Effects of external noise on threshold-induced correlations in ferromagnetic systems

In the present paper we investigate the impact of the external noise and detection threshold level on the simulation data for the systems that evolve through metastable states. As a representative model of such systems we chose the nonequilibrium athermal random-field Ising model with two types of the external noise, uniform white noise and Gaussian white noise with various different standard deviations, imposed on the original response signal obtained in model simulations. We applied a wide range of detection threshold levels in analysis of the signal and show how these quantities affect the values of exponent ${\gamma }_{S/T}$ (describing the scaling of the average avalanche size with duration), the shift of waiting time between the avalanches, and finally the collapses of the waiting time distributions. The results are obtained via extensive numerical simulations on the equilateral three-dimensional cubic lattices of various sizes and disorders.

Figure 1

An example illustrating the imposing of threshold $V_{\mathrm{th}}$ on the response signal, extraction of subavalanches, and definition of internal and external waiting time, $T_{\mathrm{w},\mathrm{int}}\left(V_{\mathrm{th}}\right)$ and $T_{\mathrm{w},\mathrm{ext}}\left(V_{\mathrm{th}}\right)$. Two avalanches (labeled by $i$ and $j$) surpassing the threshold are separated by one (blue-colored) avalanche lying below the threshold. Two subavalanches $i_1$ and $i_2$ are extracted from the avalanche $i$, and a single subavalanche $j^{\prime }$ from the avalanche $j$. Everything lying below $V_{\mathrm{th}}$ including parts of the avalanches $i$ and $j$ lying below the threshold is grayed.

Figure 2

Average avalanche size for a given avalanche duration for the system of size $1024\times 1024\times 1024$ and disorder $R=2.25$. Detecting threshold level is $V_{\mathrm{th}}=150$ in UWN case when noise function standard deviations ranges from 0 to 69.3 (panel a), and $V_{\mathrm{th}}=50$ in GWN case when noise function standard deviation ranges from 0 to 17.3 (panel b).

Figure 3

Example of the noise effect on the ${\gamma }_{S/T}$ values decrease. On the left side of the figure is presented the part of the signal with added noise (red line), while on the right side is the signal without noise (blue line). At a given threshold level for the same duration we see that the area (i.e., avalanche size) $S_T^n$, when the noise is applied, is larger than the area $S_T$ without the noise.

Figure 4

(a) Main panel: values of ${\gamma }_{S/T}$ versus detecting threshold level for the standard deviation of UWN from $\sigma =0$ to $\sigma =28.9$. Inset: Values of ${\gamma }_{S/T}$ for wider range of $\sigma$ and for $V_{\mathrm{th}}$ that ranges from 0 to 90 (i.e., the values of $V_{\mathrm{th}}$ before the plateau). (b) Main panel: values of ${\gamma }_{S/T}$ versus detecting threshold level for the standard deviation of GWN from $\sigma =0$ to $\sigma =14.4$. Inset: The same as in inset of panel (a).

Figure 5

Plateau values of ${\gamma }_{S/T}$ obtained from Fig. 4 for various noise standard deviations. Black squares represent the results for UWN, while the red circles are the results for GWN.

Figure 6

Number of occurrence of the external waiting time value $T_{\mathrm{w},\mathrm{ext}}=1051$ versus detection threshold level $V_{\mathrm{th}}$ for various standard deviations varying from $\sigma =0$ to $\sigma =57.7$. As shown in the main parts of both panels, the distributions collapse onto a single curve when presented against the threshold displaced by the shift parameter $p\left(\sigma \right)$ depending on the noise standard deviation $\sigma$. The data is obtained for the $1024\times 1024\times 1024$ system at $R=2.40$.

Figure 7

Schematic illustration of the average effect of added noise on the external waiting time: the same value of the external waiting time $T_{\mathrm{w},\mathrm{ext}}$ is found at a higher threshold level $V_{\mathrm{th}}^{\prime }$ in the response signal with added noise (red line) compared to the threshold level $V_{\mathrm{th}}$ corresponding to the same waiting time for the pure response signal (blue line).

