Spatial shape of avalanches in the Brownian force model
We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.
A large number of phenomena, as diverse as the motion of domain walls in soft magnets, fluid contact lines on rough surfaces, or strike-slip faults in geophysics, have been described by the model of an elastic interface in a disordered medium [1, 2, 3]. A prominent feature of these systems is that their response to external driving is not smooth, but proceeds discontinuously by jumps called “avalanches”. As a consequence of this ubiquitousness, much effort has been devoted to the study of avalanches, both from a theoretical and an experimental point of view [4, 5, 6, 7]. Despite this activity, there are few exact results for realistic models of elastic interfaces in random media.
An exactly solvable model for a single degree of freedom, representing the center of mass of an interface, was proposed by Alessandro, Beatrice, Bertotti and Montorsi (ABBM) [8, 9] on a phenomenological basis in the context of magnetic noise experiments. It describes a particle driven in a Brownian random force landscape. In [1, 10] it was shown that for an elastic interface with infinite-ranged elastic couplings, the motion of the center of mass has the same statistics as the ABBM model.
In this article, we study a multidimensional generalization of the ABBM model, the Brownian force model (BFM). This model, introduced in [11, 12, 13, 14], was shown to provide the correct mean-field theory describing the full space-time statistics of the velocity in a single avalanche for -dimensional realistic interfaces close to the depinning transition. Remarkably, restricted to the dynamic of the center of mass, it reproduces the ABBM model. This mean-field description is valid for an interface for with for short ranged elasticity and for long ranged elasticity.
As shown in [13, 14] the BFM has an exact “solvability property” in any dimension . It is thus a particularly interesting model to describe avalanche statistics, even beyond its mean-field applicability, i.e. for any dimension and for arbitrary (monotonous) driving. It allows to calculate the statistics of the spatial structure of avalanches, properties that the oversimplified ABBM model cannot capture. In Ref.  some finite wave-vector observables were calculated, demonstrating an asymetry in the temporal shape. Very recently the distribution of extension of an avalanche has also been calculated .
In this article we study a general discrete version of the BFM model, i.e. points coupled by an elasticity matrix in a random medium, as well as its continuum limit. In the discrete model each point experiences jumps upon driving. We derive an exact formula for the joint probability distribution function (PDF) of the jumps (the local avalanche sizes) for an arbitrary elasticity matrix. In the limit of small driving this yields a formula for the joint density of local sizes for quasi-static stationary driving near the depinning threshold. This allows us to discuss the “infinite divisibility property” of the BFM avalanche process. The obtained results are rather general and contain the full statistics of the spatial structure of avalanches. They are, however, difficult to analyze in general since they contain many variables, and thus require computing marginals (i.e. probabilities where one has integrated over most of the variables) from a joint distribution. This is accomplished here in detail for the fully-connected model. We find that in the limit of large there exist two interesting regimes. The first one corresponds to the usual picture from mean-field depinning models [3, 18], whereas the second one is novel and highlights the intermittent nature of the avalanche motion.
We then analyze the shape of avalanches, first in a discrete setting by considering few degrees of freedom. The probability exhibits an interesting saddle-point structure in phase space. We then study the continuum limit of the model. We find that the spatial shape of avalanches of fixed total size and extension , becomes, in the limit of a large aspect ratio , dominated by a saddle point. As a result, the avalanche shape becomes deterministic, up to small fluctuations, which vanish in that limit. We calculate the optimal shape of these avalanches. We then analyze the fluctuations around the saddle point. This allows us not only to quantify the shape fluctuations seen in numerical experiments, but also to obtain the mean shape for avalanches with smaller aspect ratios. We test our results with large-scale numerical simulations. While our results are obtained in the special case of an elastic line with local elasticity () the method can be extended to other dimensions and more general elasticity. Finally, we discuss the applicability of our results to avalanches in realistic, short-ranged correlated disorder. The outline of this article is as follows: Section 2 recalls the definition of the BFM model, which is first studied in a discrete setting with general, non-stationary driving. The results of [12, 13, 14] allow us to obtain the Laplace transform of the PDF of local avalanches sizes. Section 3 contains the derivation of the main result: the full probability distribution of the local avalanche sizes. Section 4 focuses on the limit of small driving, and how to obtain the avalanche density. Section 5 contains a detailed analysis of the fully-connected model. Section 6 studies avalanche shapes for interfaces with a few degrees of freedom. Section 7 contains one important application of our result, namely the deterministic shape of avalanches with large aspect ratio for an elastic line. Section 8 analyses the fluctuations around this optimal shape. Section 9 discusses the application of our results to short-ranged disorder and quasi-static driving. A series of appendices contains details, numerical verifications and some adjunct results. In particular, in C, we introduce an alternative method, based on backward Kolmogorov techniques, to calculate the joint local avalanche-size distribution, following a kick in the driving.
2 The Brownian force model
We study the over-damped equation of motion in continuous time of an “interface”, consisting of points with positions , . Each point feels a static random force and is elastically coupled to the other points by a time-independent symmetric elasticity matrix with . Each particle is driven by an elastic spring of curvature centered at the time-dependent position . The equation of motion reads
for . The are independent Brownian motions (BM) with correlations
and ; the overline denotes the average over the random forces . For definiteness we consider
We furthermore suppose that (i) the driving is always non-negative: , and (ii) the elastic energy is convex i.e. for . Under these assumptions, the Middleton theorem  guarantees that if all velocities are non-negative at some initial time: , , they remain so for all times: , .
Some explicit examples of elasticity matrices:
Throughout the rest of this article, we sometimes specify the elasticity matrix. The models studied are (where denotes the elastic coefficient):
The fully connected model:
The elastic line with short-range (SR) elasticity and periodic boundary contitions (PBCs) with
The elastic line with SR elasticity and free boundary conditions:
The general -dimensional elastic interface with PBCs, where and here is the Euclidean distance in and the elastic kernel. Long-ranged elasticity (LR) is usually described by kernels such that (i.e. in Fourier).
2.2 Velocity Theory
Supposing that we start at rest for , , then it is more convenient (and equivalent) to study the evolution of the velocity field directly. The equation of motion reads
where the are independent Gaussian white noises, with and . Equation (3) is taken in the Itô sense. Note that we replaced the original quenched noise by an annealed one , making Eq. (3) a closed equation for the velocity of the interface. The fact that (1) and (3) are equivalent (in the sense that disorder averaged observables are the same) is a non-trivial exact property of the BFM model. It was first noted for the ABBM model [8, 9] and extended to the BFM [13, 14]. It originates from the time-change property of the Brownian motion for increasing , valid as a consequence of the Middleton property . A derivation of this property is recalled in A.
2.3 Avalanche-size observables
In this article we focus on the calculation of avalanche-size observables defined in the following way. Starting from rest at as previously described, we apply a driving for during a finite time interval such that (stopped driving protocol). In response to this driving, the points move and we define the local avalanche size as , that is the total displacement of each point. We adopt the vector notation
The ’s are random variables whose statistics is encoded in the Laplace transform, also called generating function , and defined as
in terms of the solution of the “instanton” equation. The latter reads
where we have defined the dimensionless matrix
which contains all elastic and massive terms in the instanton equation. The solution of Eq. (7) which enters into Eq. (6) is the unique set of variables continuous in with the condition that all when all . The derivation of this property is recalled in a discrete setting in A. The instanton equation thus allows us in principle to express the PDF of the local avalanche sizes, as the inverse Laplace transform of . In the next section we obtain directly, without solving (7), which admits no obvious closed-form solution. We will note the average of a quantity with respect to the probability . Note that the PDF depends only on the total driving and not on the detailed time-dependence of the . This is a particularity of the BFM model.
2.4 The ABBM model
Before going further into the calculation, let us recall the result of Ref. [13, 14] that the statistical properties of the center of mass of the discrete BFM model is equivalent to that of the ABBM model. To be precise, if we write the total displacement (i.e. swept area) and total drive then, in law, we have
Here is a Gaussian white noise and .
Here is the large-scale cutoff for avalanche sizes induced by the mass term. This first result on a marginal of the joint distribution will provide a useful check of our general formula obtained below for .
