A Monotonicity of the domain-wall motion in the retarded ABBM model

Statistics of Avalanches with Relaxation, and Barkhausen Noise: A Solvable Model

Abstract

We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model of a particle in a Brownian force landscape, including retardation effects. We show that under monotonous driving the particle moves forward at all times, as it does in absence of retardation (Middleton’s theorem). This remarkable property allows us to develop an analytical treatment. The model with an exponentially decaying memory kernel is realized in Barkhausen experiments with eddy-current relaxation, and has previously been shown numerically to account for the experimentally observed asymmetry of Barkhausen-pulse shapes. We elucidate another qualitatively new feature: the breakup of each avalanche of the standard ABBM model into a cluster of sub-avalanches, sharply delimited for slow relaxation under quasi-static driving. These conditions are typical for earthquake dynamics. With relaxation and aftershock clustering, the present model includes important ingredients for an effective description of earthquakes. We analyze quantitatively the limits of slow and fast relaxation for stationary driving with velocity . The -dependent power-law exponent for small velocities, and the critical driving velocity at which the particle velocity never vanishes, are modified. We also analyze non-stationary avalanches following a step in the driving magnetic field. Analytically, we obtain the mean avalanche shape at fixed size, the duration distribution of the first sub-avalanche, and the time dependence of the mean velocity. We propose to study these observables in experiments, allowing to directly measure the shape of the memory kernel, and to trace eddy current relaxation in Barkhausen noise.

pacs:
02.50.Ey, 05.40.Jc, 75.60.Ej

I Introduction and model

i.1 Barkhausen noise

The Barkhausen noise Barkhausen (1919) is a characteristic magnetic signal emitted when a soft magnet is slowly magnetized. It can be measured and made audible as crackling through an induction coil: periods of quiescence followed by pulses, or avalanches, of random strength and duration. The statistics of the emitted signal depends on material properties and its state. By analyzing the Barkhausen signal, one can deduce for example residual stresses Gauthier et al. (1998); Stewart et al. (2004) or grain sizes Ranjan et al. (1987); Yamaura et al. (2001) in metallic materials. Understanding how particular details of the Barkhausen noise statistics depend on microscopic material properties is important for such applications.

On the other hand, Barkhausen noise pulses are just one example for avalanches in disordered media. Such avalanches also occur in the propagation of cracks during fracture Herrman and Roux (1990); Chakrabarti and Benguigui (1997); Måløy et al. (2006), in the motion of fluid contact lines on a rough surface Rolley et al. (1998); Moulinet et al. (2002); Le Doussal and Wiese (2010); Le Doussal et al. (2009a), and as earthquakes driven by motion of tectonic plates Ben-Zion and Rice (1993); Mehta et al. (2006); Fisher et al. (1997); Ben-Zion and Rice (1997). Some features of the avalanche statistics, like size and duration distributions Le Doussal and Wiese (2012a, 2013), are universal for many of these phenomena Sethna et al. (2001). Barkhausen noise is easily measurable experimentally, and provides a good way to study aspects of avalanche dynamics common to all these systems.

A first advance in the theoretical description of Barkhausen noise was the stochastic model postulated by Alessandro, Beatrice, Bertotti and Montorsi Alessandro et al. (1990a, b) (ABBM model). They proposed modeling the domain-wall position through the stochastic differential equation (SDE)

(1)

We follow here the conventions of Zapperi et al. (2005) and Colaiori et al. (2007). is the saturation magnetization, and the external field which drives the domain-wall motion. A typical choice is a constant ramp rate , , which leads to a constant average domain-wall velocity Alessandro et al. (1990a). is the demagnetizing factor characterizing the strength of the demagnetizing field generated by effective free magnetic charges on the sample boundary Alessandro et al. (1990a); Colaiori (2008). The domain-wall motion induces a voltage proportional to its velocity , which is the measured Barkhausen noise signal. Here is a random local pinning force. It is assumed to be a Brownian motion, i.e. Gaussian with correlations

This choice may seem unnatural, since the physical disorder does not exhibit such long-range correlations. It is only recently that it has been shown Le Doussal and Wiese (2012b, a, 2013) that the “ABBM guess” emerges as an effective disorder to describe the avalanche motion of the center-of-mass of the interface, denoted , in the mean-field limit of the field theory of an elastic interface with internal dimensions. This correspondence holds both for interfaces driven quasi-statically Le Doussal and Wiese (2012b, 2013), and for static interfaces at zero temperature Le Doussal and Wiese (2012a). The mean-field description is accurate above a certain critical internal dimension . For , a systematic expansion in using the functional renormalization group yields universal corrections to the scaling exponents Nattermann et al. (1992); Chauve et al. (2001); Le Doussal et al. (2002) and avalanche size Le Doussal and Wiese (2012a, 2013) and duration Le Doussal and Wiese (2012b, 2009a, 2013) distributions.

