Violation of the fluctuation–dissipation theorem in glassy systems: basic notions and the numerical evidence

Generalized fluctuation-dissipation relation and effective temperature upon heating a deeply supercooled liquid

We show that a generalized fluctuation-dissipation relation applies upon instantaneously increasing the temperature of a deeply supercooled liquid. This has the same two-step shape of the relation found upon cooling the liquid, but with opposite violation, indicating an effective temperature that is lower than bath temperature. We show that the effective temperature exhibits some sensible time dependence and that it retains its connection with the partitioned phase space visited in aging. We underline the potential relevance of our numerical results for experimental studies of the fluctuation-dissipation relation in glassy systems.

Stochastic thermodynamics, fluctuation theorems and molecular machines

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

Nonequilibrium and information: the role of cross correlations

We discuss the relevance of information contained in cross correlations among different degrees of freedom, which is crucial in nonequilibrium systems. In particular we consider a stochastic system where two degrees of freedom X{1} and X{2}-in contact with two different thermostats-are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation theorem (FDT) are both related to the cross correlation between X{1} and X{2}. Information about such cross correlation may be lost when single-variable reduced models for X_{1} are considered. Two different procedures are typically applied: (a) one totally ignores the coupling with X{2}; and (b) one models the effect of X{2} as an average memory effect, obtaining a generalized Langevin equation. In case (a) discrepancies between the system and the model appear both in entropy production and linear response; the latter can be exploited to define effective temperatures, but those are meaningful only when time scales are well separated. In case (b) linear response of the model well reproduces that of the system; however the loss of information is reflected in a loss of entropy production. When only linear forces are present, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting FDT violations.

Nonlinear response theory for Markov processes: simple models for glassy relaxation

The theory of nonlinear response for Markov processes obeying a master equation is formulated in terms of time-dependent perturbation theory for the Green's functions and general expressions for the response functions up to third order in the external field are given. The nonlinear response is calculated for a model of dipole reorientations in an asymmetric double well potential, a standard model in the field of dielectric spectroscopy. The static nonlinear response is finite with the exception of a certain temperature T_{0} determined by the value of the asymmetry. In a narrow temperature range around T_{0}, the modulus of the frequency-dependent cubic response shows a peak at a frequency on the order of the relaxation rate and it vanishes for both low frequencies and high frequencies. At temperatures at which the static response is finite (lower and higher than T_{0}), the modulus is found to decay monotonously from the static limit to zero at high frequencies. In addition, results of calculations for a trap model with a Gaussian density of states are presented. In this case, the cubic response depends on the specific dynamical variable considered and also on the way the external field is coupled to the kinetics of the model. In particular, a set of different dynamical variables that gives rise to identical shapes of the linear susceptibility and only to different temperature dependencies of the relaxation times is considered. It is found that the frequency dependence of the nonlinear response functions, however, strongly depends on the particular choice of the variables. The results are discussed in the context of recent theoretical and experimental findings regarding the nonlinear response of supercooled liquids and glasses.

Modified fluctuation-dissipation and Einstein relation at nonequilibrium steady states

Starting from the pioneering work of Agarwal [G. S. Agarwal, Zeitschrift für Physik 252, 25 (1972)], we present a unified derivation of a number of modified fluctuation-dissipation relations (MFDR) that relate response to small perturbations around nonequilibrium steady states to steady-state correlations. Using this formalism we show the equivalence of velocity forms of MFDR derived using continuum Langevin and discrete master equation dynamics. The resulting additive correction to the Einstein relation is exemplified using a flashing ratchet model of molecular motors.

Quantifying reversibility in a phase-separating lattice gas: an analogy with self-assembly

We present dynamic measurements of a lattice gas during phase separation, which we use as an analogy for self-assembly of equilibrium ordered structures. We use two approaches to quantify the degree of reversibility of this process: First, we count events in which bonds are made and broken; second, we use correlation-response measurements and fluctuation-dissipation ratios to probe reversibility during different time intervals. We show how correlation and response functions can be related directly to microscopic (ir)reversibility and we discuss the time dependence and observable dependence of these measurements, including the role of fast and slow degrees of freedom during assembly.