3 Derivation of the avalanche-size distribution in the BFM
For simplicity we now switch to dimensionless units. We define
where . The instanton equation (7) now reads
The generating functional is given by
In the following we drop the tildes on dimensionless quantities to lighten notations, and explicitly indicate when we restore units. For the ABBM model, it was possible to explicitly solve the instanton equation for the generating function . The inverse Laplace transform was then computed, leading to (10). Here this route is hopeless because Eq. (12) admits no simple closed-form solution. We instead compute directly the probability distribution using a change of variables in the inverse Laplace transform (ILT):
where “” denotes the imaginary unit number to avoid confusion with indexes. The first formula is the ILT where we left unspecified the multi-dimensional contour of integration . In the second line we used the expression of in terms of from (12), as well as the dimensionless version of (6). Changing variables from to , the contours of integration are chosen to obtain a convergent integral, see second line of Eq. (14). This makes this derivation an educated guess, which however is verified in B. We also give another derivation for a special case in C. To pursue the derivation, the Jacobian is written using Grassmann variables as
Reorganizing the order of integrations and changing , we write
Integrating on leads to
Finally, using , the integration over the Grassmann variables can be expressed as a determinant, leading to our main result
Here is the elasticity matrix. This is the joint distribution expressed in dimensionless
units (11). The expression in the original units is recovered by
substituting , and in (18)
while keeping fixed
Note that for zero coupling, , Eq. (18) becomes : the different points are decoupled and one retrieves independent ABBM models. Non-trivial tests of the formula are performed in B. One general property is that the average local size is . This average gives the shape of the interface in the large-driving limit. When uniformly in , it is easy to see by expansion of the above formula that where are (correlated) Gaussian random variables.
We show in C, using different methods, that when the driving is in the form of kicks,
Interpretation: Some features of our main result can be understood as follows. Consider the equation of motion (3). Upon integration from to we obtain
If we could replace the sum of white noises by a gaussian random variable
then we would obtain (18), but with a slightly different determinant given by the replacement in in (18). However, the replacement (21) is not legitimate because the variables are correlated in time. The determinant in (18) takes care of that correlation.
Probability distribution of the shape
Even if it is far from being obvious on Eq. (18), we know from Section 2.4 that the probability distribution of is given by (10) with . This allows us to define the probability distribution of the shape of an avalanche, given its total size : Consider with , such that . The probability distribution of the variables, given that the avalanche has a total size is
4 Avalanche densities and quasi-static limit
The goal of this section is to define and calculate avalanche densities. These allow us to describe the intermittent motion of the interface in the regime of small driving, small. The dependence of the PDF, , on the driving is denoted by a subscript . We first study the jumps of the center of mass described by the ABBM model.
4.1 Center of mass: ABBM
For the ABBM model (and for the total size in the BFM model) the avalanche-size PDF is given by
where is the total driving. The limit of small driving is very non-uniform. In the sense of distributions, its limit is a delta distribution at ,
However, this hides a richer picture and a separation of scales between typical small avalanches and rare large ones . If one defines , the PDF of has a well-defined limit given by
which is indeed normalized to unity . Hence avalanches of sizes are typical ones. However, all positive integer moments of are infinite. This indicates that these small avalanches, though typical, do not contribute to the moments of , which are finite and controlled by rare but much larger avalanches which we now analyze. In the limit of small , there remains a probability of order to observe an avalanche of order . For fixed one has
This defines the density (per unit ) of avalanches. These are the “main” avalanches with , which are also called “quasi-static” avalanches (see below and Section 9). The density is not normalizable because of the divergence at small , but all its integer moments are finite and contain all the weight in that limit, i.e. . In particular, implies .
We now show that the avalanche density contains more information and controls the moments even for finite , a property that follows as a consequence of being the PDF of an infinitely divisible process. This is best seen on its Laplace transform
The “infinite-divisibility property” indeed follows: and such that
where denotes the convolution operation. Hence is a sum of independent random variables for all . The ABBM avalanche process can thus be interpreted as a Poisson-type jump process (a Levy process) with jump density . In general the density can be defined as for fixed (i.e. it does not hold in the sense of distributions), and the relation between and is
The takes care of the divergence at small . This allows us to write the relation between and , expanding (27) in powers of , as
Taking derivatives w.r.t. , this decomposition shows that the (positive integer) moments of are entirely controlled by , for arbitrary fixed (beyond the small- limit). In this sum the term of order can be interpreted as the contribution to the total displacement of the interface (after a total driving ) of a -avalanche (quasi-static avalanche) event (of order ). The convolution structure in (30) shows that these events are statistically independent in the ABBM model. In this model however, this interpretation only holds at the level of moments. The accumulation of infinitesimal jumps, manifest in the non-normalizable divergence of at small prevents us to extend this interpretation to the probability itself, see D for a discussion.