Figure 8

The curves in insets show the distribution of number of occurrences for $T_{\mathrm{w},\mathrm{ext}}=490$ versus $V_{\mathrm{th}}$ for three different systems of linear sizes $L=512, L=724,$ and $L=1024$ and disorder $R=2.40$, while in the main panels are presented the collapses of the curves from the insets when Eq. (3) is applied for the UWN (a) and GWN (b). Standard deviation ranges from $\sigma =0$ to $\sigma =52$ for UWN, and from $\sigma =0$ to $\sigma =10$ for GWN.

Figure 9

Shift parameter values $p\left(\sigma \right)$ obtained from $1024\times 1024\times 1024$ systems with $R=2.25,2.40,2.55$ when UWN and GWN are applied. The data fitted to the function Eq. (4) with fitting values presented in Tables 1 and 2.

Figure 10

In the main figure of panel (a) is presented shifted collapse of the external waiting time distributions obtained from system with $L=2508,R=2.24,V_{\mathrm{th}}=126$ with added UWN of standard deviation $\sigma$ shown in inset. In panel (b) is shown collapse of internal waiting time distributions obtained from the same system with added GWN.

Figure 11

Shift functions $f\left(\sigma \right)$ (in main parts) and $g\left(\sigma \right)$ (in insets) for the distributions of external waiting time in panel (a) and internal waiting time in panel (b). The values of shift functions are those used to perform the collapsing shown in main panels of Fig. 10 of the distributions from the insets of that figure. In the UWN case values are fitted to the power-law function $\varphi \left(\sigma \right)\sim {\sigma }^s$, while in the GWN case they are fitted to the error function, $\varphi \left(\sigma \right)\sim \mathrm{erf}\left(\sigma \right)$.

Figure 12

In the insets of panels (a) and (b) are presented shift functions $f\left(\sigma \right)$ and $g\left(\sigma \right)$ for the external waiting time distributions obtained for four systems such that their dimensions $L$, disorders $R$, and threshold levels $V_{\mathrm{th}}$ satisfy conditions Eq. (6): $L=1448,R=2.27,V_{\mathrm{th}}=75,L=2046,R=2.25,V_{\mathrm{th}}=102,L=2508,R=2.24,V_{\mathrm{th}}=126$, and $L=3072,R=2.231,V_{\mathrm{th}}=153$. On the main panels are shown the collapses of the shift functions divided by $V_{\mathrm{th}}$ as functions of $\sigma /{\sigma }_{\mathrm{th}}$, fitted to the proposed forms Eqs. (8) and (9). The best fit parameters are given in Tables 3 and 4. Panels (c) and (d): the same as panels (a) and (b) but for the internal waiting time.

Figure 13

Main figures of panels (a) and (b) show collapses of the distributions of external (a) and internal (b) waiting times obtained from the same systems as the data shown in Fig. 12 (noncollapsed data is shown in insets) but with such a noise that the ratio between standard deviation of added UWN and imposed threshold $\rho$ ranges from 0 to 0.6. Values of shift functions are calculated using fitting parameters given in Tables 3 and 4. The same is presented in panels (c) and (d) but for GWN and $\rho$ in the range from 0 to 0.3.

Figure 14

Main figures of panels (a) and (b) show shifted collapse of the distributions of duration obtained from the same system as the data shown in Fig. 10 when UWN (a) and GWN (b) is added, while noncollapsed distributions, much less affected than the waiting time distributions by the presence of noise from the employed range, are shown in insets.

Figure 15

Shift function $f\left(\sigma \right)$ for the distributions of duration. The values of shift functions are those used to perform the collapsing shown in main panels of Fig. 14. In both UWN and GWN cases the values of shift function are fitted to the power-law function $\varphi \left(\sigma \right)\sim {\sigma }^s$. The values of parameter $s$ are $1.04\pm 0.05$ in UWN and $0.96\pm 0.06$ GWN case.