The dynamics of a system quenched into a state with lamellar order and subject to an uniform shear flow is solved in the large-N limit. The description is based on the Brazovskii free energy and the evolution follows a convection-diffusion equation. Lamellas order preferentially with the normal along the vorticity direction. Typical lengths grow as gamma t(5/4) (with logarithmic corrections) in the flow direction and logarithmically in the shear direction. Dynamical scaling holds in the two-dimensional case while it is violated in D=3.

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.

Off-equilibrium response function in the one-dimensional random-field Ising model

A thorough numerical investigation of the slow dynamics in the d=1 random-field Ising model in the limit of an infinite ferromagnetic coupling is presented in this paper. Crossovers from the preasymptotic pure regime to the asymptotic Sinai regime are investigated for the average domain size, the autocorrelation function, and staggered magnetization. By switching on an additional small random field at the time t(w) the linear off-equilibrium response function is obtained, which displays as well the crossover from the nontrivial behavior of the d=1 pure Ising model to the asymptotic behavior where it vanishes identically.

Slow relaxation in the large-N model for phase ordering

The basic features of the slow relaxation phenomenology arising in phase ordering processes are obtained analytically in the large-N model through the exact separation of the order parameter into the sum of thermal and condensation components. The aging contribution in the response function chi(ag)(t,t(w)) is found to obey a pattern of behavior, under variation of dimensionality, qualitatively similar to the one observed in Ising systems. There exists a critical dimensionality (d=4) above which chi(ag)(t,t(w)) is proportional to the defect density rho(D)(t), while for d<4 it vanishes more slowly than rho(D)(t) and at d=2 does not vanish. As in the Ising case, this behavior can be understood in terms of the dependence on dimensionality of the interplay between the defect density and the effective response associated to a single defect.

Interface fluctuations, bulk fluctuations, and dimensionality in the off-equilibrium response of coarsening systems

The relationship between statics and dynamics proposed by Franz, Mezard, Parisi, and Peliti (FMPP) for slowly relaxing systems [Phys. Rev. Lett. 81, 1758 (1998)] is investigated in the framework of nondisordered coarsening systems. Separating the bulk from interface response we find that for statics to be retrievable from dynamics the interface contribution must be asymptotically negligible. How fast this happens depends on dimensionality. There exists a critical dimensionality above that the interface response vanishes like the interface density and below that it vanishes more slowly. At d=1 the interface response does not vanish leading to the violation of the FMPP scheme. This behavior is explained in terms of the competition between curvature-driven and field-driven interface motion.

Steady state of microemulsions in shear flow

Steady-state properties of microemulsions in shear flow are studied in the context of a Ginzburg-Landau free-energy approach. Explicit expressions are given for the structure factor and the time correlation function at the one-loop level of approximation. Our results predict a four-peak pattern for the structure factor, implying the simultaneous presence of interfaces aligned with two different orientations. Due to the peculiar interface structure a nonmonotonous relaxation of the time correlator is also found.

Slow dynamics and aging in a constrained diffusion model

We carry out a complete analysis of the schematic diffusive model recently introduced for the description of supercooled liquids and glassy systems above the glass temperature. The model is described by a trivial equilibrium measure and the presence of kinetics constraints is mimicked through a rapidly decreasing mobility at high particle density. The governing equation describing a sudden quench process is investigated analytically in a mean field approach and by means of numerical simulations. For deep quenches a long lasting off-equilibrium dynamics is observed in dense systems before equilibration is achieved, where time translational invariance lacks and the system ages. The kinetics is slow in this time domain since the average particle diffusivity D decreases in time, as opposed to the standard diffusion case of a constant D, that is recovered only in equilibrium. The autocorrelation function decays slower than an exponential, falling in mean field as an enhanced power law. The linear response function is computed and the modalities of the break-down of the fluctuation dissipation theorem are analytically investigated, showing that an effective temperature can be defined which slowly approaches the bath temperature from above.

Phase separation of binary mixtures in shear flow: A numerical study

The phase-separation kinetics of binary fluids in shear flow is studied numerically in the framework of the continuum convection-diffusion equation based on a Ginzburg-Landau free energy. Simulations are carried out for different temperatures both in d=2 and 3. Our results confirm the qualitative picture put forward by the large-N limit equations studied by Corberi et al. [Phys. Rev. Lett. 81, 3852 (1998)]. In particular, the structure factor is characterized by the presence of four peaks whose relative oscillations give rise to a periodic modulation of the behavior of the rheological indicators and of the average domains sizes. This peculiar pattern of the structure factor corresponds to the presence of domains with two characteristic thicknesses, whose relative abundance changes with time.

Structure and rheology of binary mixtures in shear flow

Results are presented for the phase separation process of a binary mixture subject to a uniform shear flow quenched from a disordered to a homogeneous ordered phase. The kinetics of the process is described in the context of the time-dependent Ginzburg-Landau equation with an external velocity term. The large-n approximation is used to study the evolution of the model in the presence of a stationary flow and in the case of an oscillating shear. For stationary flow we show that the structure factor obeys a generalized dynamical scaling. The domains grow with different typical length scales Rx and R( perpendicular), respectively, in the flow direction and perpendicularly to it. In the scaling regime R( perpendicular) approximately t(alpha( perpendicular)) and Rx approximately gammat(alpha(x)) (with logarithmic corrections), gamma being the shear rate, with alpha(x)=5/4 and alpha( perpendicular)=1/4. The excess viscosity Deltaeta after reaching a maximum relaxes to zero as gamma(-2)t(-3/2). Deltaeta and other observables exhibit logarithmic-time periodic oscillations which can be interpreted as due to a growth mechanism where stretching and breakup of domains occur cyclically. In the case of an oscillating shear a crossover phenomenon is observed: Initially the evolution is characterized by the same growth exponents as for a stationary flow. For longer times the phase-separating structure cannot align with the oscillating drift and a different regime is entered with an isotropic growth and the same exponents as in the case without shear.