Linear response subordination to intermittent energy release in off-equilibrium aging dynamics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Linear response subordination to intermittentenergy release in off-equilibrium aging dynamics
Simon Christiansen and Paolo SibaniInstitut for Fysik og Kemi, SDU, DK5230 Odense MNovember 20, 2018
Abstract
The interpretation of experimental and numerical data describingoff-equilibrium aging dynamics crucially depends on the connection be-tween spontaneous and induced fluctuations. The hypothesis that lin-ear response fluctuations are statistically subordinated to irreversibleoutbursts of energy, so-called quakes , leads to predictions for averagesand fluctuations spectra of physical observables in reasonable agree-ment with experimental results [see e.g. Sibani et al., Phys. Rev.B74:224407, 2006]. Using simulational data from a simple but rep-resentative Ising model with plaquette interactions, direct statisticalevidence supporting the hypothesis is presented and discussed in thiswork. A strict temporal correlation between quakes and intermittentmagnetization fluctuations is demonstrated. The external magneticfield is shown to bias the pre-existent intermittent tails of the mag-netic fluctuation distribution, with little or no effect on the Gaussianpart of the latter. Its impact on energy fluctuations is shown to be neg-ligible. Linear response is thus controlled by the quakes and inheritstheir temporal statistics. These findings provide a theoretical basis foranalyzing intermittent linear response data from aging system in thesame way as thermal energy fluctuations, which are far more difficultto measure. pacs
How spontaneous fluctuations and linear response are related in off-equilibriumthermal dynamics is an open problem of considerable interest, as linear re-sponse measurements [1, 2, 3, 4] are the main in-road into the rich phe-nomenology of aging systems. Highlighting this issue are recent obser-1ations that aging dynamics is intermittent [5, 6, 7, 8]: rare but largefluctuations with an exponential size distribution punctuate much smallerequilibrium-like fluctuations with a Gaussian size distribution. Simula-tional [9, 10, 11, 12] and experimental [8] evidence in different areas supportsthe idea that intermittent changes of magnetization and other observablesare induced by, and hence statistically subordinated to, intermittent andirreversible outburst of heat, so called quakes . The same hypothesis leadsto a widely discussed asymptotic logarithmic time re-parameterization ofthe aging dynamics (see e.g. Ref. [13]). To ascertain whether or not othermechanisms than subordination could produce these effects, direct statisticalevidence is called for.In the following, statistical subordination and closely related issues arenumerically investigated, using as a test-bed an Ising model, which is sim-ple and yet constitutes a bona fide instance of a complex aging system [11,14, 15]. Intermittent changes in magnetization are shown to obey the samestatistics as the quakes, and the magnetic field is shown to have negligibleinfluence on the energy relaxation. Together, these two findings imply thataging dynamics is quake-driven. The fluctuation statistics is then investi-gated in detail, with the model properties in good agreement with previousinvestigations of intermittent heat flow [7, 11] and of magnetic linear re-sponse intermittency [8, 12]. Quakes in this model are shown to be nearlyuncorrelated events, which are described by a Poisson process whose averageincreases logarithmically in time. Additionally, a linear system-size depen-dence of the average number of quakes is found. This confirms that quakesare spatially localized events, a property also found directly in this work viaa real space analysis. The temperature dependence of the average numberof quakes is very weak, except at the lowest temperatures. This hints to ahierarchical structure in the energy landscape of the thermalized domainsspawning the quakes.
In the model, N Ising spins, σ i = ± H = − X P ijkl σ i σ j σ k σ l + Hη ( t w − t ) X i σ i . (1)The first sum runs over the elementary plaquettes of the lattice, includingfor each the product of the four spins located at its corners. The second2 −2.5−2.4−2.3−2.2−2.1−2 t µ e T = 1.5 −100 −50 0 5010 −4 −3 −2 −1 t w = 1000; t obs = 10000; δ t = 100 δ E P D F Figure 1: (left) : (colour on line) The average energy per spin, µ e , is plottedversus the system age t , with the magnetic field switched on at times t w =200 , , , , t w = 2000. The magnetic field has a quite smalleffect on µ e . (right) : The PDF of the energy changes δE over a time δt =100, with (stars) and without (circles) a magnetic field. Both PDFs featurea central Gaussian part of zero average and intermittent tails. The PDFsare nearly identical, except for the largest and rarest events. The simulationtemperature is T = 1 . P σ i to an externalmagnetic field. As expressed by the Heaviside step function η ( t w − t ), thefield changes instantaneously at t = t w from zero to H >
0. Previousinvestigations of the model’s properties in zero field show a low temperatureaging 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. inthe temperature range 0 . < T < .
