Response of non-equilibrium systems at criticality: ferromagnetic models in dimension two and above

Relaxation time asymmetry in stator dynamics of the bacterial flagellar motor

The bacterial flagellar motor is the membrane-embedded rotary motor, which turns the flagellum that provides thrust to many bacteria. This large multimeric complex, composed of a few dozen constituent proteins, is a hallmark of dynamic subunit exchange. The stator units are inner-membrane ion channels that dynamically bind to the peptidoglycan at the rotor periphery and apply torque. Their dynamic exchange is a function of the viscous load on the flagellum, allowing the bacterium to adapt to its local environment, although the molecular mechanisms of mechanosensitivity remain unknown. Here, by actively perturbing the steady-state stator stoichiometry of individual motors, we reveal a stoichiometry-dependent asymmetry in stator remodeling kinetics. We interrogate the potential effect of next-neighbor interactions and local stator unit depletion and find that neither can explain the observed asymmetry. We then simulate and fit two mechanistically diverse models that recapitulate the asymmetry, finding assembly dynamics to be particularly well described by a two-state catch-bond mechanism.

Vortex Dynamics in a Compact Kardar-Parisi-Zhang System

We study the dynamics of vortices in a two-dimensional, nonequilibrium system, described by the compact Kardar-Parisi-Zhang equation, after a sudden quench across the critical region. Our exact numerical solution of the phase-ordering kinetics shows that the unique interplay between nonequilibrium and the variable degree of spatial anisotropy leads to different critical regimes. We provide an analytical expression for the vortex evolution, based on scaling arguments, which is in agreement with the numerical results, and confirms the form of the interaction potential between vortices in this system.

Aging Wiener-Khinchin theorem and critical exponents of 1/f^{β} noise

The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum 〈S_{t_{m}}(ω)〉 where t_{m} is the measurement time. For processes with an aging autocorrelation function of the form 〈I(t)I(t+τ)〉=t^{Υ}ϕ_{EA}(τ/t), where ϕ_{EA}(x) is a nonanalytic function when x is small, we find aging 1/f^{β} noise. Aging 1/f^{β} noise is characterized by five critical exponents. We derive the relations between the scaled autocorrelation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of 1/f^{β} noise. We illustrate our results for blinking-quantum-dot models, single-file diffusion, and Brownian motion in a logarithmic potential.

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.

Aging and coarsening in isolated quantum systems after a quench: Exact results for the quantum O(N) model with N → ∞

The nonequilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with O(N) symmetry, starting from a ground state in the disordered phase. In the limit of infinite N, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. Our analytical predictions are confirmed by the numerical solution of the corresponding equations. Moreover, we find that the same scaling functions, yet with different exponents, also describe the coarsening dynamics for quenches below the dynamical critical point.

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.

Aging and fluctuation-dissipation ratio in a nonequilibrium q-state lattice model

A generalized version of the nonequilibrium linear Glauber model with q states in d dimensions is introduced and analyzed. The model is fully symmetric, its dynamics being invariant under all permutations of the q states. Exact expressions for the two-time autocorrelation and response functions on a d-dimensional lattice are obtained. In the stationary regime, the fluctuation-dissipation theorem holds, while in the transient the aging is observed with the fluctuation-dissipation ratio leading to the value predicted for the linear Glauber model.

Nonequilibrium fluctuation-dissipation relations of interacting Brownian particles driven by shear

We present a detailed analysis of the fluctuation-dissipation theorem (FDT) close to the glass transition in colloidal suspensions under steady shear using mode coupling approximations. Starting point is the many-particle Smoluchowski equation. Under shear, detailed balance is broken and the response functions in the stationary state are smaller at long times than estimated from the equilibrium FDT. An asymptotically constant relation connects response and fluctuations during the shear driven decay, restoring the form of the FDT with, however, a ratio different from the equilibrium one. At short times, the equilibrium FDT holds. We follow two independent approaches whose results are in qualitative agreement. To discuss the derived fluctuation-dissipation ratios, we show an exact reformulation of the susceptibility which contains not the full Smoluchowski operator as in equilibrium, but only its well defined Hermitian part. This Hermitian part can be interpreted as governing the dynamics in the frame comoving with the probability current. We present a simple toy model which illustrates the FDT violation in the sheared colloidal system.

Fluctuation dissipation relations in stationary states of interacting Brownian particles under shear

The fluctuation dissipation theorem is studied close to the glass transition in colloidal suspensions under steady shear. Shear breaks detailed balance in the many-particle Smoluchowski equation and gives response functions in the stationary state which are smaller at long times than estimated from the equilibrium fluctuation dissipation theorem. During the final shear-driven decay, an asymptotically constant relation connects response and fluctuations, restoring the form of the fluctuation dissipation theorem with, however, a ratio different from the equilibrium one.

Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Ising model

The two-dimensional kinetic Ising model, when exposed to an oscillating applied magnetic field, has been shown to exhibit a nonequilibrium, second-order dynamic phase transition (DPT), whose order parameter Q is the period-averaged magnetization. It has been established that this DPT falls in the same universality class as the equilibrium phase transition in the two-dimensional Ising model in zero applied field. Here we study the scaling of the dynamic order parameter with respect to a nonzero, period-averaged, magnetic "bias" field, H(b) for a DPT produced by a square-wave applied field. We find evidence that the scaling exponent, delta(d), of H(b) at the critical period of the DPT is equal to the exponent for the critical isotherm, delta(e), in the equilibrium Ising model. This implies that H(b) is a significant component of the field conjugate to Q. A finite-size scaling analysis of the dynamic order parameter above the critical period provides further support for this result. We also demonstrate numerically that, for a range of periods and values of H(b) in the critical region, a fluctuation-dissipation relation (FDR), with an effective temperature T(eff)(T,P,H0) depending on the period, and possibly the temperature and field amplitude, holds for the variables Q and H(b). This FDR justifies the use of the scaled variance of Q as a proxy for the nonequilibrium susceptibility, partial differential Q/partial differential H(b), in the critical region.

Fluctuation-dissipation theorem and the linear Glauber model

We obtain exact expressions for the two-time autocorrelation and response functions of the -dimensional linear Glauber model. Although this linear model does not obey detailed balance in dimensions d > or = 2, we show that the usual form of the fluctuation-dissipation ratio still holds in the stationary regime. In the transient regime, we show the occurrence of aging, with a special limit of the fluctuation-dissipation ratio, x(infinity) = 1/2, for a quench at the critical point.

Universality of fluctuation-dissipation ratios: the ferromagnetic model

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

Fluctuation-dissipation relations for Markov processes

The fluctuation-dissipation relation is calculated for stochastic models obeying a master equation with continuous time. In the general case of a nonstationary process, there appears to be no simple relation between the response and the correlation. Also, if one considers stationary processes, the linear response cannot be expressed via time-derivatives of the correlation function alone. In this case, an additional function, which has rarely been discussed previously, is required. This so-called asymmetry depends on the two times also relevant for the response and the correlation and it vanishes under equilibrium conditions. The asymmetry can be expressed in terms of the propagators and the transition rates of the master equation but it is not related to any physical observable in an obvious way. It is found that the behavior of the asymmetry strongly depends on the nature of the dynamical variable considered in the calculation of the correlation and the response. If one is concerned with a variable which randomizes with any transition among the states of the system, the asymmetry vanishes in most cases. This is in contrast to the situation for other classes of variables. In particular, for trap models of glassy relaxation, the fluctuation-dissipation ratio strongly depends on the observable and the asymmetry plays a dominant role in the determination of this ratio also if only neutral variables are considered. Some implications of a nonvanishing asymmetry with regard to the definition of an effective temperature are discussed.

Universal aging properties at a disordered critical point

We investigate, analytically near the dimension d(uc) =4 and numerically in d=3 , the nonequilibrium relaxational dynamics of the randomly diluted Ising model at criticality. Using the exact renormalization-group method to one loop, we compute the two times t, t(w) correlation function and fluctuation dissipation ratio (FDR) for any Fourier mode of the order parameter, of finite wave vector q . In the large time separation limit, the FDR is found to reach a nontrivial value X(infinity) independently of (small) q and coincide with the FDR associated to the total magnetization obtained previously. Explicit calculations in real space show that the FDR associated to the local magnetization converges, in the asymptotic limit, to this same value X(infinity). Through a Monte Carlo simulation, we compute the autocorrelation function in three dimensions, for different values of the dilution fraction p at T(c) (p) . Taking properly into account the corrections to scaling, we find, according to the renormalization-group predictions, that the autocorrelation exponent lambda(c) is independent of p . The analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state.

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 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.

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.

Nominal thermodynamic temperature in nonequilibrium kinetic Ising models

We show that a nominal temperature can be consistently and uniquely defined everywhere in the phase diagram of large classes of nonequilibrium kinetic Ising spin models. In addition, we confirm that, at critical points, the large-time "fluctuation-dissipation ratio" X( infinity ) is a universal amplitude ratio, and find in particular X( infinity ) approximately 0.33(1) and X( infinity )=1 / 2 for the magnetization in, respectively, the two-dimensional Ising and voter universality classes.

Local scale invariance as dynamical space-time symmetry in phase-ordering kinetics

The scaling of the spatiotemporal response of coarsening systems is studied through simulations of the two-dimensional (2D) and 3D Ising model with Glauber dynamics. The scaling functions are consistent with the prediction of local scale invariance, thereby suggesting the extension of dynamical scaling to a space-time dynamical symmetry.

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.

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.

Dynamic heterogeneities in the out-of-equilibrium dynamics of simple spherical spin models

The response of spherical two-spin interaction models, the spherical ferromagnet (s-FM) and the spherical Sherrington-Kirkpatrick (s-SK) model, is calculated for the protocol of the so-called nonresonant hole burning (NHB) experiment for temperatures below the respective critical temperatures. It is shown that it is possible to select dynamic features in the out-of-equilibrium dynamics of both models, one of the hallmarks of dynamic heterogeneities. The behavior of the s-SK model and the s-FM model in three dimensions is very similar, showing dynamic heterogeneities in the long-time behavior, i.e., in the aging regime. The appearance of dynamic heterogeneities in the s-SK model explicitly demonstrates that these are not necessarily related to spatial heterogeneities. For the s-FM model, it is shown that the nature of the dynamic heterogeneities changes as a function of dimensionality. With the increasing dimension, the frequency selectivity of the NHB diminishes and the dynamics in the mean-field limit of the s-FM model becomes homogeneous.