Effective confinement as origin of the equivalence of kinetic temperature and fluctuation-dissipation ratio in a dense shear-driven suspension

We study response and velocity autocorrelation functions for a tagged particle in a shear driven suspension governed by underdamped stochastic dynamics. We follow the idea of an effective confinement in dense suspensions and exploit a time scale separation between particle reorganization and vibrational motion. This allows us to approximately derive the fluctuation-dissipation theorem in a "hybrid" form involving the kinetic temperature as an effective temperature and an additive correction term. We show numerically that even in a moderately dense suspension the latter is negligible. We discuss similarities and differences with a simple toy model, a single trapped particle in shear flow.

Fluctuation-dissipation theorem in liquid and glassy glycerol: frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements

We show frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements of liquid and glassy glycerol. This allows a direct comparison of both quantities determined independently. After cooling the glycerol sample to 179 K with a cooling rate of 0.85 K/min we studied the aging time dependence of the dielectric permittivity and the polarization fluctuations using the identical glycerol sample. A cross-correlation technique allows measurements of noise levels below the amplifier noise. In the frequency range between 0.3 and 300 Hz we find the measured data to be in agreement with the fluctuation-dissipation theorem for the liquid and glassy state not depending on the aging time.

Transmission of information between complex systems: 1/f resonance

We study the transport of information between two complex systems with similar properties. Both systems generate non-Poisson renewal fluctuations with a power-law spectrum 1/f(3-μ), the case μ=2 corresponding to ideal 1/f noise. We denote by μ(S) and μ(P) the power-law indexes of the system of interest S and the perturbing system P, respectively. By adopting a generalized fluctuation-dissipation theorem (FDT) we show that the ideal condition of 1/f noise for both systems corresponds to maximal information transport. We prove that to make the system S respond when μ(S)<2 we have to set the condition μ(P)<2. In the latter case, if μ(P)<μ(S), the system S inherits the relaxation properties of the perturbing system. In the case where μ(P)>2, no response and no information transmission occurs in the long-time limit. We consider two possible generalizations of the fluctuation dissipation theorem and show that both lead to maximal information transport in the condition of 1/f noise.

Dynamics of the Blume-Capel model with quenched diluted single-ion anisotropy in the neighborhood of equilibrium states

The relaxation dynamics of a Blume-Capel model with a quenched diluted crystal field is formulated by a method combining the statistical equilibrium theory and the thermodynamics of linear irreversible processes. Using a mean-field approximation for the magnetic Gibbs free-energy production, a generalized force and a current are defined within the irreversible thermodynamics. Next the kinetic equation for the magnetization is obtained within linear response theory. Finally, the temperature dependence of the relaxation time in the neighborhood of the phase-transition points is obtained by solving the kinetic equation of the magnetization. We find that the relaxation time of the order parameter diverges near the critical and multicritical points, which corresponds to the familiar critical slowing down. On the other hand, it displays different behavior at the first-order phase transitions. It has a jump discontinuity at the first-order phase-transition temperatures. Moreover, the z dynamic critical exponent is calculated and compared with the z values obtained for a diverse class of systems, and good agreement is found with our results.

Memory effects in the relaxation of the Gaussian trap model

We investigate the memory effect in a simple model for glassy relaxation, a trap model with a Gaussian density of states. In this model, thermal equilibrium is reached at all finite temperatures and we therefore can consider jumps from low to high temperatures in addition to the quenches usually considered in aging studies. We show that the evolution of the energy following the Kovacs protocol can approximately be expressed as a difference of two monotonously decaying functions and thus show the existence of a so-called Kovacs hump whenever these functions are not single exponentials. It is well established that the Kovacs effect also occurs in the linear response regime, and we show that most of the gross features do not change dramatically when large temperature jumps are considered. However, there is one distinguishing feature that only exists beyond the linear regime, which we discuss in detail. For the memory experiment with inverted temperatures, i.e., jumping up and then down again, we find a very similar behavior apart from an opposite sign of the hump.