5, using the rejectionless Waiting TimeAlgorithm (WTM) [16]. The ‘intrinsic’ time unit of the WTM approximatelycorresponds to one Monte Carlo sweep. By choosing a high energy randomconfiguration as initial state for low temperature isothermal simulations, aneffectively instantaneous thermal quench is achieved. For each set of physicalparameters, Probability Distribution Functions (PDFs) are collected over2000 independent runs, and other statistical data over 1000 independentruns. The symbol t stands for the time elapsed from the initial quench (andfrom the beginning of the simulations). The symbol t w is the time at whichthe 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 isset to H = 0 . t > t w . The thermal energy is denoted by E , andthe magnetization by M . The average energy per spin is denoted by µ e .3
50 0 50 10010 −4 −3 −2 −1 δ M P D F t w = 1000; t obs = 10000; δ t = 100 −50 0 50 10010 −4 −3 −2 −1 δ M P D F t w = 1000; t obs = 10000; δ t = 100 ← δ E > −5
Figure 2: (colour online) (left) : The PDF of the spontaneous magneticfluctuations has a Gaussian central part and two symmetric intermittentwings. Data are sampled in the interval t w , t w + t obs . (right) : The samequantity (outer graph, circles) when a field H = 0 . t w = 1000.The left intermittent wing is clearly reduced relative to the no-field case, andthe right intermittent wing is correspondingly amplified. The inner, almostGaussian shaped, PDF (stars) is the conditional PDF obtained by excludingthe magnetic fluctuations which happen in unison with the quakes.The PDF of fluctuations in energy and magnetization are constructed usingfinite time differences of E and M , taken over short time intervals of length δt ≪ t obs . To qualify as a probe, an applied perturbation must not significantly changethe dynamics of the system investigated. Specifically for the present model,the average energy vs. time should be nearly unaffected. Secondly, fluctu-ation spectra should not undergo qualitative changes. Figure 1, left panel,shows the average energy as a function of time, with six data sets correspond-ing 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 rightpanel shows, on a logarithmic scale, the PDF of the energy fluctuations inzero 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 anintermittent tail, which carries the net heat flow out of the system. Again,only a minor difference is seen between the unperturbed and perturbedfluctuation spectra, and only for the largest and rarest of the fluctuations.Since the magnetic contribution to the average energy is negligible, quakes4 −2.5−2.48−2.46−2.44−2.42−2.4−2.38 t BS F E ne r g y T=1.5 ↑ −2.41−2.4 0 5 10 15051015051015 t w = 10 ; t obs = 10 ; T=1.5 Figure 3: (colour online) (a) : The record signal, or Best So Far energy,for a simulation at T = 1 . (b) : active spins participating in any quakes occurringduring the interval [10 , . × ] are shown in colour. All other spins areomitted. By definition, active spins change their absolute average value overfive consecutive time units by exactly 2 (blue) or by 1 . δt ’s.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 doesnot change the overall structure of the spectra: The left panel of Fig.2 showsthe PDF of the spontaneous magnetic fluctuations occurring in the interval[ t w , t w + t obs ]. Intermittent wings symmetrically extend the central Gaussianpart of the PDF. The outer curve (circles) in the right panel of the samefigure 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 andthe Gaussian part is not affected. Thus, the net average magnetizationinduced by the field, i.e. the linear response, arises through a biasing effecton the distribution of the spontaneous intermittent magnetic fluctuations.A key aging feature [7, 11] is that the excess energy trapped by theinitial quench leaves the system through intermittent quakes. Intermittentmagnetic fluctuations have a close temporal association to the quakes: Theinner curve (stars) of Fig. 2 depicts a conditional PDF obtained by filteringout the magnetic fluctuations which occur simultaneously (i.e., in practice,either within the same or within the immediately following δt ) with energyfluctuations of magnitude δE ≤ −