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

We investigate the relation between two-time multispin correlation and response functions in the nonequilibrium critical dynamics of Ising models in d=1 and d=2 spatial dimensions. In these nonequilibrium situations, the fluctuation-dissipation theorem (FDT) is not satisfied. We find FDT "violations" qualitatively similar to those reported in various glassy materials, but quantitatively dependent on the chosen observable, in contrast to the results obtained in infinite-range glass models. Nevertheless, all FDT violations can be understood by considering separately the contributions from large wave vectors, which are at quasiequilibrium and obey the FDT, and from small wave vectors where a generalized FDT holds with a nontrivial fluctuation-dissipation ratio X infinity. In d=1, we get X(infinity)=1/2 for spin observables, which measure the orientation of domains, while X(infinity)=0 for observables that are sensitive to the domain-wall motion. Numerical simulations in d=2 reveal a unique X infinity approximately equal 0.34 for all observables. Measurement protocols for X infinity are discussed in detail. Our results suggest that the definition of an effective temperature T(eff)=T/X(infinity) for large length scales is generically possible in nonequilibrium critical dynamics.

Aging at criticality in model-C dynamics

We study the off-equilibrium two-point critical response and correlation functions for the relaxational dynamics with a coupling to a conserved density (model C) of the O(N) vector model. They are determined in an epsilon=4-d expansion for vanishing momentum. We briefly discuss their scaling behaviors and the associated scaling forms are determined up to first order in epsilon. The corresponding fluctuation-dissipation ratio has a nontrivial large time limit in the aging regime and, up to one-loop order, it is the same as that of the model A for the physically relevant case N=1. The comparison with predictions of local scale invariance is also discussed.

Equivalence of driven and aging fluctuation-dissipation relations in the trap model

We study the nonequilibrium version of the fluctuation-dissipation (FD) relation in the glass phase of a trap model that is driven into a nonequilibrium steady state by external "shear." This extends our recent study of aging FD relations in the same model, where we found limiting, observable independent FD relations for "neutral" observables that are uncorrelated with the system's average energy. In this work, for such neutral observables, we find the FD relation for a stationary weakly driven system to be the same, to within small corrections, as for an infinitely aged system. We analyze the robustness of this correspondence with respect to non-neutrality of the observable, and with respect to changes in the driving mechanism.

Two-loop critical fluctuation-dissipation ratio for the relaxational dynamics of the O(N) Landau-Ginzburg Hamiltonian

The off-equilibrium purely dissipative dynamics (model A) of the O(N) vector model is considered at criticality in an epsilon=4-d>0 expansion up to O(epsilon(2)). The scaling behavior of two-time response and correlation functions at zero momentum, the associated universal scaling functions and the nontrivial limit of the fluctuation-dissipation ratio are determined in the aging regime.

Relaxation and overlap-probability function in the spherical and mean-spherical models

The problem of the equivalence of the spherical and mean-spherical models, which has been thoroughly studied and is understood in equilibrium, is considered anew from the dynamical point of view during the time evolution following a quench from above to below the critical temperature. It is found that there exists a crossover time t(*) approximately V(2/d) such that for t<t(*) the two models are equivalent, while for t>t(*) macroscopic discrepancies arise. The relation between the off equilibrium response function and the structure of the equilibrium state that usually holds for phase ordering systems is found to hold for the spherical model but not for the mean-spherical one. The latter model offers an explicit example of a system which is not stochastically stable.

Aging in ferromagnetic systems at criticality near four dimensions

We study the off-equilibrium response and correlation functions and the corresponding fluctuation-dissipation ratio for a purely dissipative relaxation of an O(N) symmetric vector model (model A) below its upper critical dimension. The scaling behavior of these quantities is analyzed and the associated universal functions are determined at first order in epsilon=4-d in the high-temperature phase and at criticality. A nontrivial limit of the fluctuation-dissipation ratio is found in the aging regime X( infinity )=1/2(1-(epsilon/4)(N+2)/(N+8))+O(epsilon(2)).

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.

Observable dependence of fluctuation-dissipation relations and effective temperatures

We study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud's trap model. We incorporate an arbitrary observable m and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of m. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature T(eff) in this model. Consequences for the wider applicability of an FDT-derived T(eff) are discussed.

Aging, phase ordering, and conformal invariance

In a variety of systems which exhibit aging, the two-time response function scales as R(t,s) approximately s(-1-a)f(t/s). We argue that dynamical scaling can be extended towards conformal invariance, thus obtaining the explicit form of the scaling function f. This quantitative prediction is confirmed in several spin systems, both for T<T(c) (phase ordering) and T = T(c) (nonequilibrium critical dynamics). The 2D and 3D Ising models with Glauber dynamics are studied numerically, while exact results are available for the spherical model with a nonconserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass.