footnote

# Avalanches and deformation in glasses and disordered systems
Contribution to the book Spin Glass Theory and Far Beyond - Replica Symmetry
Breaking after 40 Years

Alberto Rosso  Université Paris Saclay, CNRS,LPTMS, 91405, Orsay, France
James P. Sethna  LASSP, Cornell University, Ithaca, NY 14853, USA  Matthieu
Wyart  Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland

###### Abstract

In this chapter, we discuss avalanches in glasses and disordered systems, and
the macroscopic dynamical behavior that they mediate. We briefly review three
classes of systems where avalanches are observed: depinning transition of
disordered interfaces, yielding of amorphous materials, and the jamming
transition. Without extensive formalism, we discuss results gleaned from
theoretical approaches – mean-field theory, scaling and exponent relations,
the renormalization group, and a few results from replica theory. We focus
both on the remarkably sophisticated physics of avalanches and on relatively
new approaches to the macroscopic flow behavior exhibited past the
depinning/yielding transition.

##  I Introduction

Rigid systems, put under stress, will often respond through a series of
avalanches SethnaDM01 ; SethnaDP06 ; Zapperi22  with a broad distribution of
sizes and durations.111Warning: Everyday avalanches in sandpiles, or of snow
or rocks on mountains, rarely exhibit the fractal, scale invariant behavior we
study here NagelAngleOfRepose . Faults in the Earth exhibit earthquakes
spanning many decades of size Fisher1997 ; DahmenEarthquakes06 ; jagla2014 ;
dearcangelis2016  when stressed by the motion of tectonic plates. Iron placed
in a growing magnetic field will emit Barkhausen crackling noise (both
acoustic and electromagnetic) as the internal magnetic domain walls shift
between metastable states Sethna07Crossover ; DurinZapperi06 . Raindrops on
your windshield advance in bursts – their leading edges, pinned by dirt,
releasing as they grow and merge Raindrops . Avalanches also arise
microscopically in systems that usually are viewed as responding smoothly to
an external stress. Your fork, bending irreversibly when you try to cut
through a tough piece of meat, responds through many dislocation avalanches
with a distribution of sizes – measurable, for example, in experiments where
nanopillars of metal are compressed Sethna07Wire ; DahmenNanopillars ;
SethnaBDGGHKLLNQRRSZ17 ; NiZLMDSG19 ; PapanikolaouDCSUWZ12 ; Csikor07 .
Toothpaste squeezed out of a tube yields above a critical stress through a
jerky set of avalanches.

In all these systems an avalanche starts from a weak spot that becomes
unstable and slips. This triggers other instabilities nearby that in turn
infect other regions – sometimes halting quickly, and sometimes sweeping over
vast regions before halting. Some systems exhibit avalanches of all scales
(sometimes termed ‘crackling noise’) only when delicately tuned to balance
between stability (only local events) and instability (continuous flow). We
shall see that many systems can either self-organize to this balancing point,
or exhibit generic scale invariance – with events of all scales due to long-
range non-monotonic (i.e. of varying sign) interactions.

These avalanches are characteristic of systems that have a complex energy
landscape (well studied in other chapters of this book). Increasing the stress
on the system ‘tilts’ this landscape, and the avalanches represent transitions
from one metastable valley to another. For a sufficiently large tilt, rigidity
can be lost and the system flows continuously. We focus on unifying concepts
that describe both the avalanche-type response and continuous flows when it
occurs, in systems ranging from disordered magnets and granular materials to
epidemics, raindrops, and brain activity. We shall explain, without extensive
formalism, how scaling and renormalization-group methods unify all of these
systems, and elucidate the important physics that make these systems different
from one another.

Our contribution is organized as follows. In section II, we introduce a few of
the many systems exhibiting collective avalanches and crackling noise:
epidemics, depinning of lines and interfaces in random media, yielding in
amorphous solids such as glasses and foams, and dense granular flows. In the
first two cases, system-spanning avalanches occur at a critical point, while
in other cases they occur in an entire phase. In section III, we explain these
facts by introducing the notion of excitations, as regions of the material
that are about to undergo an instability. If sufficiently long-range
interactions are present, requiring stability of the material constrains the
density of excitations, which in turn implies a generic scale invariance in
the entire rigid phase. In sections IV, V and VI we introduce renormalization
group results and scaling arguments. They apply but differ in the systems
considered, allowing one to both compare these systems and build a detailed
understanding of several phenomena, including the shape of avalanches or their
connection with stationary flows when rigidity is lost. In section VII, we
conclude by emphasizing key open questions of this field.

##  II Systems considered

Avalanches appear in many areas, both in science and engineering and in other
fields. In addition to the examples above, avalanche models have been used to
describe fracture precursors in quasi-brittle material like bones and
seashells ShekhawatZS13 , bubble avalanches in foams TewariSDKLL99 , fluids
invading porous media CieplakR88 ; ortin2017  (like milk invading puffed rice
cereal url:CracklingKids  or coffee soaking into a napkin
AlavaCoffeeAvalanches ), vortex avalanches in superconductors [Zapperi22 , Ch.
9] and neutron stars StarQuakes , wars WarAvalanches , neural avalanches in
brain tissue DahmenNeurons ; beggsP03 , noise while tearing paper
AlavaPaperTearing  and while crumpling paper or candy wrappers HouleS96 ;
KramerL96 , and clapping after concert performances MichardB05 . In this
section, we introduce four phenomena we focus on: the mean field theory of
epidemics, the depinning transition of an elastic manifold in a disordered
environment, the plasticity of simple yield stress materials, and the jamming
transition in granular systems.

###  II.1 Mean-field pandemic model

Outbreaks of disease are a public-health manifestation of avalanche behavior.
An unlucky person is infected by a bird, pig, or bat, and infects one or more
surrounding people. Let $R_{0}$ denote the number of infections triggered by
each formerly infected person; $R_{0}\sim 12{-}18$ for measles, $2{-}3$ for
influenza. For a new disease where nobody is immune, $R_{0}<1$ means that the
disease will gradually disappear, $R_{0}>1$ implies that – if not stamped out
early – it will cause a global pandemic. Near $R_{0}=1$, there is a phase
transition, with avalanches (outbreaks that halt) on all scales for
$R_{0}\lesssim 1$.

This model is called the Bienaymé-Galton-Watson process bienayme1845 ;
watson1875 . It is a mean-field model for avalanches, because there is no
notion of space: everyone can infect anyone. It also describes a fully
connected Ising model in a disordered material SethnaDKKRS93 , where a spin
flips when its external field increases beyond its random threshold,
increasing the external ‘mean field’ on all the other $N$ spins enough to on
average flip $R_{0}$ neighbors. In some cases, this mean-field aspect is not a
realistic assumption: plants do not move, so crop diseases that spread locally
have avalanches that are correlated in space.

Consider the number $I_{N_{R}}$ of infected people at the time when $N_{R}$
people have stopped being infectious (either recovered or deceased). Each
person, before they recover, infects $\xi_{N_{R}}$ others, where $\xi$ is a
random integer chosen from a Poisson distribution of mean $R_{0}I_{N_{R}}$.
For simplicity, we assume these infections happen at the time when the person
recovers, implying $I_{N_{R}+1}=I_{N_{R}}+\xi_{N_{R}}$. This random walk halts
at pandemic size $S$ when $I_{S}$ first touches zero; its size $S$ is the
number of random steps for the first return to the origin. The probability of
a pandemic of size $S$ at $R_{0}=1$ thus coincides with the probability that a
random walk starting at the origin will have its first return after $S$ steps,
which can be calculated to be a power law $P(S)\sim S^{-\tau}$ with
$\tau=3/2$. Let us measure the distance to the critical point as
$r=(R_{0}-R)/R$. For $r\lesssim 0$ below threshold, the random walk starts
positive ($I_{1}=1$) but has a negative drift – it is a biased random walk.
For small avalanches near $R_{c}$, the accumulated drift $\Delta I=rS$ is
negligible compared to the typical random walk distance $I_{\text{max}}\sim
S^{1/2}$, but for large avalanches the drift dominates, making large
avalanches unusual. Equating the two, we expect the probability distribution
of avalanches $P(S,r)$ is cut off at a size $S_{\text{max}}\sim r^{-1/\sigma}$
with $\sigma=1/2$. More precisely, $P(S,r)$ can be shown to follow

$P(S,r)\sim S^{-3/2}\exp(-Sr^{2}/2)\sim
S^{-\tau}{\mathcal{P}}(S^{\sigma}r)~{}~{}~{}~{}~{}\mathrm{with}\ \
{\mathcal{P}}(X)=\exp(-X^{2}/2)\ .$ (1)

Here $\tau$ and $\sigma$ are universal critical exponents, and ${\mathcal{P}}$
is a universal scaling function of the invariant scaling ratio
$X=S^{\sigma}r$. Universality signifies that changing our epidemic model to a
more realistic stochastic SIR model (a ‘compartmental’ model discussing the
susceptible, infected, and recovered populations), or a model with
superspreaders, does not change the values for $\tau$, $\sigma$, and
${\mathcal{P}}$ that describe large epidemics. However, some features like
cities can change the behavior away from mean-field in qualitative ways.

Scaling functions arise Sethna22  whenever more than two parameters and
properties are involved (here, probability $P$, depending on size $S$, and
distance $r$ to the critical point). Other examples include avalanche duration
versus time as a function of $r$, scaling depending on the system size $L$,
crossover scaling Sethna07Crossover ; ChenZS15 , singular corrections to
scaling, etc. We shall study a universal scaling function for the average
temporal shape of avalanches, which has two properties depending on two
parameters, in section IV.3.

Finally, note that to obtain avalanches of all scales requires to tune a
parameter like $R_{0}$ to a critical point. However, in various cases this
parameter can spontaneously evolve to that value, a phenomenon called self
organized criticality bak1987self . In our disease outbreak model, imagine we
add a term which describes the societal response to widespread illness –
wearing masks, getting immunized, avoiding large gatherings. As outbreaks
become large and alarming, these responses will tend to reduce the effective
interactions, tuning $R_{0}$ down until just below the threshold for a
pandemic ZapperiPandemicSOC .

###  II.2 Depinning transitions

Figure 1: Depinning of a one-dimensional manifold in a two dimensional
environment. An increasing external field causes the manifold to locally
destabilize, moving from one configuration to another in an avalanche (large
gray region, of area $S$). If there are long-range forces in the problem, this
event may cause other distant segments to destabilize as well (small gray
regions to right).

View Fig. 1 as a one-dimensional elastic interface, pushed through a disorder
environment by applying a force per unit length $f$. Examples of this
situation abound222One can also view it as a two-dimensional disordered
crystal with a jagged dislocation line pinned on impurities, or a polymer
pinned on a disordered surface, or an oil-water interface in porous rock
CieplakR88 .: it could represent a magnetic film, magnetized ‘up’ below the
jagged interface, and ‘down’ above the interface with a forcing that depends
on the applied magnetic field DurinZapperi06 ; or the triple line formed at
the edge of a drop on your windshield, forced by gravity Raindrops . These
phenomena are well described as the propagation of a single-valued front
$u(r,t)$, where $r$ and $u$ are the horizontal and vertical coordinates of the
interface. Neglecting inertial effects, the equation of motion of our front
might be modeled as a disordered partial differential equation of this class
Fisher98 ; kardar1998 :

$\partial_{t}u(r,t)=f_{\text{el}}\left[u\right]+f_{\text{dis}}\left[u(r,t),x\right]+f\
.$ (2)

Here $f_{\text{el}}$ is the elastic force trying to restore the straight
interface, $f_{\text{dis}}$ is the force due to the random potential, and $f$
is the external drive. The solution of this class of equations shows a
continuous phase transition with the velocity playing the role of the order
parameter and the force acting as the control parameter larkin1970 ;
fisher1985 ; middleton1992a ; giamarchi2006 ; joerg2021theory . In particular,
below a critical depinning force $f_{\text{c}}$ the steady velocity is zero.

Avalanches: If we increase that external field for $f<f_{\text{c}}$, we can
destabilize a weakly-pinned region, triggering an avalanche (see Fig. 1). Just
as in the epidemic model, below $f_{\text{c}}$ these avalanches have a size
distribution $P(S,f)\sim S^{-\tau}\mathcal{P}(S(f_{\text{c}}-f)^{1/\sigma})$,
becoming scale free when the force approaches $f=f_{\text{c}}$. Individual
avalanches smaller than the cutoff
$S_{\text{max}}=(f_{\text{c}}-f)^{1/\sigma}$ will spread over a spatial extent
$\sim S^{1/d_{f}}$, where $d_{f}=1/(\sigma\nu)$. They will have a duration in
time $\sim S^{z/d_{f}}$ that goes as their spatial extent to the power $z$.
Avalanche distributions have been extensively studied in the literature
alessandro1990 ; colaiori2008 ; ledoussal2009c ; ledoussal2009b ; rosso2009 ;
le2013avalanche ; delorme2016 .