In the BFM, “the infinite-divisibility property” of the avalanche process is even richer, since avalanches occur at different positions along the interface. Let us define the -th “elementary” driving which applies only to site , i.e. , and denote the corresponding size-PDF as . Consider now the PDF for the general driving, . From the structure of its LT, see (13), as a product of exponential factors linear in the , this PDF can be written as a convolution for ,
An avalanche in the BFM can thus be understood as a superposition of avalanches independently generated by each local driving .
As for the ABBM model (center of mass), the structure of the LT of the PDF shows that each of these elementary jump processes is infinitely divisible. We define the avalanche density generated by the driving on the -th point as
where as in the previous case, this equality is to be understood point-wise in the variables. Consider the functions of which appear in Eq. (13) and satisfy Eq. (12). It is the analogue of appearing in (27) for the ABBM model and we thus conjecture the generalization of (29),
This allows us to write an equation relating to similar to (30) (see D). The subtleties linked with the accumulation of small avalanches and the non-normalizability of , are the same as in the previous case, which is also reminiscent of the fact that the limit of small driving of is very non-uniform, as we now detail. Consider with and fixed: the limit of is again given (in the sense of distributions) by . More precisely, in this small- regime, almost all avalanches are : with the distributed according to
as can be seen from an examination of (18) in that regime. The PDF was defined in (25). One sees that the regime contains all the probability, and that for these very small avalanches the local sizes are statistically independent.
The remaining probability to observe large avalanches is encoded in the densities ,
As before, the positive integer moments are entirely controlled by . A more general expression, which illustrates that these large avalanches occur according to a Poisson process, is given in D.
We now give exact expressions for these densities. For a general elasticity matrix, the expression of is obtained from Eq. (18), and contains a determinant. Remarkably, one can compute this determinant in various cases, leading to the following result
where depends on the chosen elasticity matrix:
Fully connected model:
Linear chain with periodic boundary conditions:
Linear chain with free boundary conditions:
PDF of the shape in the small-driving limit
As we just detailed, the small-driving limit of exhibits a complicated structure due to the accumulation of small avalanches. The situation is very different for the PDF of the shape of the interface conditioned to a given total size (22). This conditioning naturally introduces a small-scale cutoff that simplifies the small driving limit with which reads
This limit holds in the sense of distributions, and defines a normalized probability distribution. This indicates that the only small-scale divergence present in originates from the direction uniformly in , in agreement with the conjecture (33).
5 Fully-connected model
Structure of the PDF and marginals
For each , it is a probability distribution, that corresponds to the (dimensionless, with ) PDF of the avalanches of one particle in a Brownian force landscape (ABBM model), interacting with one parabolic well through the force and with another parabolic well through the force . Formula (38) is thus reminiscent of the fact that the various sites interact with one another only through the center of mass of the interface. This simple structure permits a direct evaluation of various marginals of (38) of the type (local sizes on sites and total size). This is done in E. Here we focus on the joint PDF of the total size , and the single-site local avalanche size . Its explicit form is
Of interest is the participation ratio of a given site to the total motion. Its average is . Its second moment, conditioned to the total size , is easily extracted from (40),
We now study the limit of a large number of sites in Eq. (40). There are (at least) two relevant regimes depending on how the driving scales with .
First regime: (“many avalanches”):
Consider the case with fixed. In this case, typical values of are of order . Consider (empirical mean avalanche-size ), which is distributed according to
The joint probability , is given by Eq. (40) (with the change of variable ), and admits the large- limit
Hence the jump of the center of mass becomes peaked at , while the individual sites keep a broader jump distribution. The local avalanche statistics is the same as the one for a particle submitted to the parabolic driving force and to the elastic force from the center of mass of the interface, . This observation extends to any number of particles with respect to : in the large- limit, the particles become independently distributed according to the law (5). This picture is the “mean-field” regime usually studied in fully-connected models [3, 18], and here derived in a rigorous way. Note that in this case, due to a cancellation in (41), the participation ratio scales as which shows that is typically of order .
Second regime: small driving (“single avalanche”)
We now focus on the regime with fixed. In this case is typically of order 1 and is distributed according to
We now compute, using (40), the joint PDF of and in the scaling regime fixed,
The first factor is reminiscent of the density of avalanches and contains a non-normalizable divergence . However (40) implies a cutoff on smal