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Physics and Mathematics of Disordered Systems

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Extreme avalanches in the ABBM model

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Extreme value statistics is becoming a popular and well-studied field. Just like the sums of random variables exhibit universality described by the central limit theorem, maxima of random variables also obey a universal distribution, the generalized extreme-value distribution. This is interesting for studying e.g. extreme events in finance (see this paper by McNeil and Frey) and climate statistics (see e.g. this physics paper in Science and this climatology paper).

Having refereed a related paper recently, I’d like to share some insights on the statistics of extreme avalanches in disordered systems. An example are particularly large jumps of the fracture front when slowly breaking a disordered solid. A simple but reasonable model for such avalanches is the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model, originally invented for describing Barkhausen noise. I already touched upon it in a previous blog post, and will focus on it again in the following.
I’ll show an exact formula for the distribution of the maximum avalanche size in an interval, and connect this result to the universal extreme value distribution when considering a large number of avalanches.

Brief review: The ABBM model

To briefly recap (see my previous post for details), the ABBM model consists in a particle at position u(t) pulled on a random landscape by a spring. A key assumption of the model is that the force resulting from the disordered potential is a Brownian Motion in u. This allows computing many observables exactly.
For example, when the force due to the spring increases by f, it is well-known (see e.g. arxiv:0808.3217 and the citations therein) that the resulting displacement S of the particle follows the distribition

\displaystyle P_f(S) = \frac{f}{2\sqrt{\pi} S^{3/2}}\exp\left[-\frac{(kS-f)^2}{4S}\right].    (1)

Here, k is the spring constant. For simplicity I took a Brownian random force landscape with variance \sigma=1 here, but the results are straightforward to generalize. This result is basically the distribution of the first-passage time of a Brownian motion with drift k at a given level f. In this context it is also known as the Bachelier-Levy formula (see also my post on first-passage times).
For small forces, f \to 0, and weak springs, k \to 0, (1) becomes a power-law distribution P(S) \sim S^{-3/2}.

The largest avalanche in the ABBM model

Now let us consider particularly large avalanches. When applying a force f, the probability to have a total displacement S \leq S_0 is

\displaystyle F^{tot}_f(S_0) := \int_0^{S_0}\mathrm{d}S\,P_f(S).    (2)

Note, however, that the total displacement in this case is the sum of many avalanches triggered by infinitesimal steps in f. So how do we obtain the probability F_f(S_0) that all avalanches triggered during this ramp-up of the force are smaller than S_0? We decompose it in n intervals of size f/n, and let n \to \infty:

\displaystyle F_f(S_0) = \lim_{n\to\infty} \left[F^{tot}_{f/n}(S_0)\right]^n.    (3)

Note that we use here the Markovian property of the Brownian Motion, which ensures that avalanches on disjoint intervals are independent. Only this property permits us to take the n-th power to obtain the cumulative distribution for the n intervals; on any non-Markovian landscape the avalanches in these intervals would be correlated and things would be much more complicated.

Combining equations (1), (2) and (3) we can compute F_f(S_0) explicitly:

\begin{array}{rl}  F_f(S_0) =& \lim_{n\to\infty} \left[F^{tot}_{f/n}(S_0)\right]^n = \exp\left[-f\partial_f\big|_{f=0}\int_{S_0}^{\infty}\mathrm{d}S\,P_f(S)\right] \\  =& \displaystyle \exp\left[-f\left(\frac{e^{-k^2 S_0/4}}{\sqrt{\pi S_0}} - \frac{k}{2}\text{erfc} \frac{k\sqrt{S_0}}{2}\right)\right].  \end{array}    (4)

This satisfies the normalization expected of a cumulative distribution function: For S_0 \to 0, F_f(S_0)\to 0 and for S_0 \to \infty, F_f(S_0)\to 1.
The probability distribution of the maximal avalanche size S_{max} is correspondingly

\displaystyle P_f(S_{max}) = \partial_{S_0}\big|_{S_0=S_{max}}F_f(S_0).    (5)

Eqs. (4) and (5) are a nice closed-form expression for the size distribution of the largest avalanche during the force increase by f!

From few to many avalanches

As one goes to infinitesimal force steps f \to 0, only a single avalanche is triggered. Then it is clear from our construction and eqs. (4), (5) that P_f(S_{max}) \to P_f(S) as defined in (1). So, as expected, when considering a single avalanche the maximal avalanche size and the total avalanche size coincide.

On the other hand, for large force steps f \to \infty, the ramp-up of the force triggers many independent avalanches. The total displacement S is then the sum of many independent avalanche sizes S_1...S_n. Thus, by the central limit theorem, one expects to find a Gaussian distribution for S. We can see this explicitly from (1): The expectation value is \left\langle S \right\rangle = f/k, and the fluctuations around it are \delta S := S-\left\langle S \right\rangle. From (1) one finds that they scale like \delta S \sim \sqrt{f/k^3}. The normalized fluctuations d := \delta S \sqrt{k^3/f} have the distribution

\displaystyle P(d) = \frac{1}{2\sqrt{\pi}}\exp\left(-\frac{d^2}{4}\right) + \mathcal{O}(f^{-1/2}).     (6)

For large f, we are indeed left with a Gaussian distribution for the normalized fluctuations d. This is easily checked numerically, see figure 1 below.