Flow properties are singular: For $f>f_{\text{c}}$, the interface will begin
moving with a mean velocity $v\sim(f-f_{\text{c}})^{\beta}$ in a jerky
fashion, with correlated jumps that mimic the avalanches found just below
$f_{\text{c}}$. Similarly to equilibrium continuous phase transitions, the
collective nature of this intermittent dynamics arises from the existence of a
correlation length $\xi\sim|f-f_{\text{c}}|^{-\nu}$ that diverges when the
force approaches $f_{\text{c}}$.

Fractal structure: At the depinning threshold, the interface is fractal, with
a roughness that is characterized by a typical vertical height change
$u(r,t)-u(r^{\prime},t)\sim(r-r^{\prime})^{\zeta}$ corresponding to a height-
height correlation function
$C(r)=\langle(u(r^{\prime})-u(r^{\prime}+r))^{2}\rangle\sim r^{2\zeta}$.

The elastic force $f_{\text{el}}$ on a domain wall can involve long-range
interactions Zoia07 . This is the case for the triple line of a raindrop on
glass joanny1984 ; Raindrops  (and also of crack fronts propagating in brittle
materials gao1989 ; alava2006 ). The elastic force on the droplet interface
must include the surface tension energy of the wiggly drop shape imposed by
the jagged air-water interface $u(r)$, hence $f_{\text{el}}\sim\int
d^{D}r^{\prime}(u(r^{\prime})-u(r))/(r-r^{\prime})^{D+\alpha}$, with
$\alpha=1$ and $D=1$ joanny1984 ; gao1989 ; bonamy2008 ; ledoussal2006 ;
bonamy2011 . A second important example is provided by the sliding of
frictional interfaces or of localized shear bands in amorphous materials
dahmen2009micromechanical ; antonaglia2014bulk ; wright2016experimental ;
wright2018slip . Here, if inertia, velocity weakening or visco-elastic effects
can be neglected, the dynamics is depinning-like with $D=2$ and $\alpha=1$.

The presence of long range elasticity does not change the qualitative picture
of depinning given above, but for $\alpha<2$ the value of the exponents
$\beta,\zeta,\tau$ and $\sigma$ are modified; for $\alpha\geq 2$ short range
exponents remain correct. Also, long range interactions can trigger distant
weak spots to yield, leading to avalanches with disconnected pieces (see Fig.
1), called clusters. Their sizes, distances and extensions display power law
behaviors as for the full avalanche, but with new exponents.

Note that many physical systems self-organize at their depinning transitions –
naturally exhibiting an emergent scale invariant avalanche behavior and
fractal front geometries. For example, in magnets, the weak but long-range
magnetic dipole fields lead to demagnetizing forces (Zapperi22, , Sec. 8.2)
that decrease the effective driving force as the front advances. Theoretical
depinning models add an extra parabolic term to Eq. (2) in order to model this
self-organization.

In section IV we will explain the relationship between avalanches below
$f_{\text{c}}$, stationary flow above $f_{\text{c}}$, and the fractal
structure of the interface at $f_{\text{c}}$, and how these properties are
affected by long-range interactions.

###  II.3 Plasticity in amorphous materials

Other disordered systems display out of equilibrium phase transitions induced
by an external drive. For example, a yielding transition is observed in foams,
emulsions and metallic glasses when a stress, $\Sigma$, is applied. For a
small stress these materials deform as solids, but for stresses $\Sigma$ above
a yield stress $\Sigma_{\text{c}}$ they flow as liquids. In the liquid phase,
the strain rate vanishes non-linearly Herschel1926 ; ovarlez2013rheopexy ,
$\dot{\epsilon}\sim(\Sigma-\Sigma_{\text{c}})^{\beta}$. This behavior is
similar to depinning (albeit with a larger flow exponent $\beta$), but here
the system exhibits generic scale invariance. Scale invariant avalanches start
well below $\Sigma_{\text{c}}$.

In the solid phase one observes plastic instabilities, called shear
transformations Argon79 , involving irreversible rearrangements of few bubbles
or droplets. As illustrated in Fig. 4C, this local rearrangement induces a
large, complex nearby rearrangement and a far-field power-law decay. This
stress redistribution then can trigger other shear transformation zones that
are kicked above their stability threshold.

Thus, as in depinning, this first instability can be the epicenter of an
avalanche Maloney04 ; Salerno12 . The connection with the depinning transition
can be made explicit considering the growth of local strain $\epsilon(r,t)$ at
a given stress $\Sigma$. Its evolution can be written Lin14  in the form of
Eq. (2) for elastic interfaces driven with a force $f$. A crucial difference
between yielding and depinning concerns the far-field kernel, which is of
Eshelby type and not only features long range decay, but also333Monotonic, or
‘abelian’ interactions, also lead to no-passing theorems with fascinating
implications middleton1992a ; SethnaDKKRS93 . is non-monotonic Baret02 ;
Picard2004 . For example in 2D the elastic kernel can be written as

$f_{el}[\epsilon(r,t)]\sim\int dr\frac{\cos 4\phi}{r^{2}}\delta\epsilon$ (3)

Here, $r$ is the distance from the shear transformation, $\phi$ the angle
associated to the position $\vec{r}$, and $f_{el}$ is the component of the
stress tensor projected on the direction of the applied shear.444A more
realistic description of plastic interactions is tensorial
budrikis2017universal , yet the scalar approximation described in the main
text does not affect critical properties near the transition. The presence of
the positive and negative interactions from the cosine term is what makes
$f_{el}$ non-monotonic middleton1992a  – the transition increases the stresses
for some neighbors, and decreases the stress for others. Indeed, after a
plastic instability, the stress redistribution is on average zero: some
regions see their stress increase, while in others the stress diminishes. This
effect does not occur for the pandemic model or for the depinning transition,
where the motion of a region of the interface can only destabilize other
regions. For a yielding amorphous material, the stress that a given spot feels
in time rises and falls, so that survival without going unstable becomes
increasingly unlikely for the particularly weak spots. In section III we will
see that the resulting pseudogap in the probability of regions ready to yield
gives scale-invariant avalanches in the entire yielding solid phase – the
system shows generic scale invariance.