Increased accuracy of ligand sensing by receptor diffusion on cell surface

The physical limit with which a cell senses external ligand concentration corresponds to the perfect absorber, where all ligand particles are absorbed and overcounting of same ligand particles does not occur. Here, we analyze how the lateral diffusion of receptors on the cell membrane affects the accuracy of sensing ligand concentration. Specifically, we connect our modeling to neurotransmission in neural synapses where the diffusion of glutamate receptors is already known to refresh synaptic connections. We find that receptor diffusion indeed increases the accuracy of sensing for both the glutamate α-Amino-3-hydroxy-5-Methyl-4-isoxazolePropionic Acid (AMPA) and N-Methyl-D-aspartic Acid (NMDA) receptor, although the NMDA receptor is overall much noisier. We propose that the difference in accuracy of sensing of the two receptors can be linked to their different roles in neurotransmission. Specifically, the high accuracy in sensing glutamate is essential for the AMPA receptor to start membrane depolarization, while the NMDA receptor is believed to work in a second stage as a coincidence detector, involved in long-term potentiation and memory.

Aging effects manifested in the potential-energy landscape of a model glass former

We present molecular dynamics simulations of the binary Kob-Andersen Lennard-Jones model, a model glass former, and consider the distributions of inherent energies and metabasins during aging. In addition to the typical protocol of performing a temperature jump from a high temperature to a low destination temperature, we consider the temporal evolution of the energy distributions after an "up-jump," i.e., from a low to a high temperature. In this case, for times on the order of the relaxation time the distribution of metabasin energies exhibits a transient two-peak structure for a small system (65 particles). For a large system (1000 particles) no such two-peak structure appears because it is averaged out. The simulations show, however, that a clear signal of an intermediate two-peak structure survives in the thermodynamic limit, namely that the inherent energy distribution width goes through a maximum at intermediate times for temperature up jumps; no such maximum is found for temperature down jumps. These findings are qualitatively rationalized in terms of a simple trap model with a Gaussian distribution of energies, assuming that the liquid may be divided into non-interacting regions each of which is described by a trap model.

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.

Crystallization of hard disks induced by a temperature gradient

While uniform temperature has no effect on equilibrium properties of hard-core systems, its gradient might substantially change their behavior. In particular, in hard-disk system subject to temperature difference ΔT disks are repelled from the hot boundary of the system and accumulate at the cold one. Using event-driven molecular dynamics simulations we show that for sufficiently large ΔT or coverage ratio ρ∗, crystal forms at the cold boundary. In this spatially inhomogeneous system a significant decrease of diffusivity of disks clearly marks the stationary interface between liquid and crystal. Such a behavior is also supported through calculation of the radial distribution function and the bond order parameter. Simulations show that for this nonequilibrium system the equipartition of energy holds and velocity obeys the Boltzmann distribution.

Entropy-based characterizations of the observable dependence of the fluctuation-dissipation temperature

The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly on three different stochastic models, where observable dependence of the fluctuation-dissipation temperature arises only when the uniformity of the phase-space distribution is broken. The first model is an energy transport model on a ring, with biased local transfer rules. In the second model, defined on a fully connected geometry, energy is exchanged with two heat baths at different temperatures, breaking the uniformity of the phase-space distribution. Finally, in the last model, the system is connected to a zero temperature reservoir, and preserves the uniformity of the phase-space distribution in the relaxation regime, leading to an observable-independent temperature.

How do self-assembling polymers and gels age compared to glasses?

