Avalanche statistics from data with low time resolution

Extracting avalanche distributions from experimental microplasticity data can be hampered by limited time resolution. We compute the effects of low time resolution on avalanche size distributions and give quantitative criteria for diagnosing and circumventing problems associated with low time resolution. We show that traditional analysis of data obtained at low acquisition rates can lead to avalanche size distributions with incorrect power-law exponents or no power-law scaling at all. Furthermore, we demonstrate that it can lead to apparent data collapses with incorrect power-law and cutoff exponents. We propose new methods to analyze low-resolution stress-time series that can recover the size distribution of the underlying avalanches even when the resolution is so low that naive analysis methods give incorrect results. We test these methods on both downsampled simulation data from a simple model and downsampled bulk metallic glass compression data and find that the methods recover the correct critical exponents.

Figure 1

(a) The time derivative stress vs time series for a typical section of simulation signal from the simple mean-field model (from Ref. [18]). The inset shows the stress vs time curve. (b) A demonstration of what we call the “traditional” avalanche detection procedure. The top figure is a stress-time curve with two avalanches and the bottom is its time derivative. The avalanches are detected when the time derivative drops below a threshold (the dashed line in the bottom figure). The circles and squares indicate the starts and ends of the avalanches, respectively. The avalanche size is the total decrease in stress, as indicated in the top figure.

Figure 2

Duration distributions for several values of the sampling time $t_s.$ The line represents the MFT prediction $C\left(T\right)\sim T^{-1}$ [18]. The apparent power-law changes with sampling time $t_s$, even for values of $t_s$ much less than the maximum avalanche duration $T_a\approx 100.$ The simulation parameters used were number of cells $N={10}^4,$ coupling between cells $J=1,$ loading spring stiffness $K={10}^{-2},$ and displacement rate $v_d={10}^{-5}.$

Figure 3

(a) A portion of a stress-time series from a simulation of the mean-field model (from Ref. [18]). The circles mark every 20th data point and thus correspond to a lower-resolution signal with sampling time $t_s=20$ (in units of the simulation time step $\delta t=1$). The size of the large event on the left side of the plot is measured fairly accurately by the size of the stress drop in the low-resolution signal. As the inset shows, a small stress drop does not have a corresponding drop in the low resolution signal. (b) A portion of the stress-time curve from the simulation, tilted according to Eq. (13). Subtracting out the elastic stress increase makes the small avalanches visible as drops in the signal. The inset shows the stress-time curve before tilting.

Figure 4

Number of avalanches collected versus sampling time (in units of the simulation time step $\delta t=1$) from a simulation of the mean-field model (from Ref. [18]). The simulation parameters used were number of cells $N={10}^4,$ coupling between cells $J=1,$ loading spring stiffness $K={10}^{-2},$ and displacement rate $v_d={10}^{-5}.$ The line shows the predicted power law $1-\tau =-1/2$ from Eq. (11).

Figure 5

Measured avalanche size CCDFs at full resolution $\left(t_s=1\right)$ and at $t_s=100$ (about one-tenth of the average interevent time $T_i$) for both the untilted and the tilted stress-time series. The tilted data give a CCDF that is very close to the one obtained at full resolution, whereas the untilted signal gives a distribution that appears rounded and the power law is difficult to identify by eye. The simulation parameters used were number of cells $N={10}^4,$ coupling between cells $J=1,$ loading spring stiffness $K={10}^{-2},$ and displacement rate $v_d={10}^{-5}.$

Figure 6

A portion of a stress-time series from a simulation of the mean-field model (from Ref. [18]), tilted as described in Eq. (13). The circles mark every 2000th data point and thus correspond to a very low-resolution signal with sampling time $t_s=2000.$ This sampling time is about twice the inverse avalanche nucleation rate, so most increments contain at least one avalanche. As a result, measuring avalanches as successive drops in the tilted stress will give stress drops of very long duration, corresponding to many successive underlying avalanches. However, the stress drops measured between successive sample points will be close in size to the underlying avalanches, although sometimes several avalanches may be merged, as is the case for the first and last time step of size $t_s$ of the inset plot. The simulation parameters used were number of cells $N={10}^4,$ coupling between cells $J=1,$ loading spring stiffness $K={10}^{-2},$ and displacement rate $v_d={10}^{-5}.$

Figure 7

Mean values of the measured size and duration vs sampling time from the analysis (shown in Fig. 1) of the tilted signal. The curves are the predicted values from Eqs. (15) (top) and (18) (bottom).

Figure 8

Avalanche size CCDFs from simulations of the mean-field model (from Ref. [18]) at full resolution and from tilted stress-time curves at low resolution $t_s=1000$ (roughly two times the average interevent time $T_i$) using the analysis described in Fig. 1 and the improved method for very low-resolution data described in Sec. 6c that defines the sample stress drops defined in Eq. (19) (after tilting) as avalanche sizes. The distribution from the improved method is much closer to the full-resolution distribution and the same rough power-law behavior can be seen, but it is still somewhat distorted due to avalanche merging. The simulation parameters used were number of cells $N={10}^4,$ coupling between cells $J=1,$ loading spring stiffness $K={10}^{-2},$ and displacement rate $v_d={10}^{-5}.$

Figure 9

(a) Data collapse of the size CCDF [Eq. (7)] for different $K$ values from a simulation of the mean-field model (from Ref. [18]) at full resolution. The collapse uses MFT exponents $\tau =3/2$ and $\alpha =2$ [18]. (b) An apparent collapse for $t_s=500$ using the analysis shown in Fig. 1. Here the collapse yields different exponents $\tau =1$ and $\alpha =1.5.$ (c) A collapse with MFT exponents $\tau =3/2$ and $\alpha =2$ for $t_s=100\ll T_i,$ using the the tilted signal described in Sec. 6b. (d) A successful collapse with MFT exponents $\tau =3/2$ and $\alpha =2$ for $t_s=1000\approx T_i,$ using the sample stress drops defined in Eq. (19). Insets are the original distribution before the rescaling.

Figure 10

(a) Avalanche size distributions measured at different resolutions from simulations of the mean-field model (from Ref. [18]) and (b) from BMG experiment (from Ref. [13]). The trend lines are power laws with exponent $1-\tau =-1/2$ predicted by MFT. In both cases, the analysis described in Fig. 1 gives avalanche size distributions that become increasingly narrow at low resolution. Thus, low resolution can obscure power-law distributed jumps in the underlying signal. In the experimental data, a pronounced noise regime is visible for the higher resolutions, but is averaged out at lower resolutions.

Figure 11

Number of avalanches collected versus sampling time $t_s$ from the experimental data. The line is a power law with exponent $1-\tau =-1/2$ from Eq. (11).

Figure 12

The bottom curve is the avalanche size CCDF measured using the analysis described in Fig. 1 on the full-resolution (100-kHz) experimental data. The scaling regime follows a power law with exponent $1-\tau =-1/2$, in agreement with MFT. The top curve is the CCDF measured with the analysis when the same data are drastically downsampled to 2 Hz. The power law is no longer apparent. The middle curve is the CCDF measured using the improved method outlined in Sec. 6a, where the sample stress drops are analyzed from a tilted signal [see Eq. (19)]. The $-1/2$ power law apparent in the 100-kHz data is recovered from the downsampled data.