For the particular case of magnetic domain walls, the predictions of the ABBM model are well verified experimentally in certain ferromagnetic materials, for example FeSi alloys Alessandro et al. (1990b); O’Brien and Weissman (1994); Durin and Zapperi (2000). These are characterized by long-range dipolar forces decaying as between parts of the domain wall a distance apart. This leads Cizeau et al. (1997) to a critical dimension coinciding with the physical dimension of the domain wall. In this kind of systems, as expected, the mean-field approximation is reasonably well satisfied. Measurements on other types of ferromagnets, for example FeCoB alloys Durin and Zapperi (2000) indicate a universality class different from the mean-field ABBM model. This may be explained by short-range elasticity, and a critical dimension . To describe even the center of mass mode in this class of domain walls, one needs to take into account the spatial structure of the domain wall. Predictions for roughness exponents Chauve et al. (2001); Le Doussal et al. (2002) and avalanche statistics Le Doussal and Wiese (2012b, 2009a, a, 2013) for this non-mean-field universality class have been obtained using the functional renormalization group.

On the other hand, even for magnets in the mean-field universality class, a careful measurement of Barkhausen pulse shapes Spasojevic et al. (1996); Kuntz and Sethna (2000); Sethna et al. (2001); Zapperi et al. (2005) shows that they differ from the simple symmetric shape predicted by the ABBM model Papanikolaou et al. (2011); Le Doussal and Wiese (2012b). This hints at a more complicated equation of motion than the first-order overdamped dynamics usually considered for elastic interfaces in disordered media.

In a physical interface, there may be additional degrees of freedom. One example was studied in Lecomte et al. (2009); Barnes et al. (2012). Other examples include deformations of a plastic medium, or eddy currents arising during the motion of a magnetic domain wall. For viscoelastic media, these can be modeled by a memory term which is non-local in time Marchetti et al. (2000); Marchetti (2005). At mean-field level, this is equivalent to a model with dynamical stress overshoots Schwarz and Fisher (2001). Such memory terms may lead to interesting new phenomena, like coexistence of pinned and moving states Marchetti et al. (2000); Marchetti (2005); Le Doussal et al. (2008). A similar memory term, non-local in time, is argued in Zapperi et al. (2005) to describe the dissipation of eddy currents in magnetic domain-wall dynamics,

(2)

The response function , derived by solving the Maxwell equations in a rectangular sample Bishop (1980); Zapperi et al. (2005); Colaiori et al. (2007); Colaiori (2008), is

(3)

are relaxation times for the individual eddy current modes,

They depend on the sample width , thickness , permeability and conductivity . (2) and (3) correspond to Eqs. (13), (17) and (21) in Colaiori et al. (2007); we refer the reader there for details of the derivation.

Zapperi et al. Zapperi et al. (2005) showed numerically that avalanche shapes in the model (2) are asymmetric. They concluded that eddy-current relaxation may be one way of explaining the experimentally observed skewness of Barkhausen noise pulses. They also argue that similar relaxation effects may be relevant for other physical situations where asymmetric pulse shapes are observed2.

A simplification of Eq. (3) occurs when considering only the leading contributions for small and large relaxation times3. Then one obtains Zapperi et al. (2005) a natural generalization of the ABBM equation (1):

(4)

Here, is the longest relaxation time of the eddy-current modes, . and are damping coefficients given in Zapperi et al. (2005).

i.2 The ABBM model with retardation

For the remainder of this work, we adopt the conventions used in the study of elastic interfaces. Let us introduce a more general model than (4),

(5)

which describes a particle driven in a force landscape , with retardation. At this stage is arbitrary. Here is a general memory kernel with the following properties:

  1. (without loss of generality, since a constant may be absorbed into the parameter ).

  2. as .