Distribution of the total displacement for increasing force steps.

Figure 1: Distribution of the total displacement S for increasing force steps f, as given in eq. (1). Observe how the distribution converges to the Gaussian limit eq. (6) indicated by black dotted lines.

So what happens with the maximal avalanche size S_{max} for large steps f \to \infty? S_{max} is now the maximum of many independent avalanche sizes S_1...S_n, and as mentioned in the introduction we expect its distribution to be a universal extreme value distribution.
Since only large S_0 are relevant in (4), the exponent can be approximated by

\displaystyle \frac{e^{-k^2 S_0/4}}{\sqrt{\pi S_0}} - \frac{k}{2}\text{erfc} \frac{k\sqrt{S_0}}{2} \approx \frac{4}{k^2 \sqrt{\pi} S_0^{3/2}}e^{-k^2 S_0/4}.    (7)

Inserting this back into (4), we see that for large f the distribution P_f(S_{max}) is centered around S_f := \frac{4}{k^2} \log f. The cumulative distribution function (4) is well approximated by

\displaystyle F_f(S_{max}) \approx \exp\left[- \frac{2}{k^2 \sqrt{\pi} S_f^{3/2}}e^{-k^2 (S_{max}-S_f)/4} \right].    (8)

This is, up to rescaling, the cumulative distribution function of the Gumbel extreme-value distribution e^{-e^{-x}}. It is easy to check this universal asymptotic form numerically, see figure 2 below. Note that here the convergence here is much slower than for the total displacement shown in figure 1, since the typical scale for S_{max} only grows logarithmically with f.

Distribution of the size of the largest avalanche

Figure 2: Distribution of the size of the largest avalanche S_{max} for force steps of increasing size f, as given by eq. (5). Observe how the distribution converges to the universal Gumbel limit in eq. (8) indicated by black dotted lines.

For some applications, the limit of a very soft spring, k \to 0, is important. I leave the details to the reader but the main picture is that the exponential decay for large S_0 in eq. (6) is replaced by a power law S_0^{-1/2}. Correspondingly, the universal extreme-value distribution observed for large force steps f is no longer the Gumbel distribution (8) but instead a Fréchet distribution.

Side note: The minimal avalanche size

One may be tempted to approach similarly the problem of the minimal avalanche size for a slow ramp-up of the applied force. However, this is not well-defined: Due to the roughness of the Brownian force landscape, as we increase the force more and more slowly, the size of the smallest avalanche decreases more and more. Hence, its distribution will always be discretization-dependent and will not yield a finite result such as eq. (4) in the continuum limit.

All this gives a consistent picture of the maximal avalanche in the ABBM model. I find it really nice that it is so simple, knowing the avalanche size distribution (1), to express the distribution of the size of the largest avalanche in closed form and understand how it behaves!

Written by inordinatum

August 19, 2014 at 9:53 pm

The Alessandro-Beatrice-Bertotti-Montorsi model

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When a magnet is submitted to a slowly varying external magnetic field, its magnetization changes not smoothly, but in discrete jumps. These avalanches can be made audible using an induction coil. The resulting crackling signal is called Barkhausen noise. By analysing various features of this signal one can deduce information on material properties, for example residual stress or defect sizes, which is important for applications such as non-destructive testing. In this post, I will discuss a simple model describing the physics of Barkhausen noise. I will explain some of its predictions, including the stationary signal distribution, and sizes and durations of the avalanches.

Stochastic differential equation of the ABBM model

Example of signal generated by ABBM model

Example of signal generated by ABBM model

As you probably know, a ferromagnetic material with zero net magnetization consists of many magnetic domains. Inside one domain, the spins are oriented in parallel (thus each domain has a non-vanishing magnetization), however the magnetizations of different domains are randomly aligned and cancel out on average.
We will be interested in so-called soft magnets. In these materials domain walls can move quite freely, until they encounter a defect. This means they have a wide hysteresis loop. The dominant mechanism for magnetization is the motion of domain walls (and not the changing of magnetization inside one domain, as for hard magnets).

Alessandro, Beatrice, Bertotti and Montorsi (for details, see ref. [1]) model the change in magnetization under an external field through the motion of a single domain wall transverse to the magnetic field. They propose the following stochastic differential equation for the domain wall position u(t):

\displaystyle \Gamma \partial_t u(t) = H(t) -k u(t) + F(u(t))

Here, H(t) is the external applied magnetic field, and \Gamma is the domain wall relaxation time. The term -ku(t) is the so-called demagnetizing field, which keeps the domain wall from moving indefinitely. F(u) is a random pinning force, which depends on the position of the particle u(t). In the ABBM model, it is assumed to be a realization of Brownian motion. For more details on the motivation of the individual contributions, see e.g. the review in ref. [2].

