Linear response subordination to intermittent energy release in off-equilibrium aging dynamics

The interpretation of experimental and numerical data describing off-equilibrium aging dynamics crucially depends on the connection between spontaneous and induced fluctuations. The hypothesis that linear response fluctuations are statistically subordinated to irreversible outbursts of energy, so-called quakes, leads to predictions for averages and fluctuations spectra of physical observables in reasonable agreement with experimental results [see e.g. Sibani et al., Phys. Rev. B74:224407, 2006]. Using simulational data from a simple but representative Ising model with plaquette interactions, direct statistical evidence supporting the hypothesis is presented and discussed in this work. A strict temporal correlation between quakes and intermittent magnetization fluctuations is demonstrated. The external magnetic field is shown to bias the pre-existent intermittent tails of the magnetic fluctuation distribution, with little or no effect on the Gaussian part of the latter. Its impact on energy fluctuations is shown to be negligible. Linear response is thus controlled by the quakes and inherits their temporal statistics. These findings provide a theoretical basis for analyzing intermittent linear response data from aging system in the same way as thermal energy fluctuations, which are far more difficult to measure.


Introduction
How spontaneous fluctuations and linear response are related in off-equilibrium thermal dynamics is an open problem of considerable interest, as linear response measurements [1,2,3,4] are the main in-road into the rich phenomenology of aging systems. Highlighting this issue are recent obser-vations that aging dynamics is intermittent [5,6,7,8]: rare but large fluctuations with an exponential size distribution punctuate much smaller equilibrium-like fluctuations with a Gaussian size distribution. Simulational [9,10,11,12] and experimental [8] evidence in different areas supports the idea that intermittent changes of magnetization and other observables are induced by, and hence statistically subordinated to, intermittent and irreversible outburst of heat, so called quakes. The same hypothesis leads to a widely discussed asymptotic logarithmic time re-parameterization of the aging dynamics (see e.g. Ref. [13]). To ascertain whether or not other mechanisms than subordination could produce these effects, direct statistical evidence is called for.
In the following, statistical subordination and closely related issues are numerically investigated, using as a test-bed an Ising model, which is simple and yet constitutes a bona fide instance of a complex aging system [11,14,15]. Intermittent changes in magnetization are shown to obey the same statistics as the quakes, and the magnetic field is shown to have negligible influence on the energy relaxation. Together, these two findings imply that aging dynamics is quake-driven. The fluctuation statistics is then investigated in detail, with the model properties in good agreement with previous investigations of intermittent heat flow [7,11] and of magnetic linear response intermittency [8,12]. Quakes in this model are shown to be nearly uncorrelated events, which are described by a Poisson process whose average increases logarithmically in time. Additionally, a linear system-size dependence of the average number of quakes is found. This confirms that quakes are spatially localized events, a property also found directly in this work via a real space analysis. The temperature dependence of the average number of quakes is very weak, except at the lowest temperatures. This hints to a hierarchical structure in the energy landscape of the thermalized domains spawning the quakes.

Model and method
In the model, N Ising spins, σ i = ±1 are placed on a cubic lattice with periodic boundary conditions. They interact through the plaquette Hamiltonian The first sum runs over the elementary plaquettes of the lattice, including for each the product of the four spins located at its corners. The second term describes the coupling of the total magnetization σ i to an external magnetic field. As expressed by the Heaviside step function η(t w − t), the field changes instantaneously at t = t w from zero to H > 0. Previous investigations of the model's properties in zero field show a low temperature aging regime [14,15], during which energy leaves the system intermittently, and at a rate falling off as the inverse time [11].
The present simulations are all performed within the aging regime, i.e. in the temperature range 0.5 < T < 2.5, using the rejectionless Waiting Time Algorithm (WTM) [16]. The 'intrinsic' time unit of the WTM approximately corresponds to one Monte Carlo sweep. By choosing a high energy random configuration as initial state for low temperature isothermal simulations, an effectively instantaneous thermal quench is achieved. For each set of physical parameters, Probability Distribution Functions (PDFs) are collected over 2000 independent runs, and other statistical data over 1000 independent runs. The symbol t stands for the time elapsed from the initial quench (and from the beginning of the simulations). The symbol t w is the time at which the field is switched on, while t obs = t − t w stands for the 'observation' time, during which data are collected. The value of the external field is set to H = 0.3 for t > t w . The thermal energy is denoted by E, and the magnetization by M . The average energy per spin is denoted by µ e . The left intermittent wing is clearly reduced relative to the no-field case, and the right intermittent wing is correspondingly amplified. The inner, almost Gaussian shaped, PDF (stars) is the conditional PDF obtained by excluding the magnetic fluctuations which happen in unison with the quakes.
The PDF of fluctuations in energy and magnetization are constructed using finite time differences of E and M , taken over short time intervals of length δt ≪ t obs .