###  II.4 Elasto-plastic models of depinning and yielding

Models of depinning and plastic yielding (sections II.2 and II.3) often
involve coupled networks of sites that each slip or jump to new configurations
when pushed by one another or an overall stress field. Coupled block-spring
models were introduced in the study of earthquakes by Carlson and Langer
CarlsonL89  without explicit disorder. We can model a depinning system with a
$D$-dimensional lattice of blocks connected by springs, moving in a vertical
direction $u$ perpendicular to the lattice. We add a force threshold
$\sigma^{y}_{i}=-f_{\text{dis}}(u_{i},r_{i})$ to each block that depends
randomly on its vertical position, and have each block move by one unit,
$u_{i}\to u_{i}+1$, when it goes unstable. This produces a kick that can
destabilize one or more of its neighbors, triggering an avalanche. Adding
longer range springs with strength $G_{ij}\sim r_{i}-r_{j})^{-(1+\alpha)}$
gives us a long-range depinning model, and adding springs
$G_{ij}=\cos(4\phi)/(r_{i}-r_{j})^{2}$ gives us a model of yielding in
amorphous materialsBaret02 ; Picard2004 ; nicolas2018deformation ; cao2018a .
We shall return to this class of models again when we discuss mean-field
theories of depinning in sections IV.1 and V.1.

###  II.5 Jamming transition

How can a crowded, dense collection of particles manage to move and avoid each
other as they are sheared? This question is relevant for the glass transition
(when a liquid becomes an amorphous solid), the study of the flow of
pedestrians PedestrianJamming  or (unexpectedly) the optimization of neural
networks WyartNeuralNetworkJamming . Initially it was motivated by granular
materials, which can be solid or liquid depending on density and shear stress.
They present a yield stress $\Sigma_{\text{c}}$ separating these two phases,
which for cohesionless materials must be proportional to the pressure $p$ (the
only stress scale in the problem) Andreotti13 . From the ratio
$\mu_{\text{c}}=\Sigma_{\text{c}}/p$, one obtains the angle of repose
$\Theta_{\text{c}}=\arctan{\mu_{\text{c}}}$, at which a layer of sand stops
flowing. The stress ratio $\mu_{\text{c}}$ and the resulting angle of
repose555As mentioned above, sandpiles of frictional particles present two
characteristic angles: $\Theta_{\text{c}}$ where flow stops, and
$\Theta_{\text{start}}>\Theta_{\text{c}}$ where flow starts NagelAngleOfRepose
; nowak2005maximum ; perrin2019interparticle ; Dijksman11 ;
mowlavi2021interplay ; DeGiuli17a ; Zapperi22 .  $\Theta_{\text{c}}$
determines a critical point. For angles just above $\Theta_{\text{c}}$, flow
can occur despite the material being almost as dense as in the solid phase.
Yet to avoid each other, the motion of the particles must be very cooperative
Pouliquen04 ; Olsson10 ; During14  with long-range correlations or ‘eddies’,
as illustrated in Fig. 2A. Also, the flow curve is singular at
$\mu_{\text{c}}$ Andreotti13 ; Olsson11 ; degiuli14d ; perrin2021nonlocal  as
illustrated in Fig. 2B. For example in dense suspensions $\mu({\cal
J})-\mu_{\text{c}}\sim{\cal J}^{\beta}$, where ${\cal
J}\equiv\eta_{0}\dot{\epsilon}/p$ is a dimensionless ratio of the shear rate
$\dot{\epsilon}$, the viscosity $\eta_{0}$ of the suspending fluid, and the
particle pressure $p$ (see section V.2).

Figure 2: Jamming: Critical properties of experimental granular flows.
Fluctuating velocity of particles at the free surface of a granular flow in an
inclined plane geometry of angle (a) $21^{o}$ degree close to
$\Theta_{\text{c}}=20^{o}$, showing long-range spatial correlations and (b)
far away from it $\Theta=26^{o}$, where the flow is faster, less dense, and
where these correlations are absent. From [Pouliquen04 ]. (c) Macroscopic
friction coefficient $\mu$ versus dimensionless shear rate ${\cal J}$ for
different layer thicknesses $h$, showing an exponent $\beta\approx 0.35$. This
experiment is performed is a drum geometry with short-range electrostatic
repulsion between suspensed particles, to ensure that they are effectively
frictionless. We shall study the singular flow rates produced by the
avalanches for depinning, yielding, and jamming in Secs. IV, V.2, and VI.2.
From [perrin2021nonlocal ].

Progress was made by considering an ideal system: frictionless particles with
repulsive, finite-range interactions (e.g., slippery rubber balls) Liu10 ;
Hecke10 . Imagine their positions to be initially random, then moved to
locally minimize the energy. As illustrated in Fig. 3, for a large enough
initial packing fraction of particles, this procedure leads to an amorphous
solid, similar to a structural glass. For a small density, it leads to a gas
of particles. At some threshold value $\phi_{\text{c}}$, the particles are
barely touching (a good model of granular materials) OHern03 . Key findings at
$\phi_{\text{c}}$ are that (i) there exists a $\mu_{\text{c}}\approx 0.05$
below which the material is solid Peyneau08 ; Lespiat11 . For
$\mu>\mu_{\text{c}}$, flow occurs and the flow curve is singular, capturing
quantitatively the experimental results of Fig. 2C. (ii) Avalanches are
observed for $\mu<\mu_{\text{c}}$, as shown in Fig. 3C Combe00 . (iii) Some
structural properties display singular behavior. In particular, the
distribution $P(f)$ of contact forces $f$, or the distribution $g(h)$ of
interstices between particles, are characterized by non-trivial exponents at
small arguments Lerner13a :

$\displaystyle P(f)$ $\displaystyle\sim$ $\displaystyle f^{\theta}\ \
\hbox{with}\ \theta\approx 0.44\ ,$ (4) $\displaystyle g(h)$
$\displaystyle\sim$ $\displaystyle h^{-\gamma}\ \ \hbox{with}\ \gamma\approx
0.38\ .$ (5)

Figure 3:  (a) Unjammed assembly of particles at low packing fraction
$\phi<\phi_{\text{c}}$ (b) Jammed packing with $\phi>\phi_{\text{c}}$. (c) At
$\phi_{\text{c}}$, as the stress anisotropy $\mu$ is increased, the strain
$\epsilon$ displays a devil staircase: there are sequence where no
rearrangements occur (horizontal line) followed by sudden jumps in strain or
‘avalanches’ that are power-law distributed. We will associate the presence of
such a crackling phase with its marginal stability. From Ref. Combe00 .

