Gaps between avalanches in one-dimensional random-field Ising models

We analyze the statistics of gaps ($\mathrm{\Delta }H$) between successive avalanches in one-dimensional random-field Ising models (RFIMs) in an external field $H$ at zero temperature. In the first part of the paper we study the nearest-neighbor ferromagnetic RFIM. We map the sequence of avalanches in this system to a nonhomogeneous Poisson process with an $H$-dependent rate $\rho \left(H\right)$. We use this to analytically compute the distribution of gaps $P\left(\mathrm{\Delta }H\right)$ between avalanches as the field is increased monotonically from $-\infty$ to $+\infty$. We show that $P\left(\mathrm{\Delta }H\right)$ tends to a constant $\mathcal{C}\left(R\right)$ as $\mathrm{\Delta }H\to 0^{+}$, which displays a nontrivial behavior with the strength of disorder $R$. We verify our predictions with numerical simulations. In the second part of the paper, motivated by avalanche gap distributions in driven disordered amorphous solids, we study a long-range antiferromagnetic RFIM. This model displays a gapped behavior $P\left(\mathrm{\Delta }H\right)=0$ up to a system size dependent offset value $\mathrm{\Delta }{H}_{\text{off}}$, and $P\left(\mathrm{\Delta }H\right)\sim {(\mathrm{\Delta }H-\mathrm{\Delta }{H}_{\text{off}})}^{\theta }$ as $\mathrm{\Delta }H\to H_{\text{off}}^{+}$. We perform numerical simulations on this model and determine $\theta \approx 0.95\left(5\right)$. We also discuss mechanisms which would lead to a nonzero exponent $\theta$ for general spin models with quenched random fields.

Figure 1

The increase in magnetization per site $M$ in the random-field Ising model at zero temperature as the external field $H$ is increased monotonically from $-\infty$ to $+\infty$. The jumps in magnetization of size $s$ correspond to avalanches in the system and occur at certain values of the external field $\{H_1<H_2<H_3...\}$. We study the gaps $\mathrm{\Delta }{H}_i=H_{i+1}-H_i$ between successive avalanches.

Figure 2

A schematic representation of avalanches in the disordered Ising system. On the left are two realizations of the system at a particular value of the external field $H$. The white region represents spins that have already flipped from $-1$ to $+1$. The colored areas depict clusters of spins that flip together (avalanche) and have yet to undergo a failure. The number of such regions in each configuration is denoted by $I_N(H,\left\{h_i\right\})$, where $N$ is the total number of spins in the system. When the field is incremented by a value $\mathrm{\Delta }H$, some of these regions undergo failures at different values of $H$ (represented by stars). The green regions on the right represent these clusters postavalanche. In the limit of large $N$, the correlations between events tends to zero, and each of these events can be treated as being independently drawn from an underlying distribution $\rho \left(H\right)$.

Figure 3

Distribution of gaps between avalanches $P\left(\mathrm{\Delta }H\right|H)$ in the nearest-neighbor ferromagnetic RFIM with uniform disorder at $R=5$ and $H=2.9$ for ${10}^7$ realizations of the disorder. The bold lines represent analytical results computed using Eqs. (5), (11), and (17). The points represent data obtained from simulations. We find a very good agreement between our analytical results and those obtained from the simulations. The discontinuity in the distribution occurs at $\mathrm{\Delta }H=2J-R-H=0.1$ [Eq. (17)], and reflects the discontinuity in the underlying disorder distribution.

Figure 4

Distribution of gaps between avalanches $P\left(\mathrm{\Delta }H\right)$ in the nearest-neighbor ferromagnetic RFIM with uniform disorder for different $R$. The bold lines represent analytical results computed using Eqs. (8), (11), and (17). The points represent data obtained from simulations. The data have been averaged over ${10}^7$ realizations. We find a very good agreement between our analytical results and those obtained from the simulations. Inset: The saturation value $\mathcal{C}\left(R\right)={lim}_{\mathrm{\Delta }H\to 0^{+}}P\left(\mathrm{\Delta }H\right)$ for different values of $R$.

Figure 5

Distribution of gaps between avalanches $P\left(\mathrm{\Delta }H\right|H=1)$ in the nearest-neighbor ferromagnetic RFIM with exponential disorder at $R=5$ and $H=1$ for ${10}^7$ realizations of the disorder. The bold lines represent analytical results computed using Eqs. (5), (11), and (19). The points represent data obtained from simulations. The data have been averaged over ${10}^7$ realizations. We find a very good agreement between our analytical results and those obtained from the simulations.

Figure 6

Distribution of gaps between avalanches $P\left(\mathrm{\Delta }H\right)$ in the nearest-neighbor ferromagnetic RFIM with exponentially distributed random fields for different $R$. The bold lines represent analytical results computed using Eqs. (11), (8), and (19). The points represent data obtained from simulations. The data have been averaged over ${10}^7$ realizations. We find a very good agreement between our analytical results and those obtained from the simulations. Inset: The saturation value $\mathcal{C}\left(R\right)={lim}_{\mathrm{\Delta }H\to 0^{+}}P\left(\mathrm{\Delta }H\right)$ for different values of $R$.

