# inordinatum

Physics and Mathematics of Disordered Systems

## The Alessandro-Beatrice-Bertotti-Montorsi model

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

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.