Note that the random pinning force is quenched, i.e. depends on the particle position u(t) and not directly on the time t. A time-dependent random force would be a model for thermal noise (instead of localized defects).

Simulating the stochastic differential equation above yields a trajectory (see the figure on the right) which is very similar to the results of Barkhausen noise measurements. Due to the specific properties of Brownian motion, the ABBM model is easy to treat analytically. I will now discuss several observables which can be computed analytically: The stationary distribution of domain wall velocities, and the distributions of avalanche sizes and durations.

Stationary domain wall velocity distribution

To obtain a stationary state of domain wall motion, one ramps up the external field H linearly:
\displaystyle H(t) = c t.
Then, the instantaneous domain wall velocity \dot{u} has a stationary distribution given by
\displaystyle P(\dot{u}) = \frac{e^{-\dot{u}}\dot{u}^{-1+c}}{\Gamma(c)}.
Here I use dimensionless units k = \Gamma = 1. One way to derive this result is by solving the Fokker-Planck equation associated to the SDE above (as was done by ABBM in ref. [1]).
This distribution is interesting since it exhibits two different kinds of behaviour: For c < 1, P(\dot{u}=0)=\infty, meaning that the domain wall is pinned at near zero velocity most of the time. On the other hand, for c > 1, P(\dot{u}=0)=0 and the motion is smooth.
In the following I will focus on the case c<1, where we have intermittent avalanches.

Avalanche statistics

One way to obtain information on avalanches in the ABBM model is mapping the SDE above to a path integral (the Martin-Siggia-Rose formalism). This is done e.g. in references [3] and [4] below, where the resulting path integral is solved for any external field H(t). Probably the simplest way to define an avalanche is to apply a step-like field, H(t) = w \theta(t). The instantaneous increase at t=0 from H=0 to H=w triggers precisely one avalanche. Its size S and duration T are distributed according to (again in dimensionless units):
\displaystyle P(S) = \frac{w}{\sqrt{4\pi}S^{3/2}}e^{-\frac{(w-S)^2}{4S}},
\displaystyle P(T) = \frac{w}{(2\sinh T/2)^2}e^{-\frac{w}{e^T-1}}.
For small w and small avalanches, these distributions display a large power-law regime where one has
P(S) \sim S^{-3/2}, \quad P(T) \sim T^{-2}.
These power laws indicate that avalanches in the ABBM model are scale-free: There are both extremely small and extremely large ones (between the microscopic cutoff scale given by w and the macroscopic cutoff scale given by k=1).

Universality (or the lack of it)

The exponents of the power-law regimes in P(S) and P(T) above are universal for mean-field models of elastic interfaces. They do not depend on material properties or on details of the dynamics, but only on the fact that one has sufficiently long range interactions between different parts of the interface. These exponents are well-verified experimentally for magnets falling into the mean-field universality class.

Being universal, they also apply to other elastic interfaces with long-range interactions: Some even argue that the P(S) \sim S^{-3/2} behaviour is related to the Gutenberg-Richter distribution of earthquake moments. The avalanche size S would correspond to the earthquake moment, related to its magnitude M (the number reported in newspapers) via M \propto \log_{10} S. The exponent 3/2 for P(S) would give a Gutenberg-Richter b-value of b\approx 0.75, which is not too far off from the observed one.

On the other hand, I find it a little overly simplistic to try and find universal aspects of completely disparate physical systems. We know after all, that motion of magnetic domain walls and of earthquakes is not the same thing — so maybe the more interesting physics is in their differences, rather than their similarities.
A more detailed analysis of avalanches thus requires going beyond just power-law exponents. Several more refined observables — like mean shapes of avalanches — have been proposed to that end. It has been shown (see ref. [5]) that they are sensitive to the details of the dynamics. In my view, the interesting question (which is still not completely answered) is: What features should one look at, in order to determine if a signal is Barkhausen noise or something else? What can one learn from it about the microscopic disorder in one particular sample by listening to the Barkhausen noise it emits?

Outlook and References

If there is interest, in the future I may extend this blog post to a Wikipedia article, since I believe the model is simple but frequently used. It is still a field of active research, thus the list of references is certainly incomplete. Let me know if you want to add anything!

[1] B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, “Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. I. Theory,” J. Appl. Phys., vol. 68, no. 6, p. 2901, 1990.

[2] F. Colaiori, “Exactly solvable model of avalanches dynamics for Barkhausen crackling noise,” Adv. Phys., vol. 57, no. 4, p. 287, Jul. 2008.

[3] P. Le Doussal and K. J. Wiese, “Distribution of velocities in an avalanche,” Europhys. Lett., vol. 97, no. 4, p. 46004, Apr. 2012.

[4] A. Dobrinevski, P. Le Doussal, and K. J. Wiese, “Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model,” Phys. Rev. E, vol. 85, no. 3, p. 18, Mar. 2012.

[5] S. Zapperi, C. Castellano, F. Colaiori, and G. Durin, “Signature of effective mass in crackling-noise asymmetry,” Nature Physics, vol. 1, no. 1, p. 46, Oct. 2005.

Written by inordinatum

October 11, 2012 at 8:37 pm