Results
To qualify as a probe, an applied perturbation must not significantly change the dynamics of the system investigated. Specifically for the present model, the average energy vs. time should be nearly unaffected. Secondly, fluctuation spectra should not undergo qualitative changes. Figure 1, left panel, shows the average energy as a function of time, with six data sets corresponding to the perturbation switched on at t w values ranging from t w = 200 to t w = 2000. As expected, the field has little impact on the energy. The right panel shows, on a logarithmic scale, the PDF of the energy fluctuations in zero field (circles) and when a field is switched on at t w = 1000 (stars). Gaussian energy fluctuations of zero average are flanked on the left by an intermittent tail, which carries the net heat flow out of the system. Again, only a minor difference is seen between the unperturbed and perturbed fluctuation spectra, and only for the largest and rarest of the fluctuations. Since the magnetic contribution to the average energy is negligible, quakes the dissipation of the excess energy entrapped in the initial configuration, see Fig. 1, in the same as in an unperturbed system would do [11]. While a magnetic field induces a non-zero average magnetization, it does not change the overall structure of the spectra: The left panel of Fig.2 shows the PDF of the spontaneous magnetic fluctuations occurring in the interval [t w , t w +t obs ]. Intermittent wings symmetrically extend the central Gaussian part of the PDF. The outer curve (circles) in the right panel of the same figure depicts the PDF obtained when the field is turned on at t w = 1000. The positive intermittent tail is enhanced, the negative tail is reduced and the Gaussian part is not affected. Thus, the net average magnetization induced by the field, i.e. the linear response, arises through a biasing effect on the distribution of the spontaneous intermittent magnetic fluctuations.
A key aging feature [7,11] is that the excess energy trapped by the initial quench leaves the system through intermittent quakes. Intermittent magnetic fluctuations have a close temporal association to the quakes: The inner curve (stars) of Fig. 2 depicts a conditional PDF obtained by filtering out the magnetic fluctuations which occur simultaneously (i.e., in practice, either within the same or within the immediately following δt ) with energy fluctuations of magnitude δE ≤ −5. The filtering threshold utilized is near The normalized correlation between τ k and τ k+n (stars) is plotted versus n on a logarithmic scale. The line is a guide to the eye. We see that consecutive data points (n = 1) are only weakly correlated, and that the correlation decays exponentially for n > 1. This is in reasonable qualitative agreement with the theoretical approximation C(n) = δ n,0 .
the onset of the intermittent behavior of the heat flow PDF, as seen in Fig 1. The filtering produces a nearly Gaussian PDF, demonstrating that quakes and intermittent magnetic fluctuations are synchronous events. In summary, the magnetic fluctuations which contribute to the linear response are subordinated to the quakes which dissipate the excess energy stored in the initial configuration. Visual inspection shows that energy traces of the p-spin model have a fluctuating part superimposed onto a monotonic step-wise decay, the latter given by the function r E (t) = min y<t E(t). This function is called record signal, or Best So Far (BSF) energy. The BSF energy is plotted in the main left panel of Fig. 3. The insert shows the full trace for a shorter interval of time. The right panel of Fig. 3  single quake typically involves small clusters only.