Figure 7

A schematic representation of states in the spin model [up (down) arrows correspond to $S_i=+1(-1)$]. Starting from the ground (Néel) state (Initial), a block of spins numbered $1,2\cdots L$ is flipped. The $A$ term represents the interaction of a spin $k$ with the spins to its left within the block and $B$ represents the interaction of this spin with spins to its left outside the block. The cost of flipping this block of spins can be made arbitrarily small as $L\to \infty$, for any finite disorder.

Figure 8

Distribution of avalanche (changes in spin configuration) sizes $P\left(s\right)$ in the long-range antiferromagnetic RFIM for a range of model parameters. The data have been averaged over ${10}^5$ realizations of the quenched disorder. We find that this follows a fast-decaying exponential distribution, consistent with the fact that there is no long-range order in the system.

Figure 9

Statistics of gaps between avalanches (changes in spin configuration) for a lattice of size $N=1000$ with $\alpha =\frac{1}{4}$ and a range of disorder strengths $R$. The plot shows two distinct types of avalanches (i) that change the total magnetization and (ii) that leave the magnetization unchanged. There is a crossover from type (ii) to type (i) dominated regions at $\mathrm{\Delta }{H}_{\text{off}}$. The data have been averaged over $5\times {10}^4$ realizations of the quenched disorder.

Figure 10

Statistics of gaps between avalanches (changes in magnetization) for a lattice of size $N=1000$ with $\alpha =\frac{1}{4}$ and a range of disorder strengths $R$. The distribution shows a nonzero $\theta =0.95\left(5\right)$ independent of $R$ as $\mathrm{\Delta }H\to \mathrm{\Delta }{H}_{\text{off}}^{+}$. The data have been averaged over $5\times {10}^4$ realizations of the quenched disorder. Inset: The same data displayed in linear scale.

Figure 11

Statistics of gaps between avalanches (changes in magnetization) for a lattice of size $N=1000$ with disorder strength $R=\frac{1}{4}$ and various interaction ranges $\alpha$. The distribution shows a nonzero $\theta =0.95\left(5\right)$ independent of $\alpha$ as $\mathrm{\Delta }H\to \mathrm{\Delta }{H}_{\text{off}}^{+}$. The data have been averaged over $5\times {10}^4$ realizations of the quenched disorder. Inset: The rescaled distributions show a universal behavior for small $\mathrm{\Delta }H$, with the best fit value $\gamma =0.50\left(2\right)$.

Figure 12

Scaling of $P\left(\mathrm{\Delta }H\right)$ with system size in the small $\mathrm{\Delta }H$ regime for $\alpha =\frac{1}{4}$ and $\frac{1}{2}$ at fixed disorder strength $R=0.25$. The distribution shows a nonzero $\theta =0.95\left(5\right)$ independent of model parameters as $\mathrm{\Delta }H\to \mathrm{\Delta }{H}_{\text{off}}^{+}$. The data have been averaged over $5\times {10}^4$ realizations of the quenched disorder. The $\alpha =\frac{1}{2}$ plots have been shifted to the left by one decade to aid visibility. Inset: The scaling of the same data using the scaling ansatz provided in Eq. (29) with the best fit value $\gamma =0.50\left(2\right)$ shows a very good collapse for a range of model parameters in the small $\mathrm{\Delta }H$ regime.

Figure 13

Evolution of the local fields $h_{e,i}$ at each site, as the system undergoes an avalanche event (left) for the nearest-neighbor ferromagnetic RFIM and (right) the long-range antiferromagnetic RFIM. Since the stable spin configurations are governed by $S_i=\text{sgn}\left(h_{e,i}\right)$, positive local fields correspond to spins $+1$. The smallest negative $h_{e,i}$ governs the distance (gap) to the next avalanche. In the case of the nearest-neighbor ferromagnetic RFIM, since each avalanche destabilizes the neighboring spins, their $h_{e,i}\mathrm{s}$ move closer to 0. This effectively decreases the roughness of the membrane. For the long-range model, each avalanche roughens the membrane, depleting the density of near-failure regions.

Figure 14

Illustration of a Bethe lattice with coordination number $z=3$, along with two of its associated Cayley subtrees with four generations ($r=4$). The site at $r=0$ is the origin. The shaded region denotes a subtree $T_X$ rooted at $X$ with a parent node $Y$. The linear chain, considered in this paper, corresponds to $z=2$.

Figure 15

Size distribution of avalanches for the one-dimensional RFIM with random fields drawn from a uniform distribution [Eq. (15)] with $R=5$ at an external field $H=1$, for different system sizes $N$. The data have been averaged over ${10}^7$ realizations of the disorder. The bold line corresponds to the analytic expression for $N\to \infty$ computed using Eq. (C11).

Figure 16

Size distribution of avalanches for the one-dimensional RFIM with random fields drawn from an exponential distribution [Eq. (16)] with $R=5$ at an external field $H=1$, for different system sizes $N$. The data have been averaged over ${10}^7$ realizations of the disorder. The bold line corresponds to the analytic expression for $N\to \infty$ computed using Eq. (C11).