##  III Instability thresholds and stability

Avalanches are triggered at weak spots, where an abrupt nonlinear response is
triggered by an increasing field. More generally, amorphous and disordered
materials allow for low-energy local rearrangements that can have a
distribution $P(z)$ of excitations whose threshold force is $z$. It has long
been known that systems like Coulomb glasses Efros75  and spin glasses
Thouless77  will excite those rearrangements with the lowest energies to
optimize the interaction energies in finding the ground state. This removes
many of the lowest energy excitations, leaving a density $P(z)$ of excitations
that vanishes as a power law,

$P(z)\sim z^{\theta_{y}}\ ,$ (6)

as the local excitation energy $z$ goes to zero. Such a vanishing density is
called a pseudo-gap (to distinguish it from a hard gap with no excitations in
a finite energy range). Since then, it was realized that this situation is
common, and affects both avalanches and stationary flows when they occur for
large forcing.

###  III.1 Nature of excitations in the systems considered

Excitations correspond to nearly unstable rearrangements of the material. In
magnetic systems, they are spins that are weakly polarized Thouless77 . In
elastic manifolds, they correspond to small portions of the interface. Here we
focus on systems with long-range interactions, in which the notion of
excitations turns out to be more crucial.

Amorphous solids: As discussed above, instabilities in that case are shear
transformations, illustrated in Fig. 4(b). Here $z$ is the increment of shear
stress that will trigger an instability of a shear transformation zone, and
$P(z)$ is the density of zones with trigger stress $z$. As briefly mentioned
in Sec. II.3, the non-monotonicity in the long-range interactions between
transformations leads to a local stress history that varies both up and down
as the material is plastically deformed, depleting the density of the lowest
threshold transformation zones. As in the equilibrium systems, sheared
amorphous solids are found Karmakar10a ; Lin14a  to displays a pseudo-gap,
with $P(z)\sim z^{\theta_{y}}$. Interestingly, $\theta_{y}$ varies
continuously under loading Lin15a . The condition $\theta_{y}>0$ can be
obtained form a stability argument Lin14a . Assume that $\theta_{y}=0$ such
that $P(x=0)=P_{0}>0$. Consider a shear transformation occurring at the
origin, leading to a kick of stress of order $\Delta\Sigma(r)\sim r^{-d}$ at a
distance $r$. The probability that a region of unit area displays an
instability in response to this kick is $\sim\Delta\Sigma(r)P_{0}$. In a
system of size $L$, the total number of new instabilities $R_{0}$ is then
obtained by integration: $R_{0}\sim
P_{0}\int_{r<L}\Delta\Sigma(r)r^{d-1}dr\sim\ln(L)$. This result implies that
an individual plastic events always completely destroy the system (since
$R_{0}\gg 1$), with probability one. We know from experience that amorphous
solids are stable to such local perturbations, thus implying $\theta_{y}>0$.

Figure 4: (a) Schematic stability diagram in configuration space. $A$ and $B$
are observables characterizing the configurations visited. The full black line
corresponds to marginal stability: It separates regions in which excitations
are stable and unstable, respectively. The arrow illustrates a dynamical
trajectory of a system that is cooling from an initial high temperature phase
by stepwise relaxation of individual elementary excitations. When the system
reaches the marginality line, the excitations become stable. If these
excitations are the main drive of the dynamics, the system slows down very
rapidly as it enters into the stable region and freezes very close to the
marginal stability line. There, low-energy excitations are abundant, and rich
dynamics, such as crackling noise, can occur. From Ref. Muller14 . (b)
Elementary excitation corresponding to the opening of a single contact in a
packing of hard discs. From Ref. Lerner13a . (c) A shear transformation in an
amorphous solid made of soft compressed particles. From Ref. Maloney06a .

Packings of frictionless hard spheres: These systems are isostatic: they just
have enough contacts to ensure the stability of the material
kane2014topological . Under loading undeformable particles, rearrangements can
only occur when a contact force goes to zero, leading to the opening of a
contact and to a global motion of all particles called a ‘floppy mode’. An
example of such an excitation is shown in Fig. 4(b). This motion stops when a
new contact is formed. This process gives a kick of stress in the entire
system, which can in turn trigger the opening of new contacts. The magnitude
of the kick can be shown to be proportional to the displacement along the
floppy mode before a new contact is formed, which occurs when a small gap
between nearly touching particles closes. Thus, this distance is controlled by
the distribution $g(h)\sim h^{\gamma}$ of gaps. Furthermore, the probability
that a kick of given magnitude opens a new contact is controlled by the
distribution of small forces $P(f)\sim f^{\theta}$. Requiring stability, i.e.
that the number of new excitations triggered by a single one does not diverge
in the thermodynamic limit (i.e. $R_{0}\leq 1$), can be shown to imply Wyart12
:

$\gamma\geq 1/(2+\theta)\ .$ (7)

As described below, infinite range interactions (independent of distance)
imply Muller14  a saturation of Eq. (7), $\gamma=1/(2+\theta)$, in agreement
with numerical estimates of $\gamma$ and $\theta$. A great success of replica
theory applied to the jamming transition is the computation of $\theta\approx
0.42$ and $\gamma\equiv 1/(2+\theta)\approx 0.41$ in infinite dimensions
Charbonneau14 . These exponents agree well with simulations in finite
dimensions Lerner13a ; Charbonneau15 , as perhaps expected from numerical
evidence GoodrichLiuUCD2 ; SartorRCnn  that the upper critical dimension is
$D_{\text{uc}}=2$ for frictionless jamming.

###  III.2  Pseudo-gaps imply crackling noise

Stability arguments do not exclude the possibility that the density of
excitations presents a hard gap (and strictly no rearrangements for a finite
range of loading). Even if gapped configurations exist, they are not found in
practice. Fig. 4(a) gives a pictorial illustration as to why it is so: for
some systems and dynamics the system essentially stops as soon as it becomes
stable, leading to a gapless marginal state. In mean-field spin glasses for
example, it can be shown that finding gapped configurations is a NP complete
problem behrens2022dis .