Experiments on gels have provided contradictory results concerning the relation between correlation and response functions during aging. To clarify this puzzle, we numerically investigate the fluctuation-dissipation plot in equilibrium polymers and in network forming gels employing two distinct observables, (i) the density Fourier transform and (ii) the single-particle potential energy, to probe (i) diffusional processes and (ii) the development of a bond network. The plot behaves very differently for the two cases. Violation from the equilibrium behavior is found only for the second observable. The experimental implications of the discovered sensitivity to the choice of the probe are discussed, in particular, with respect to the existing experimental results.

Predicting the effective temperature of a glass

We explain the findings by Di Leonardo et al. [Phys. Rev. Lett. 84, 6054 (2000)10.1103/PhysRevLett.84.6054] that the effective temperature of a Lennard-Jones glass depends only on the final density in the volume and/or temperature jump that produces the glass. This is not only a property of the Lennard-Jones liquid, but a feature of all strongly correlating liquids. For such liquids data from a single quench simulation provide enough information to predict the effective temperature of any glass produced by jumping from an equilibrium state. This prediction is validated by simulations of the Kob-Andersen binary Lennard-Jones liquid and shown not to apply for the nonstrongly correlating monatomic Lennard-Jones Gaussian liquid.

Increased accuracy of ligand sensing by receptor internalization

Many types of cells can sense external ligand concentrations with cell-surface receptors at extremely high accuracy. Interestingly, ligand-bound receptors are often internalized, a process also known as receptor-mediated endocytosis. While internalization is involved in a vast number of important functions for the life of a cell, it was recently also suggested to increase the accuracy of sensing ligand as the overcounting of the same ligand molecules is reduced. Here we show, by extending simple ligand-receptor models to out-of-equilibrium thermodynamics, that internalization increases the accuracy with which cells can measure ligand concentrations in the external environment. Comparison with experimental rates of real receptors demonstrates that our model has indeed biological significance.

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 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.

Dependence of the fluctuation-dissipation temperature on the choice of observable

On general grounds, a nonequilibrium temperature can be consistently defined from generalized fluctuation-dissipation relations only if it is independent of the observable considered. We argue that the dependence on the choice of observable generically occurs when the phase-space probability distribution is nonuniform on constant energy shells. We relate quantitatively this observable dependence to a fundamental characteristics of nonequilibrium systems, namely, the Shannon entropy difference with respect to the equilibrium state with the same energy. This relation is illustrated on a mean-field model in contact with two heat baths at different temperatures.

Fluctuation relations for diffusion that is thermally driven by a nonstationary bath

In the context of the Markovian dynamical evolution in a nonstationary thermal bath, we construct a family of fluctuation relations for the entropy production that are not verified by the work performed on the system. We exhibit fluctuation relations, which are global versions either of the generalized fluctuation-dissipation theorem around a nonequilibrium diffusion or of the usual fluctuation-dissipation theorem for energy resulting from a pulse of temperature.

Einstein relation and effective temperature for systems with quenched disorder

According to the Einstein relation the ratio between independent measurements of fluctuations (e.g., diffusivity) and the response to a weak external field (e.g., mobility) is equal to the thermal temperature when the system is kept close to thermal equilibrium. For strongly disordered systems, which are not self-averaging this ratio is a random variable and hence in this sense the Einstein relation is not valid. Thus effective temperatures found using the fluctuation dissipation ratio are at least in some cases stochastic variables. This scenario is tested with the quenched trap model. An average over an ensemble of systems yields an averaged effective temperature, which is compared with results obtained from the mean-field continuous-time random-walk model.

Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state

A modified fluctuation-dissipation theorem for a nonequilibrium steady state is experimentally checked by studying the position fluctuations of a colloidal particle whose motion is confined in a toroidal optical trap. The nonequilibrium steady state is generated by means of a rotating laser beam which exerts on the particle a sinusoidal conservative force plus a constant nonconservative one. The modified fluctuation-dissipation theorem is perfectly verified by the experimental data. It can be interpreted as an equilibriumlike fluctuation-dissipation relation in the Lagrangian frame of the mean local velocity of the particle.