  3. for all , i.e. memory of the past trajectory always decays with time.

This model possesses a remarkable property for any such kernel and any landscape . It has monotonicity, i.e. it satisfies the Middleton theorem: For non-negative driving , after an initial transient period one has at all times. A more precise statement and a proof are given in Appendix A. It has very important consequences, both in the driven regime, and in the limit of quasi-static driving, i.e. small . In that limit it converges to the quasi-static process , where is the (forward) Middleton metastable state, defined as the smallest (leftmost) root of

(6)

It is independent of the precise form of the kernel . Hence the domain-wall position is uniquely determined by the value of the driving field , due to the monotonicity property Middleton (1992). This process exhibits jumps at a set of individual points, the avalanche locations , and the quasi-static avalanche sizes

(7)

are thus independent of the retardation kernel. What depends on the kernel is the dynamics within these avalanches, and that is studied here. The quasi-static avalanche sizes have a well-defined distribution which has been computed for a particle in various force landscapes Le Doussal and Wiese (2009b); Dobrinevski et al. (2012) and for the non-trivial case of a -dimensional elastic interface using functional RG methods Le Doussal and Wiese (2009a); Le Doussal et al. (2009b); Rosso et al. (2009). As long as the dynamics obeys the Middleton theorem, the avalanche-size distribution remains independent of the details of the dynamics Le Doussal and Wiese (2013).

While monotonicity holds for any , in this article we focus on the case of the Brownian force landscape which can be solved analytically. As in the standard ABBM model, we choose the effective random pinning force to be a random walk, i.e. Gaussian with correlator given by (I.1). 4 We call this the ABBM model with retardation. In view of the application to Barkhausen noise, the parameter describes the overall strength of the force exerted by eddy currents on the domain wall. For , (5) reduces to the equation of the standard ABBM model in the conventions of Le Doussal and Wiese (2009b); Dobrinevski et al. (2012).

The retarded ABBM model is particularly interesting in view of the monotonicity property. Other ways of generalizing the ABBM model to include inertia, e.g. by a second-order derivative Le Doussal et al. (2012), do not inherit this property from the standard ABBM model. This makes the ABBM model with retardation very special, and it will be important for its solution in section III.

When considering the particularly interesting case of exponential relaxation motivated in Zapperi et al. (2005), we set

(8)

is the longest time scale of eddy-current relaxation, as discussed above. In this approximation, (5) can be re-written as two coupled, local equations for the domain-wall velocity , and the eddy-current pressure ,

(9)
(10)
(11)

Although most of our quantitative results will be derived for this special case only, most qualitative features carry over to more general kernels with sufficiently fast decay.

By rescaling , and in Eq. (5) (for details, see section III.1), one finds the characteristic time scale and length scale of the standard ABBM model (). They set the scales for the durations and sizes of the largest avalanches. There are of course avalanches of smaller size (up to some microscopic cutoff if one defines it). The velocity scale is and one can define a renewal time for the large avalanches as , the limit of quasi-static driving being , equivalent to . In the retarded ABBM model (8) one introduces an additional memory time scale and various regimes will emerge depending on how compares with the other time scales (whose meaning will be changed).

Eq. (11) then describes a depinning model with relaxation, i.e. one can think of the disorder landscape as relaxing via the additional degree of freedom . This is a feature of interest for earthquake models as discussed below. In this context one considers the limit of well separated time scales, .

Other features of Barkhausen noise predicted for the ABBM model with retardation are quite different from those of the standard ABBM model. Zapperi et al. Zapperi et al. (2005) already realized that the inclusion of eddy currents leads to a skewness in the avalanche shape. In this article, we go further and discuss changes in the avalanche statistics. The relaxation of eddy currents introduces an additional slow time scale into the model. This leads to avalanches which stretch further in time. In particular, avalanches following a kick (or, more generally, stopped driving) never terminate, by contrast with the standard ABBM model. This is because of the exponentially decaying retardation kernel, which never vanishes 5. Avalanche sizes however, are not changed by retardation in the limit of quasi-static driving, as discussed above. In that limit, retardation leads to a break-up of avalanches into sub-avalanches, which can also be called aftershocks. Avalanches at continuous driving overlap stronger, and the velocity threshold for the infinite avalanche (i.e. the velocity no longer vanishes) is decreased. We now describe these effects in detail and formulate more precise statements.

i.3 Protocols

Let us first review qualitatively the main situations that we will study, and define the terminology.