The existence of a pseudo-gap implies that the rearrangements must be
collective – avalanche like, with an initial rearrangement triggering at least
some large events, leading usually to what we call crackling noise, without
tuning or self-organizing to a critical point. Following [Muller14 ], consider
for example an amorphous solid of volume $N$, and assume that Eq. (6) applies.
During an adiabatic loading, there will be periods without plasticity: their
magnitude $\delta\Sigma$ is given by $z_{min}$, the stability of the least
stable excitation. Standard extreme value statistics arguments Karmakar10a ;
Lemaitre07a  lead to the typical value $z_{min}\sim
N^{-1/(1+\theta_{y})}\gg\frac{1}{N}$. It implies that the rate of plastic
events is sub-extensive: a system twice as large does not display twice as
many plastic events. Assume that no crackling occurs: then for local
excitations, a plastic event involves ${\cal O}(1)$ excitations and thus
dissipates an energy ${\cal O}(1)$, which is independent of $N$. For a global
change of stress $\Delta\Sigma$, the energy dissipated is thus
$\sim\Delta\Sigma/\delta\Sigma\sim N^{1/(1+\theta_{y})}\ll N$, i.e. it is sub-
extensive. Thus, in the absence of crackling, dissipation per unit volume must
vanish in the thermodynamic limit. Plastic materials displaying a finite
density of dissipation (which we expect to be the generic situation for
disordered materials) must then display crackling. In particular for
dissipation to be extensive, one must have $\langle S\rangle\sim
N^{\theta_{y}/(1+\theta_{y})}$.

We return to infinite range interactions (such as frictionless hard spheres in
any dimensions, or the SK model of spin glasses). In that case, it can be
shown that if inequalities like Eq. (7) are not saturated Muller14 , then an
excitation has a vanishing probability to trigger a second one in the
thermodynamic limit: crackling cannot occur. But we have seen before that
crackling noise must occur (to allow for an extensive dissipation), thus our
hypothesis is incorrect: we conclude that the bound in Eq. (7) must be
saturated when infinite-range interactions are present. Note that saturation
does not occur for power-law interactions, e.g. in amorphous solids.

Ultimately, this approach leads to a classification of glassy systems based on
the range of the interaction of their excitations. If sufficiently long-range
and non-monotonic, then a pseudo-gap must be present, and crackling occurs.
Instead for short-range interactions, crackling occurs only at a critical
point and not in the entire glassy phase. As we shall see below, the pseudo-
gap exponent also characterizes the flowing phase, when it exists.

##  IV Theoretical Results: Depinning

###  IV.1 Mean-field theory of depinning

Depinning (see Eq. (2)) is characterized by the competition between a decaying
elastic force, $f_{\text{el}}\sim\int
d^{D}r(u(r^{\prime})-u(r))/|r-r^{\prime}|^{D+\alpha}$ that flattens the
interface and a disorder force of the $D+1$ dimensional medium that tries to
roughen it. Dimensional analysis can help to determine the result of this
competition, if we assume that the interface is self-affine, namely that after
a dilatation $x\to bx$, its transverse fluctuations grow as $b^{\zeta}u(x)$.
After the dilatation, the elastic force is rescaled as $b^{\zeta-\alpha}$ for
$\alpha<2$ (for shorter-ranged forces it turns out to scale as $b^{\zeta-2}$)
and the short-range disorder force turns out to be rescaled as
$b^{D/2-\zeta/2}$. Using a Flory type of argument one has to balance the two
forces in order to estimate the roughness exponent nattermann1990 ;
agoritsas2016driven . Thus, the interface results flat ($\zeta=0$) at the
upper critical dimension $D_{\text{uc}}=2\alpha$ for $\alpha<2$, matching
nicely with the value $D_{\text{uc}}=4$ above $\alpha>2$.

Above $D_{\text{uc}}$ disorder is not able to roughen the interface, which
turns out to slide with a velocity linear in the force above threshold
($\beta=1$), and display mean field avalanches ($\tau=3/2$, $\sigma=1/2$, and
$z=\alpha$), as discussed below. Below $D_{\text{uc}}=2\alpha$, the interfaces
are rough ($\zeta>0$), the velocity grows with a power $\beta$ less than one,
with avalanche exponents $\tau$, $\sigma$, and $z$ smaller than those in mean-
field.

This Flory argument correctly predicts $D_{\text{uc}}$, a result that can be
also obtained treating the disorder as a perturbation larkin1970 ; tanguy1998
; chauve2000 ; cao2018 . Unfortunately both arguments fail in predicting the
exact value of the exponents below $D_{\text{uc}}$. Indeed, for purely linear
models, the Flory balance works perfectly at each length scale. However, any
nonlinear terms will couple the different length scales, and thus demand a
renormalization group approach.666In systems with ‘KPZ’-type nonlinearities
Flory’s arguments do not work in any dimension. In this case it remains
controversial if it exists an upper critical dimension above which interfaces
are flat.

A mean-field theory for depinning can be formed using the block-spring model
of section II.4, but having every pair of sites $x_{i}$ and $x_{j}$ connected
by a spring $G_{ij}\equiv G/N$, with $N$ the number of blocks. After each
block slips, all blocks receive a kick (force increment) of $G/N$. Hence, the
dynamics are precisely the same as that in the Bienaymé-Galton-Watson process
we saw in the pandemic model. The value of the force $f$ controls the
parameter $R_{0}$, which becomes critical ($R_{0}=1$) at $f=f_{c}$. Because
all of the kicks are in the same direction, the interaction is monotonic, and
there is no pseudogap. Above $f_{c}$ a stationary flow is present with a
permanent fraction of unstable blocks $\sim|f-f_{c}|^{\beta}$, with $\beta=1$.