Negative fluctuation-dissipation ratios in the backgammon model

We analyze fluctuation-dissipation relations in the backgammon model: a system that displays glassy behavior at zero temperature due to the existence of entropy barriers. We study local and global fluctuation relations for the different observables in the model. For the case of a global perturbation we find a unique negative fluctuation-dissipation ratio that is independent of the observable and which diverges linearly with the waiting time. This result suggests that a negative effective temperature can be observed in glassy systems even in the absence of thermally activated processes.

Apparent violation of the fluctuation-dissipation theorem due to dynamic heterogeneity in a model glass-forming liquid

Here we study the relation between the mobility and the translational diffusion in supercooled two-dimensional polydisperse colloidal liquids, using numerical simulations. We find an apparent violation of the Einstein-Smoluchowski (ES) relation D=kB T micro (D: diffusion constant; mu: mobility; kB; Boltzmann's constant; T: temperature). The violation is a direct consequence of the fact that it is difficult for a driven particle to enter a jammed region with high order due to its yield stress. The degree of this apparent ES violation is controlled solely by the characteristic size of slow jammed regions, xi. Our finding implies that the characteristic time of this problem is not the structural relaxation time tau alpha but the lifetime of dynamic heterogeneity, tau xi. A supercooled liquid can be regarded to be ergodic only over tau xi, which may be the slowest intrinsic time scale of the 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.

Macroscopic expression connecting the rate of energy dissipation with the violation of the fluctuation response relation

A direct connection between the magnitude of the violation of the fluctuation response relation (FRR) and the rate of energy dissipation is presented in terms of field variables of nonequilibrium systems. Here, we consider the density field of a colloidal suspension either in a relaxation process or in a nonequilibrium steady state driven by an external field. Using a path-integral representation of the temporal evolution of the density field, we find an equality that relates the magnitude of the violation of the FRR for scalar and vector potentials of the velocity field to the rate of energy dissipation for the entire system. Our result demonstrates that the violation of the FRR for field variables captures the entropic component of the dissipated free energy.

Temperature of systems out of thermodynamic equilibrium

Two phenomenological approaches are currently used in the study of the vitreous state. One is based on the concept of fictive temperature introduced by Tool [J. Res. Natl. Bur. Stand. 34, 199 (1945)] and recently revisited by Nieuwenhuizen [Phys. Rev. Lett. 80, 5580 (1998)]. The other is based on the thermodynamics of irreversible processes initiated by De Donder at the beginning of the last century [L'Affinite (Gauthier-Villars, Paris, 1927)] and recently used by Moller et al. for a thorough study of the glass transition [J. Chem. Phys. 125, 094505 (2006)]. This latter approach leads to the possibility of describing the glass transition by means of the freezing-in of one or more order parameters connected to the internal structural degrees of freedom involved in the vitrification process. In this paper, the equivalence of the two preceding approaches is demonstrated, not only for glasses but in a very general way for any system undergoing an irreversible transformation. This equivalence allows the definition of an effective temperature for all systems departed from equilibrium generating a positive amount of entropy. In fact, the initial fictive temperature concept of Tool leads to the generalization of the notion of temperature for systems out of thermodynamic equilibrium, for which glasses are just particular cases.

Nonequilibrium critical dynamics of the two-dimensional Ising model quenched from a correlated initial state

The universality class, even the order of the transition, of the two-dimensional Ising model depends on the range and the symmetry of the interactions (Onsager model, Baxter-Wu model, Turban model, etc.), but the critical temperature is generally the same due to self-duality. Here we consider a sudden change in the form of the interaction and study the nonequilibrium critical dynamical properties of the nearest-neighbor model. The relaxation of the magnetization and the decay of the autocorrelation function are found to display a power law behavior with characteristic exponents that depend on the universality class of the initial state.