(i) stationary driving: The driving velocity is constant, , and the distributions of the domain-wall velocity and of the eddy-current pressure reach a steady state, which we study. If is large enough the velocity will never vanish and one has a single infinite avalanche, also called “continuous motion”. At smaller the velocity will sometimes vanish. That defines steady state avalanches. These are more properly called sub-avalanches of the infinite avalanche since at finite they immediately restart. Only in the limit they become well separated in time and can then be called steady state avalanches.

(ii) Avalanches following a kick: We consider an initial condition at prepared to lie in the “Middleton attractor” at , as discussed above. It can be obtained by driving the system monotonously in the far past with , until memory of the initial condition is erased; then let it relax for a long time with until time . Hence the initial condition is . At one changes the external magnetic field instantaneously by , i.e. sets . For , the external field does not change anymore, thus a kick in the driving velocity corresponds to a step in the applied force. At the system has settled again into the Middleton attractor at because of the properties discussed above. One can thus consider the total motion to define a single avalanche following a kick, which is thus unambiguously defined. The total size is the same as in the absence of retardation. We will ask about the total duration (which becomes infinite) and whether the velocity has vanished at intermediate times, i.e. whether the avalanche has broken into sub-avalanches.

Avalanches following a kick are called non-stationary avalanches (since driving is non-stationary). However, in the limit of they become identical to the steady-state avalanches obtained by stationary driving discussed above (conditioned to start at ).

(a) Standard ABBM model.

(b) ABBM model with retardation, .

(c) ABBM model with retardation, .

(d) ABBM model with retardation, .

Figure 1: Splitting of an avalanche into sub-avalanches through the retardation mechanism. We have set and and we vary the relaxation time .

i.4 Organization of this article

The remainder of this article is structured as follows: In section II, we discuss in more detail the phenomenology and the qualitative physics of the ABBM model with retardation. We discuss the splitting of a quasi-static avalanche into  sub-avalanches, and the effects of retardation on the stationary and the non-stationary dynamics.

In section III we explain how the probability distribution of observables linear in the domain-wall velocity can be computed by solving a non-linear, non-local “instanton” equation. By this, the stochastic model is mapped onto a purely deterministic problem of non-linear dynamics. This is a generalization of the method developed in Le Doussal and Wiese (2012b); Dobrinevski et al. (2012) for the standard ABBM model with arbitrary driving.

Section IV discusses how the explicit form of the memory kernel can be extracted in an experiment from the response to a kick.

Section V is devoted to an analysis of the instanton equations in the limit . This means that eddy currents relax much more slowly than the domain wall moves. In this limit, we obtain the stationary distributions of the eddy-current pressure and domain-wall velocity, as well as their behavior following an instantaneous kick in the driving field. The instanton solution reflects the two time scales in the problem: A short time scale, on which eddy currents build up but do not affect the dynamics, and a long time scale, on which they relax quasi-statically. We prove that, even after the driving has stopped, the velocity never becomes zero permanently.

In section VI we discuss the fast-relaxation limit . In this limit, eddy currents relax much faster than the domain wall moves. The instanton solutions again exhibit two time scales, but now eddy currents are irrelevant for the long-time asymptotics. Qualitative results (like the fact that the domain-wall motion never stops entirely) are in agreement with those for the slow-relaxation limit, considered in section V.

In section VII we discuss non-stationary avalanches following an instantaneous kick in the driving. In particular, we compute their average shape at fixed size.

In section VIII, we show how to include an absorbing boundary in the instanton solution of section III. This is required for treating avalanches during stationary driving. We then derive the distribution of avalanche durations in the standard ABBM model at finite driving velocity, , and the leading corrections for weak relaxation and . We also show numerical results for more general situations, and give some conjectures on the modification of size and duration exponents by retardation effects.

Last, in section IX, we summarize our results. We discuss how they can be used to learn more about the dependence of Barkhausen noise on eddy current dissipation.

Ii Physics of the model and summary of the results

ii.1 Quasi-static driving: Sub-avalanches and aftershocks

Consider the system either under stationary driving at , or following an infinitesimal kick as discussed above, and call the starting time of the avalanche. The main physics can be understood from figure 1 and keeping in mind the equations (11).

