Unexpected universality in static and dynamic avalanches

We find that some equilibrium systems and their nonequilibrium counterparts actually show the same jerky response or avalanche behavior on many scales in response to slowly changing external conditions. In other words, their static and dynamic avalanches behave statistically the same. This suggests that their critical properties are much more generally applicable than previously assumed. In this case, systems far from equilibrium may be used to predict equilibrium critical behavior and vice versa.

Figure 1

(Color online) Disorder-dependent avalanche behavior in RFIM. Main panel: magnetization curves in equilibrium (d,e,f) and nonequilibrium (a,b,c) below, near, and above the critical disorder $R_{\text{c}}$. Insets: cross sections of three-dimensional (3D) systems showing all the avalanches (denoted by different colors) occurring during those magnetization processes. The calculation is done on 3D Gaussian zt-RFIM with system size ${64}^3$. Nonequilibrium: (a) $R=2.0$, (b) $R=2.224$, and (c) $R=2.6$. Equilibrium: (d) $R=2.25$, (e) $R=2.45$, and (f) $R=2.9$. Note that $R_{\text{c}}^{\text{neq}}=2.16\pm 0.03$ and $R_{\text{c}}^{\text{eq}}=2.28\pm 0.01$ for 3D Gaussian RFIM [11, 17].

Figure 2

(Color online) Scaling functions of integrated avalanche size distributions. The original $D_{\text{int}}\left(S,R\right)$ curves for static and dynamic avalanches at different disorders and different $\rho \left(h\right)$’s are calculated in 3D with system size ${64}^3$ and are averaged up to 100 initial random-field configurations. In the legend, the subscripts stand for equilibrium (e) or nonequilibrium (n) and the type of $\rho \left(h\right)$: Gaussian (g), Lorentzian (l), parabolic (p), and uniform (u). (a) For Gaussian, Lorentzian, and parabolic $\rho \left(h\right)$’s, using critical exponents $\left(\tau +\sigma \beta \delta \right)=2.03$ and $\sigma =0.24$, 24 curves collapse onto each other, up to nonuniversal critical disorders ($R_{\text{c}}^{\text{nG}}=2.16$, $R_{\text{c}}^{\text{eG}}=2.29$; $R_{\text{c}}^{\text{nL}}=1.92$, $R_{\text{c}}^{\text{eL}}=2.08$; $R_{\text{c}}^{\text{nP}}=4.84$, $R_{\text{c}}^{\text{eP}}=5.0$) and overall scale factors. The thick black curve through the collapse is the universal scaling function ${\overline{\mathcal{D}}}_{-}^{\text{int}}\left(X\right)$ of nonequilibrium Gaussian RFIM [11]. (b) For uniform $\rho \left(h\right)$, eight curves collapse onto each other, up to nonuniversal critical disorders ($R_{\text{c}}^{\text{nU}}=4.64$, $R_{\text{c}}^{\text{eU}}=4.46$) and overall scale factors. The collapse yields critical disorders $\left(\tau +\sigma \beta \delta \right)=2.08\pm 0.02$ and $\sigma =0.52\pm 0.03$. Both the critical exponents and the scaling function are different from those of the Gaussian $\rho \left(h\right)$.

Figure 3

(Color online) Scaling functions of integrated avalanche correlation function. The original $G_{\text{int}}\left(x,R\right)$ curves for static and dynamic avalanches at different disorders and different $\rho \left(h\right)$’s are calculated in 3D with system size ${64}^3$ and are averaged up to 100 initial random-field configurations. (a) For Gaussian, Lorentzian, and parabolic $\rho \left(h\right)$’s, 24 curves collapse onto each other, using $d+\beta /\nu =3.07$ and $\nu =1.37$ [11]. (b) For uniform $\rho \left(h\right)$, eight curves collapse onto each other with $d+\beta /\nu =3.0\pm 0.3$ and $\nu =0.8\pm 0.2$. Note that those collapses are up to the same critical disorders as used in Fig. 2.

Figure 4

(Color online) Schematic phase diagram of zt-RFIM with $k$-spin-flip dynamics and demagnetization dynamics. Dashed lines stand for the first-order phase transitions occurring at the critical field $H_{\text{c}}$. Note that for both demagnetization dynamics and ground-state evolution $\left(k=\infty \right)$, $H_{\text{c}}=0$ due to symmetry. (Inset) Schematic $M\left(H\right)$ curves associated with different dynamics.