###  IV.2 Finite dimension and Functional RG

Table 1: Critical exponents and scaling relations for depinning systems of $D$ dimensional interfaces in $D+1$ dimensions, assuming no nonlinearities of the qKPZ type and elastic forces $f_{\text{el}}\sim\int d^{D}r(u(r^{\prime})-u(r))/|r-r^{\prime}|^{D+\alpha}$. The last column shows one-loop functional renormalization group calculations narayan1992 ; nattermann1992 ; narayan1993 ; ertas1994  with $D_{\text{uc}}=\min(2\alpha,4)$ and $\epsilon=D_{\text{uc}}-D$. Above $D_{\text{uc}}$ behavior is mean-field. For $D<D_{\text{uc}}$, exponents are measured numerically with a precision at the second digit. Note that depinning models do not have a pseudogap, so $\theta_{y}=0$.  Depinning | Observable | $D=1$ | $D=1$ | $D=2$ | FRG
---|---|---|---|---|---
exponent |  | $\alpha=2$ | $\alpha=1$ | $\alpha=2$ | $D=D_{\text{uc}}-\epsilon$
$z$ | $t(L)\sim L^{z}$ | 1.43 | 0.77 | 1.56 | $D_{\text{uc}}/2-2\epsilon/9$
$\zeta$ | $u(x)\sim x^{\zeta}$ | 1.25 | 0.39 | 0.75 | $\epsilon/3$
$\tau$ | $P(S)\sim S^{-\tau}$ | $\tau=2-\alpha/(D+\zeta)$ | $3/2-\epsilon/(3D_{\text{uc}})$
$\nu$ | $\xi\sim|f-f_{c}|^{-\nu}$ | $\nu=1/(\alpha-\zeta)$ | $2/D_{\text{uc}}+4\epsilon/(3D_{\text{uc}}^{2})$
$\beta$ | $v\sim|f-f_{c}|^{\beta}$ | $\beta=\nu(z-\zeta)$ | $1-\epsilon/(9D_{\text{uc}})$

Table 1 gives a summary of exponents for the depinning of $D$-dimensional
manifolds in $D+1$ dimensions. The last three rows in the table give a set of
scaling relations that allow one to express all the exponents in terms of two,
$\zeta$ and $z$, valid below $D_{\text{uc}}$. For example, the velocity above
threshold can be expressed as the ratio between avalanche displacement and
avalanche duration, namely
$v\sim\xi^{\zeta}/\xi^{z}\sim|f-f_{c}|^{-\nu(\zeta-z)}$, giving the equation
for $\beta$. The last column in Table 1 gives one-loop functional RG (FRG)
results for the exponents narayan1992 ; nattermann1992 ; narayan1993 ;
ertas1994  close to the upper critical dimension. Two-loop chauve2001 ;
ledoussal2002 ; ledoussal2004  and three-loop calculations for the static
problem are now available wiese2018field ; husemann2018field .

The RG approach captures the large-scale scale invariant properties of a
population of configurations. The traditional RG studies the flow in a space
of parameters (field $H$, temperature $T$, …) which yield an effective coarse-
grained free energy; the free energy encodes the Gibbs weight of different
configurations. Here, the weight of the dynamical trajectories is encoded into
the Martin-Siggia-Rose action, which averages trajectories over disorder. The
resulting RG involves a full function: the force-force disorder correlator,
$\Delta(u-u^{\prime})$, where $u,u^{\prime}$ are two different positions of
the center of mass of the interface. The fixed point $\Delta^{*}$ can be
computed by FRG as a perturbation in $\epsilon=D_{\text{uc}}-D$ and has a
peculiar cusp at the origin $u=u^{\prime}$ which has been experimentally
measured wiese2021force . Indeed, when an avalanche occurs, much of the
interface is left pinned at the same position (same disorder), but the
disorder force changes abruptly in the regions which have slipped forward
(Fig. 1), leading to the cusp.

For physical systems in $D=1$ and $D=2$, the best estimation of depinning
exponents come from numerical calculations leschhorn1997 ; tanguy1998 ;
rosso2002 ; rosso2003 ; duemmer2005 ; duemmer2007 ; ferrero2013 , summarized
in the first two rows of Table 1. The two-loop expansion chauve2001 ;
ledoussal2002 ; ledoussal2004 ; rosso2007 , however, settled a qualitative
question — showing that the interface at thermal equilibrium has a different
roughness and universality class from that of the interface at depinning. Even
more surprisingly, FRG chauve2000  is also able to describe the thermally
activated slow dynamics of magnetic domain walls driven by a magnetic field
much smaller than the critical force. Most of the predictions for this creep
regime have been tested both experimentally lemerle1998 ; caballero2018  and
numerically ferrero2020 . See also the FRG results for the avalanche shape in
section IV.3.

Critical exponents are neither the only quantity of interest nor the simplest
to observe in experiments laurson2009 ; bares2019 . The value of the critical
force $f_{c}$ and its universal fluctuations bolech2004 ; demery2014 ;
fedorenko2006  are important for applications. In the following we will see in
some details some other important example of such observables.

###  IV.3 The average avalanche time series

Figure 5: Avalanche, and average temporal shape, from Ref.
[PapanikolaouBSDZS11 ]. (A) Experimental signal during an avalanche (jagged),
with noise filtered out (wavy), rather than using thresholding (dotted). The
dashed line shows the signal $V(t|T)$ averaged over all avalanches of similar
duration. (B) Average temporal shape in experiment and predicted by mean-field
theory (Eq. (8)) for various avalanche durations.

A longstanding challenge for the scaling theory of dynamical disordered
systems was the rather symmetrical predictions for the average temporal shape
KuntzS00 ; MehtaMDS02 , which disagreed with a much larger asymmetry in
experiments SethnaDM01 . This eventually was explained as an effect of eddy
currents ZapperiCCD05 . Figure 5A shows the time-dependent mean front velocity
during a mean-field magnetic avalanche in a thin magnetic film
PapanikolaouBSDZS11 , avoiding the effects of eddy currents. Note the
characteristic irregular, fractal shape – almost stopping several times. (If
it had stopped, it would form a smaller avalanche – big avalanches are just
smaller avalanches that did not stop a few times.) Note also the dashed curve
– an average $V(t|T)$ over all the experimental avalanches whose durations $T$
were similar to this one. The universal scaling prediction for this average
shape is KuntzS00 ; SethnaDM01

$V(t|T,r)=T^{d_{f}/z-1}{\mathcal{V}}(t/T,Tr^{z\nu})=T{\mathcal{V}}(t/T,Tr)\ .$
(8)

Figure 5B shows a scaling collapse of average avalanche shapes for different
durations (top) and the universal scaling prediction from mean-field
PapanikolaouBSDZS11 ; LeDoussalWieseAverageTemporalShape  (bottom). The
average shape is an inverted parabola for avalanches small compared to the
cutoff $T_{\mathrm{max}}\sim r^{-z\nu}=1/r$, and has an explicit analytical
form for longer durations in mean-field theory (Fig. 5B).