In Fig. 1a we represent the usual construction for in the standard ABBM as the left-most solution of the equation (6) (in the figure we set ). Assuming this construction indicates the position of the domain wall as a function of on time scales of order . At the solution jumps from to corresponding to an avalanche of size ; the latter occurs on the much faster time scale . During the avalanche the velocity (setting ) is given by the difference in height between the line and the landscape , providing a graphic representation of the motion. The velocity vanishes at and . For illustration we have represented a force landscape which is ABBM like at large scales but smooth at small scales. For the continuous ABBM model the construction is repeated at all scales and one has avalanches of all smaller sizes.

Let us now add retardation, setting , and varying the memory time . The graphical construction corresponding to Eq. (11) is represented in Fig. 1b to 1d. The difference in height is now the sum of and (in the Figure we chose ), which evolves according to the second equation in Eq. (11). It can be rewritten as

(12)

Hence increases from , initially as (since ). Thus the curve versus starts with a negative slope .

Another way to see this is to note that for , the second equation of (11) gives

(13)

Inserting this into the first equation of (11), we obtain

(14)

Effectively, for short times the mass is modified from . Thus, while is fixed, the end of the first sub-avalanche is determined not by the roots of , but by the roots of . Equivalently, in the landscape , instead of looking at intersections with the horizontal curve , we should look at intersections with , a line with slope .

At the point where this curve intersects first the landscape we get a point where first vanishes. This defines the size of the first sub-avalanche. If is small this usually occurs near the end, but if is larger the original avalanche (called main avalanche) is divided – in size – in a sequence of sub-avalanches . The number of sub-avalanches in the main avalanche is finite for a smooth landscape, and infinite for the continuous Brownian landscape. The total size is however the same as for , due to the Middleton theorem. For instance in the landscape of figure 1d, the main avalanche is divided into three large sub-avalanches, and for the continuous Brownian landscape the intermediate segments are also divided into smaller sub-avalanches, at infinitum. Figure 1 illustrates the correlation between the sub-avalanche structure (in ) and the realization of the random landscape, where larger hills favor the breakup into sub-avalanches. Note also that in intermediate regions where is very small, starts decreasing again (it decreases whenever ). The effective driving seen by the particle then becomes and increases. This mechanism triggers a new sub-avalanche, and so on.

To obtain the dynamics one must solve the equations (11), which we do below. For the standard ABBM model Dobrinevski et al. (2012), and in the mean-field theory of the elastic interface Le Doussal and Wiese (2012b, 2013), it was seen that an avalanche terminates with probability 1, i.e.  for . This allowed defining and computing the distribution of avalanche durations Le Doussal and Wiese (2012b); Dobrinevski et al. (2012), and their average shape Papanikolaou et al. (2011); Le Doussal and Wiese (2012b); Dobrinevski et al. (2012).

In presence of retardation, and for an exponential kernel, the avalanche duration defined in the same way becomes infinite. Inside one avalanche, the velocity becomes zero infinitely often, but is then pushed forward again by the relaxation of the eddy-current pressure. Thus, an avalanche in the ABBM model with retardation splits into an infinite number of sub-avalanches, delimited by zeroes of . Each sub-avalanche has a finite size and duration with (the same size as in the standard ABBM model), but .

Below we study in detail two limits:

In the slow-relaxation limit the duration of the largest sub-avalanches remains of order , while the total duration is of order . This leads to the estimate that the main avalanche breaks into significant (i.e. non-microscopic) sub-avalanches.

For the fast-relaxation limit ( here) . The correction to the domain-wall velocity is small in this limit, and vanishes as (in contrast to the limit discussed above). In fact, the correction due to retardation amounts to a rescaling of the velocity as .

Of course, in presence of driving, the total duration is not strictly infinite since at some time-scale the driving will kick in again, and lead to another main avalanche, itself again divided in sub-avalanches and so on. We can call that scale again but its precise value may differ from the estimate for the case .