The quake statistics of this model should be amenable to a description as a Poisson process with average [17,18,19,7] n I (t ′ , t) = α ln(t/t ′ ). ( Assuming that quakes can be treated as instantaneous events, which occur at times t 1 , . . . t q . . ., a testable property equivalent to Eq. 2, is that the logarithmic differences (rather than linear differences) τ q def = ln(t q /t q−1 ) are independent and identically distributed stochastic variables, sharing the exponential distribution Prob(τ q > x) = exp(−αx). ( A simple choice to evaluate the τ q is to identify the quake times t q with the steps of the BSF energy. Admittedly, this introduces an unwanted dependence on the data sampling frequency, when the latter is too high: the same transition from one metastable configuration to another may then be registered as multiple quakes. This leads to over-counting, and as discussed below, to spurious correlations appearing between successive quakes. From the empirical series of {τ q } q=1,2,... collected for each trajectory, the correlation function is estimated as C(k) = τ q τ q+k q − τ q q τ q+k q . The result is then averaged over all trajectories. Systematic and statistical errors in the quake identification procedure lead to deviations from the theoretically expected Kronecker delta C(k)/C(0) = δ k,0 .
In the left panel of Fig. 4 the distribution of the τ q (dots) is well approximated by an exponential decay over nearly three decades (line). For large values of the abscissa, deviations likely stem from a statistical undersampling of rare events. For x ≈ 0 (the first three points are excluded from the fit) deviations indicate that successive intervals of equal or near equal duration appear more frequently than theoretically expected. This can arise when the interval between successive events is spuriously affected by the sampling time. The correlation function C(k) of the logarithmic time differences, normalized to C(k = 0) = 1, is shown in the right panel of the figure. After a drop to approximately 1/10 at k = 1, the function tapers off exponentially with increasing k. The (modest) residual correlation confirms that successive events are occasionally mis-classified as quakes.
A spatially extended glassy system with short range interactions is expected to contain a number, say α, of independently relaxing thermalized domains, with a small and slowly growing characteristic size [20,21]. Assuming that quakes originate independently within the domains, the observed fluctuations integrate the effect of α independent Poisson processes, whence α equals the logarithmic rate of quakes. Equivalently, its value can be estimated from the exponential distribution of the logarithmic waiting times τ q , see Fig. 4, as done in the present case.
The number of domains, and hence α, must increase linearly with the system size, as demonstrated in the left panel of Figure 5. The temperature dependence (or lack thereof) of α is related to the geometrical properties of the configuration space, or energy landscape, of each domain. For record sized thermal fluctuations to elicit attractor changes, the energy landscape must be scale invariant [12]. As a consequence, changing the temperature should not change α at all. Note however that scale invariance cannot hold below a cut-off value where the granularity of the energy values makes itself felt. In the present model, the numerically smallest energy change following a single spin flip is δE = ±4, i.e. unlike models with Gaussian quenched disorder, the granularity is important. The right panel of Figure 5 shows that α has a modest temperature dependence for 1 ≤ T ≤ 2, i.e. for a major part of the range where aging behavior is observed. For lower temperatures, a clear T dependence is visible.

Summary and conclusions
Direct numerical evidence has been provided that intermittent magnetic fluctuations are statistically subordinated to a certain type of events, quakes, which dissipate the excess energy trapped in the initial configuration. The external field does not alter the temporal statistics of the quakes and of the spontaneous magnetic fluctuations. It only slightly biases the size distribution of the latter. Therefore, the field can rightly be considered as a probe of the unperturbed off-equilibrium aging dynamics. In agreement with previous investigations of other models [12] and experiments [8], the temporal statistics of intermittent energy and magnetization changes, is shown to be well described by Poisson process.
Considering that aging dynamics is widely insensitive to details of the microscopic interactions, it seems reasonable to assume that the above findings are valid beyond the plaquette model itself. It should therefore be possible to analyze a wide range of intermittent linear response data from complex dynamical systems precisely as done for the intermittent heat flow data in Ref. [7,11].