Temperature of a Hamiltonian system given as the effective temperature of a nonequilibrium steady-state Langevin thermostat

In nonequilibrium steady states (NESS) far from equilibrium, it is known that the Einstein relation is violated. Then, the ratio of the diffusion coefficient to the mobility is called an effective temperature, and the physical relevance of this effective temperature has been studied in several works. Although the physical relevance is not yet completely clear, it has been found that the role of an effective temperature in NESS is indeed analogous to that of the temperature in equilibrium systems in a number of respects. In this paper, we find further evidence establishing this analogy. We employ a nonequilibrium Langevin system as a thermostat for a Hamiltonian system and find that the kinetic temperature of this Hamiltonian system is equal to the effective temperature of the thermostat.

Fluctuation-dissipation ratios in the dynamics of self-assembly

We consider two seemingly very different self-assembly processes: formation of viral capsids and crystallization of sticky disks. At low temperatures, assembly is ineffective, since there are many metastable disordered states, which are a source of kinetic frustration. We use fluctuation-dissipation ratios to extract information about the degree of this frustration. We show that our analysis is a useful indicator of the long-term fate of the system, based on the early stages of assembly.

Fluctuation-dissipation theorem for event-dominated processes

We study a system whose dynamics are driven by non-Poisson, renewal, and nonergodic events. We show that external perturbations influencing the times at which these events occur violate the standard fluctuation-dissipation prescription due to renewal aging. The fluctuation-dissipation relation of this Letter is shown to be the linear response limit of an exact expression that has been recently proposed to account for the luminescence decay in a Gibbs ensemble of semiconductor nanocrystals, with intermittent fluorescence.

Fluctuation-dissipation relation for stochastic dynamics without detailed balance

We show that the fluctuation-dissipation relation can be established for a class of stochastic dynamics that lack detailed balance. This class comprises lattice spin models whose time evolution is governed by a master equation with a one-spin-flip transition rate having the up-down symmetry. The relation is obtained by the introduction of a multiplicative perturbation of the transition rate, which reduces to the usual perturbation when detailed balance is fulfilled. As a part of the derivation we set up an equivalent two-spin-flip stochastic dynamics that conserves the magnetization.

Ontology of temperature in nonequilibrium systems

The laws of thermodynamics provide a clear concept of the temperature for an equilibrium system in the continuum limit. Meanwhile, the equipartition theorem allows one to make a connection between the ensemble average of the kinetic energy and the uniform temperature. When a system or its environment is far from equilibrium, however, such an association does not necessarily apply. In small systems, the regression hypothesis may not even apply. Herein, it is shown that in small nonequilibrium systems, the regression hypothesis still holds, though with a generalized definition of the temperature. The latter must now be defined for each such manifestation.

Efficient measurement of linear susceptibilities in molecular simulations: application to aging supercooled liquids

We propose a novel method to measure time-dependent linear susceptibilities in molecular simulations, which does not require the use of nonequilibrium simulations, subtraction techniques, or fluctuation-dissipation theorems. The main idea is an exact reformulation of linearly perturbed quantities in terms of observables accessible in a single unperturbed trajectory. We apply these ideas to supercooled liquids in a nonequilibrium aging regime. We show that previous work had underestimated deviations from fluctuation-dissipation relations in the case of a Lennard-Jones system, while our results for silica are in qualitative disagreement with earlier results.

Einstein relation generalized to nonequilibrium

The Einstein relation connecting the diffusion constant and the mobility is violated beyond the linear response regime. For a colloidal particle driven along a periodic potential imposed by laser traps, we test the recent theoretical generalization of the Einstein relation to the nonequilibrium regime which involves an integral over measurable velocity correlation functions.