Thus one main property of the retarded ABBM model is that it leads to aftershocks, a feature not contained in the standard ABBM model. The main avalanche is divided into a series of aftershocks (the sub-avalanches) which can be unambiguously defined and attributed to a main avalanche (which basically contains all of them) in the limit of small driving. This sequence of sub-avalanches is also called an avalanche cluster. The aftershocks are triggered by the relaxation of the additional degree of freedom . That in turn changes the force acting on the elastic system. Relaxation and aftershock clustering have been recognized as important ingredients of an effective description of earthquakes; the present model is a solvable case in this class. In some earthquake models considered previously, relaxation was implemented in the disorder landscape itself Jagla (2010); Jagla and Kolton (2010); Jagla (2011). Here the relaxation mechanism is simpler, which makes it amenable to an analytic treatment. Note of course that at this stage it is still rudimentary. First it is not clear how to identify the“main shock” among the sequence of sub-avalanches; while there is indeed some tendency, see e.g. Fig. 1d, that the earliest sub-avalanche is the largest, this is not necessarily true. Second, to account for features such as the decay of activity in time as a power-law (Omori law Scholz (1998)) one needs to go beyond the exponential kernel, to a power-law one. Finally, more ingredients are needed if one wants to account for other features of realistic earthquakes, such as quasi-periodicity.

ii.2 Stationary motion

In the case where the driving velocity is constant, , the distributions of the domain-wall velocity and of the eddy current pressure become stationary. The distribution of for small has a power-law form with an exponent depending on ,

(15)

There is no contribution . is a critical driving velocity, which separates several different regimes:

  1. For , the velocity never becomes zero. It is not possible to identify individual avalanches, one can say that there is a single infinite avalanche.

  2. For , the velocity vanishes infinitely often. The times delimit individual (sub-)avalanches6. Their durations and sizes have distributions and depending on the driving velocity . In section VIII we compute for the standard ABBM model and for a special case of the ABBM model with retardation. For sub-avalanches, starting at and a fixed value of the eddy-current pressure , in the limit of small and , we show that

    In particular, the pure ABBM power-law exponent is not modified for the first sub-avalanche, starting at . Since the typical goes to zero as , we conjecture that the quasi-static exponents are still given by the mean-field values , .

In sections V.3 and VI, we compute in several limiting regimes. For , i.e. eddy-current relaxation slow with respect to the domain-wall motion, we obtain in section V.3

This means that slow eddy-current relaxation decreases the critical velocity. The stronger the eddy-current pressure , the smaller becomes. On the other hand, for , i.e. fast eddy-current relaxation, we obtain in section VI

Hence, fast eddy-current relaxation also decreases the critical velocity. However, the correction in this case is small and vanishes, as the time-scale separation between and becomes stronger.

The above regimes 1 and 2 do not change qualitatively compared to the standard ABBM model. This means that features like the power-law behavior of around are robust towards changes in the dynamics, as long as it remains monotonous.

(a1) , , , , .

(a2) , , , , .

(b1) , , , , .

(b2) , , , , .

(c1) , , , , .

(c2) , , , , .

Figure 2: (Color online) Example trajectories for (thin, red) and (thick, yellow), for various parameter values. The left column (a1),(b1),(c1) corresponds to the standard ABBM model (), the right column to the model with retardation (here ). Figures (a), (b) correspond to stationary driving with a constant velocity, whereas the driving in (c) has a kick at . Observe that after a kick, in the standard ABBM model becomes zero permanently after a certain time, see figure (c1) , whereas in the ABBM model with retardation (c2) sub-avalanches restart infinitely often.

ii.3 Non-stationary driving: Response to a finite kick

Instead of continuous driving, let us now perform a kick as defined in section I.3. In the standard ABBM model, like for the quasi-static driving discussed above, this leads to an avalanche on a time scale of order , which terminates with probability 1. At some time , the domain-wall velocity becomes zero. The domain wall then stops completely, so that for all . This gives an unambiguous definition for the size and duration of the non-stationary avalanche following a kick Dobrinevski et al. (2012). Formally, this behavior is seen by computing the probability . It turns out that for any , and tends to as . The distribution (with ) for has a continuous part and a -function part: Dobrinevski et al. (2012).

In the ABBM model with retardation, the situation is different. We show in section V.4 that following a kick, so that the dynamics never terminates completely. If one defines the avalanche duration as , is infinite. This is also seen from the example trajectories in figure 2b. However, the velocity intermittently becomes zero an infinite number of times. Thus, the avalanche following a kick is split into an infinite number of sub-avalanches, just like a quasi-static avalanche discussed above.