5. The filtering threshold utilized is near5 −4 −3 −2 −1 x P ( l og ( t k /t k − ) >= x ) −1 t w = 10 t obs =10 t obs / δ t = 10 T = 1.5 C ( n ) n Figure 4: (colour online) (a) : The empirical cumulative distribution (stars)of the ’logarithmic waiting times’ τ k = ln( t k ) − ln( t k − ) is plotted on a logscale. The full line is a least square fit to y ∝ exp( − αx ), using the datapoints between the vertical lines. (b) : The normalized correlation between τ k and τ k + n (stars) is plotted versus n on a logarithmic scale. The line is aguide to the eye. We see that consecutive data points ( n = 1) are only weaklycorrelated, and that the correlation decays exponentially for n >
1. Thisis in reasonable qualitative agreement with the theoretical approximation C ( n ) = δ n, .the onset of the intermittent behavior of the heat flow PDF, as seen inFig 1. The filtering produces a nearly Gaussian PDF, demonstrating thatquakes and intermittent magnetic fluctuations are synchronous events. Insummary, the magnetic fluctuations which contribute to the linear responseare subordinated to the quakes which dissipate the excess energy stored inthe initial configuration.Visual inspection shows that energy traces of the p-spin model have afluctuating part superimposed onto a monotonic step-wise decay, the lattergiven by the function r E ( t ) = min y
10 at k = 1, the function tapers offexponentially with increasing k . The (modest) residual correlation confirmsthat successive events are occasionally mis-classified as quakes.A spatially extended glassy system with short range interactions is ex-pected to contain a number, say α , of independently relaxing thermalizeddomains, with a small and slowly growing characteristic size [20, 21]. Assum-ing that quakes originate independently within the domains, the observedfluctuations integrate the effect of α independent Poisson processes, whence α equals the logarithmic rate of quakes. Equivalently, its value can be es-timated 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 thesystem size, as demonstrated in the left panel of Figure 5. The temperaturedependence (or lack thereof) of α is related to the geometrical properties ofthe configuration space, or energy landscape, of each domain. For recordsized thermal fluctuations to elicit attractor changes, the energy landscapemust be scale invariant [12]. As a consequence, changing the temperatureshould not change α at all. Note however that scale invariance cannot holdbelow a cut-off value where the granularity of the energy values makes itselffelt. In the present model, the numerically smallest energy change followinga single spin flip is δE = ±
4, i.e. unlike models with Gaussian quencheddisorder, the granularity is important. The right panel of Figure 5 showsthat α has a modest temperature dependence for 1 ≤ T ≤
2, i.e. for a majorpart of the range where aging behavior is observed. For lower temperatures,a clear T dependence is visible. Direct numerical evidence has been provided that intermittent magnetic fluc-tuations are statistically subordinated to a certain type of events, quakes,8hich dissipate the excess energy trapped in the initial configuration. Theexternal field does not alter the temporal statistics of the quakes and of thespontaneous magnetic fluctuations. It only slightly biases the size distribu-tion of the latter. Therefore, the field can rightly be considered as a probeof the unperturbed off-equilibrium aging dynamics. In agreement with pre-vious investigations of other models [12] and experiments [8], the temporalstatistics of intermittent energy and magnetization changes, is shown to bewell described by Poisson process.Considering that aging dynamics is widely insensitive to details of themicroscopic interactions, it seems reasonable to assume that the above find-ings are valid beyond the plaquette model itself. It should therefore bepossible to analyze a wide range of intermittent linear response data fromcomplex dynamical systems precisely as done for the intermittent heat flowdata in Ref. [7, 11].
Acknowledgemnts
Financial support from the Danish Natural Sciences Research Council isgratefully acknowledged. The authors are indebted to the Danish Centerfor Super Computing (DCSC) for computer time on the Horseshoe Cluster,where most of the simulations were carried out.
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