Violation of the fluctuation-dissipation theorem in a protein system

We report the results of molecular dynamics simulations of the protein myosin carried out with an elastic network model. Quenching the system, we observe glassy behavior of a density correlation function and a density response function that are often investigated in structure glasses and spin glasses. In the equilibrium, the fluctuation-response relation, a representative relation of the fluctuation-dissipation theorem, holds that the ratio of the density correlation function to the density response function is equal to the temperature of the environment. We show that, in the quenched system that we study, this relation can be violated. In the case that this relation does not hold, this ratio can be regarded as an effective temperature. We find that this effective temperature of myosin is higher than the temperature of the environment. We discuss the relation between this effective temperature and energy transduction that occurs after ATP hydrolysis in the myosin molecule.

Strong dynamical heterogeneities in the violation of the fluctuation-dissipation theorem in spin glasses

We analyze numerically the violation of the fluctuation-dissipation theorem (FDT) in the +/-J Edwards-Anderson (EA) spin-glass model. Using single spin probability densities we reveal the presence of strong dynamical heterogeneities, which correlate with ground-state information. The physical interpretation of the results shows that the spins can be divided into two sets. In 3D, one set forms a compact structure which presents a coarseninglike behavior with its characteristic violation of the FDT, while the other asymptotically follows the FDT. Finally, we compare the dynamical behavior observed in 3D with 2D.

Intensive thermodynamic parameters in nonequilibrium systems

Considering a broad class of steady-state nonequilibrium systems for which some additive quantities are conserved by the dynamics, we introduce from a statistical approach intensive thermodynamic parameters (ITPs) conjugated to the conserved quantities. This definition does not require any detailed balance relation to be fulfilled. Rather, the system must satisfy a general additivity property, which holds in most of the models usually considered in the literature, including those described by a matrix product ansatz with finite matrices. The main property of these ITPs is to take equal values in two subsystems, making them a powerful tool to describe nonequilibrium phase coexistence, as illustrated on different models. We finally discuss the issue of the equalization of ITPs when two different systems are put into contact. This issue is closely related to the possibility of measuring the ITPs using a small auxiliary system, in the same way as temperature is measured with a thermometer, and points at one of the major difficulties of nonequilibrium statistical mechanics. In addition, an efficient alternative determination, based on the measure of fluctuations, is also proposed and illustrated.

Measurement of effective temperatures in an aging colloidal glass

We study the thermal fluctuations of an optically confined probe particle, suspended in an aging colloidal suspension, as the suspension transforms from a viscous liquid into an elastic glass. The micron-sized bead forms a harmonic oscillator. By monitoring the equal-time fluctuations of the tracer, at two different laser powers we determine the temperature of the oscillator, T(o). In the ergodic liquid the temperatures of the oscillator and its environment are equal, while in contrast, in a nonequilibrium glassy phase we find that T(o) substantially exceeds the bath temperature.

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.

Aging in a free-energy landscape model for glassy relaxation. II. Fluctuation-dissipation relations

Several fluctuation-dissipation relations are investigated for a simple free-energy landscape model designed to describe the primary relaxation in supercooled liquids. The calculations of the response and of the correlation functions are performed for a quench from a high temperature to a low temperature. In the model, all dynamical quantities reach equilibrium after long times, but for times shorter than the re-equilibration time they do not exhibit time-translational invariance and the fluctuation-dissipation theorem is violated. Two measures for these violations are considered. One such measure is given by the slope in a plot of the integrated response versus the correlation function and another one by the so-called fluctuation-dissipation ratio. It is found that these measures do not coincide and furthermore are not independent of the dynamical variable considered in the calculation. We propose to determine the fluctuation-dissipation ratio experimentally via measurements of the deuteron spin-lattice relaxation rate and the dielectric loss.

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.

Activated aging dynamics and negative fluctuation-dissipation ratios

In glassy materials, aging proceeds at large times via thermal activation. We show that this can lead to negative dynamical response functions and novel and well-defined violations of the fluctuation-dissipation theorem, in particular, negative fluctuation-dissipation ratios. Our analysis is based on detailed theoretical and numerical results for the activated aging regime of simple kinetically constrained models. The results are relevant to a variety of physical situations, such as aging in glass formers, thermally activated domain growth, and granular compaction.