On the other hand, the sub-avalanches become smaller and smaller with time. In section VII.1, we show that the total avalanche size following a kick of size is finite and distributed according to the same law as in the standard ABBM model Le Doussal and Wiese (2009b),

(16)

This result holds independently of the memory kernel . For infinitesimal kicks, , becomes the distribution of quasi-static avalanche sizes discussed above.

The disorder-averaged velocity following the kick decays smoothly. In the standard ABBM model, the decay is exponential Dobrinevski et al. (2012). With retardation, we show in section IV that the dependence of on is directly related to the form of the memory kernel .

Another interesting observable is the mean avalanche shape. Conventionally, it is defined at stationary driving for a sub-avalanche: One takes two neighboring zeroes and which delimit a (sub-)avalanche of duration . The mean avalanche shape is then the average of the domain-wall velocity as a function of time, in the ensemble of all such (sub-)avalanches of duration . It has been realized Zapperi et al. (2005) that the skewness of this shape provides information on the relaxation of eddy currents.

However, this definition is hard to treat analytically. Instead of considering the mean (sub-)avalanche shape at a constant duration, we discuss the mean shape of a complete avalanche (consisting of infinitely many sub-avalanches, with infinite total duration) of a fixed size , triggered by a step in the force at . In section VII.2 we give an explicit expression for this shape at fixed size, for exponential eddy-current relaxation. We show how it reflects the time scale of eddy-current relaxation.

The phenomenology discussed here is expected to be similar if instead of a kick at , one takes some arbitrary driving for , which stops at so that .

We see that the non-stationary relaxation properties of the retarded ABBM model differ qualitatively from those of the standard ABBM model. They provide a more sensitive way of distinguishing experimentally the effect of eddy currents than stationary observables at finite velocity, and allow one to identify the form of the memory kernel . In the following sections, we provide quantitative details underlying this picture.

Iii Solution of the retarded ABBM model

In this section, we apply the methods developed in Le Doussal and Wiese (2012b); Dobrinevski et al. (2012); Le Doussal and Wiese (2013); Le Doussal et al. (2012) to obtain the following exact formula for the generating functional of domain-wall velocities,

(17)

It is valid for an arbitrary monotonous driving , where is the solution of the following nonlocal instanton equation,

(18)

with boundary condition . The important observation that allows such an exact formula is that for monotonous driving, the motion in the ABBM model with retardation is still monotonous, as in the standard ABBM model (see appendix A) as discussed above.

To prove (17) we apply the same series of arguments as in the absence of retardationLe Doussal and Wiese (2012b); Dobrinevski et al. (2012); Le Doussal and Wiese (2013). Taking one derivative of Eq. (5) gives a closed equation of motion for , instead of :

(19)

is a Gaussian white noise, with . The term comes from rewriting the position-dependent white noise in terms of a time-dependent white noise,

(20)

This uses crucially the monotonicity of each trajectory.

Using the Martin-Siggia-Rose method, we express the generating functional for solutions of (19) as a path integral,

(21)

For compactness, we have noted time arguments via subscripts. We will use this notation from now on when convenient.

As in the standard ABBM model, the action (21) is linear in . Thus, the path integral over can be evaluated exactly. It gives a -functional enforcing the instanton equation (18). The only term not involving in the action is , which yields the result (17) for the generating functional. For more details, see section II in Dobrinevski et al. (2012) and sections II B-E in Le Doussal and Wiese (2013).

Similarly to the discussion in Dobrinevski et al. (2012); Le Doussal and Wiese (2013) the solution (17) generalizes to an elastic interface with internal dimensions in a Brownian force landscape (i.e. elastically coupled ABBM models). There is indeed a simple way to introduce retardation in that model to satisfy the monotonicity property. We will not study this extension here.

For the case of exponential relaxation, , Eq. (19) can be simplified to a set of two local Langevin equations for the velocity and the eddy-current pressure :

(22)
(23)

The action for this coupled system of equations is

This action is linear in and . Thus, integrating over these fields gives -functionals enforcing a set of two local instanton equations for and ,

(24)
(25)

We then obtain the generating functional for the joint distribution of velocity and eddy-current pressure ,

(26)

in terms of the solution to these two instanton equations. It reduces to (17) for .

Now, the remaining difficulty for arbitrary observables is to obtain sufficient information on the solutions of (24) and (25) with the corresponding source terms. We shall see that this is more difficult than in the standard ABBM model, but can be done for certain observables and certain parameter values.

iii.1 Dimensions and scaling

Before we proceed to compute observables, let us discuss the scaling behaviour of our model, and determine the number of free parameters. The mass can be eliminated by dividing both sides of (19) by ,

(27)

The time derivative shows that there is a natural time scale so that , where is dimensionless. The nonlinear term shows that there is a natural length scale , so that , where is dimensionless. We thus rescale velocities as

(28)

using the natural unit of velocity . Multiplying with , we get the equation

(29)

where the noise is now . Effectively, for the dynamics in terms of the primed variables we have (i.e. we have fixed the units of time and space so that ).

For the standard ABBM model, , and Eq. (III.1) is a dimensionless equation without any free parameters. To describe a signal produced by the standard ABBM model, it thus suffices to fix the velocity (amplitude) scale , and the time scale .

For the ABBM model with retardation, we have an additional time scale , on which the memory kernel in (19) changes. The ratio of to the time scale of domain-wall motion is a dimensionless parameter . Eq. (III.1) also contains a second dimensionless parameter , which gives the strength of the eddy-current pressure, as compared to the driving by the external magnetic field. We thus remain with two dimensionless parameters and , which cannot be scaled away.

From now on, we will use the rescaled (primed) variables only. To simplify the notation, we drop all primes; we thus remain with the dimensionless equation of motion

(30)

This amounts to setting in the original equation of motion, i.e. to working in the natural units for the ABBM model without retardation.

Iv Measuring the memory kernel

First, we discuss how the function in equation (5) can be measured in an experiment or in a simulation. This allows verifying the validity of the exponential approximation (4). We consider the mean velocity at following a kick by the driving field at , i.e. . Our claim is that its Fourier transform and the Fourier transform of the memory kernel are related via

(31)

where .

To show this, we apply (17) to express the mean velocity at time as

(32)

The function is the solution of (18) with . Since above we only need the term of order , and is of order itself, the nonlinear term in (18) can be neglected. In other words, the disorder does not influence the mean velocity , and to obtain , it suffices to solve the linear equation

(33)

Its solution is a function of the time difference only, , which can be obtained by taking the Fourier transform as

(34)

Here is the Fourier transform of the memory kernel. Inserting this relation into (32), the Fourier transform of the mean velocity after a kick is

(35)

which then gives (31), as claimed. In fact, it is easy to see from (32) that a more general relation holds for a kick of arbitrary shape,

(36)

where .

This relation allows one to obtain, at least in principle, the memory kernel by measuring following a kick. This permits to verify the validity of the exponential approximation (11) experimentally. It also allows to test the validity of the ABBM model. Indeed, while (36) at small is simply a linear response, the fact that it holds for a kick of arbitrary amplitude is a very distinctive property of the ABBM model. Alternatively, it may allow to determine the frequency range in which the model provides a good description of the experiment.

V The slow-relaxation limit

In order to go beyond the mean velocity and see the influence of disorder, one needs to solve the instanton equation (18) including the nonlinear term. Even in the special case of exponential relaxation, where (18) reduces to the local equations (24) and (25), their solution is complicated. However, we can analyze the latter in the slow-relaxation limit . In this limit, the relaxation of the domain wall to the next (zero force) metastable state, occurring on a time scale , is much faster than the relaxation of eddy currents (occurring on a time scale ). Using the expressions for the relaxation times derived in Zapperi et al. (2005), one sees that this is the case for very thick or very permeable samples7. To simplify the expressions, we rescale as discussed in section III.1. This amounts to setting . Thus, the time scale of domain-wall motion becomes .

In the following sections, we will compute stationary distributions of the eddy-current pressure and domain-wall velocity at constant driving , as well as their behaviour following a kick. A similar calculation for position differences at constant driving velocity is relegated to appendix B.

v.1 Stationary distribution of eddy-current pressure

Using (26), the generating functional for the eddy-current pressure , at constant driving is

(37)

is obtained from the instanton equations (24), (25) with the sources , and . From (25), one sees that evolves on a time scale . On this scale, both and have a finite limit for . In this limit they are related via

(38)
(39)

The equation (25) for reads

(40)

Replacing on both sides of this equation using Eq. (38) yields a closed equation for ,

(41)

The boundary condition at is fixed by the source, (note by causality):

(42)

Using Eq. (41), we can now compute the generating functional (37),