The same functional RG methods we discussed in section IV.2, remarkably, have
been applied Dobrinevski15avalanche  to predict the average temporal shape
both at fixed duration and fixed avalanche size for $D\leq D_{uc}$. They
predict a slight skewing of the avalanches that nicely fits the measured
shapes MehtaMDS02 ; durin2016quantitative , and also incorporate eddy currents
into the scaling predictions dobrinevski2013statistics ; durin2016quantitative
.

###  IV.4 Clusters in avalanches with long-range interactions

As we saw in Fig. 1, avalanches in systems with long-range interactions can be
disconnected into clusters. It is infeasible in most experiments to visualize
avalanches one-by onelepriol2020 , and naturally impossible to figure out
after several avalanches which clusters came from which avalanches – we cannot
directly find $\tau$, $\sigma$, etc.

At moderate rates of strain, one can measure the statistics of clusters
localized in space, which fortunately also display universal scaling maloy2006
. For example their size $S_{c}$ has a power law probability distribution
$P(S_{c})\sim S_{c}^{-\tau_{c}}$. The other avalanche exponents can be defined
for clusters as well. A solid numerical conjecture predicts the value of
cluster exponents in terms of the avalanche exponents laurson2010 . For
example the relation $\tau_{c}=2\tau-1$ should hold for $\alpha<2$. One must
note, though, that our analytical study of a simple epidemic model with long
range dispersal does not show this behavior cao2022clusters .

The key to this conjecture is the identification of a Bienaymé-Galton-Watson
process describing the statistics of the cluster number lepriol2021 . This
process has been clearly observed numerically for all $\alpha<2$, but no proof
of its existence yet exists. For $\alpha>2$ the number of clusters is strongly
suppressed; $\tau_{c}=\tau$ for forces of finite range. It is not yet clear
how $\tau_{c}$ behaves for large, but finite values of $\alpha$. A future
challenge is to understand the origin of the Bienaymé-Galton-Watson process
describing the cluster statistics.

##  V Theoretical Results: Yielding transition

###  V.1 Mean field models of yielding

The mean-field version of the elasto-plastic block-spring models for the
yielding transition (section II.4) have non-monotonic interaction matrices
$G_{ij}$ with both positive and negative values that are randomly chosen with
the same distribution and independent of $i$ and $j$ (infinite range). Let us
call $z_{i}=\sigma_{i}^{y}-\sigma_{i}$, where $\sigma_{i}$ and
$\sigma^{y}_{i}$ are respectively the shear stress and the yield stress of
that block. When $z_{i}<0$ the block yields, reducing all the other $z_{j}$ by
$G_{ij}$

In the Hebraud-Lequeux model Hebraud98 , $G_{ij}$ is assumed to follow a
Gaussian distribution of variance $\sim N^{-1/2}$, where $N$ is the number of
blocks. As a result, each variable $z_{i}$ is a sum of Gaussian random kicks,
where the walk ends when the sum hits zero – a random walk with an absorbing
condition at $z=0$. After yielding, the site is reinserted at some finite
positive $z_{i}$ value. Solving this random walk problem, we find that the
distribution $P(z)$ evolves until it vanishes linearly with $\theta_{y}=1$.
The flow curve is found to be singular at $\Sigma_{c}$, with $\beta=2$.
Interestingly jagla2015avalanche , the distribution of avalanches sizes
depends on details of how the system is loaded. For realistic loading,
$\tau\approx 1$.

In a more realistic mean-field model, the noise is not assumed to be Gaussian.
It is still i.i.d. and thus has no spatial correlations, but the distribution
of $G_{ij}$ is chosen to satisfy that of the true propagator. For a propagator
that vanishes with distance as a power-law, $P(G)$ is then power-law
distributed Lemaitre07 . In that case, $z_{i}$ follows a Levy flight with a
Levy coefficient $\mu=1$. The pseudo-gap is still governed by an absorbing
condition: it is found that after a quench at zero stress, $\theta_{y}=1/2$
Lin16 . However, here $\theta_{y}$ continues to evolve as the system is
strained; it starts by rapidly decreasing, and then increases. This has been
observed in finite dimensional elasto-plastic models Lin15a  and in molecular
dynamics of amorphous solids ji2019theory ; ozawa2018random . Flow curves are
again singular, with $\beta=1$ lin2018microscopic . In this model, aging
responses following a quench (a minimization of the energy from some random
configuration at $t=0$) can be computed parley2022mean , and are found to be
similar to experimental ones.

###  V.2 Scaling theory for yielding

In this section we will provide a link between the avalanche physics at the
mesoscale and the flow curve at the macroscale, for systems with non-monotonic
long-range interactions (such as amorphous materials yielding under stress,
section III).

Stationarity: Consider applying a quasi-static strain in a system of linear
size $L$ at the yielding transition. The stress $\Sigma$ ramps up under
increasing strain, and drops abruptly during avalanches. On average, these
must cancel out Maloney04 ; Karmakar10a ; Salerno13 . An avalanche of size $S$
will lead to a stress drop in the system that is proportional to
$\delta\Sigma\sim\langle S\rangle/L^{d}$. The mean avalanche size is $\langle
S\rangle\sim\int^{S_{\text{max}}}S\times S^{-\tau}dS\propto
S_{\text{max}}^{2-\tau}$, where $S_{\text{max}}\sim L^{d_{f}}$ is the largest
avalanche that can fit into the system. The stress rise between avalanches is
determined by the pseudogap in the distribution of threshold stresses. As
discussed in section III, the weakest site will yield after the stress
increases by $\sim L^{-d/(1+\theta_{y})}$. Equating the average drop and the
average increment, one obtainsLin14 :

$\tau=2-\frac{\theta_{y}}{\theta_{y}+1}\frac{d}{d_{f}}\ .$ (9)

Dynamics in the flowing regime: As one approaches the critical stress
$\Sigma_{c}$ from above in the flowing regime, there is a diverging length
scale $\xi\sim(\Sigma-\Sigma_{c})^{\nu}$. Below this length scale, both in
depinning transitions Fisher98  and in the yielding transition Lin14  one can
roughly describe the motion as a continuously evolving collection of
avalanches of spatial length $\xi$, covering space. The net strain rate
$\mathcal{\dot{\epsilon}}$ is the number of avalanches $\sim 1/\xi^{d}$ times
the strain release $S\sim\xi^{d_{f}}$ per avalanche divided by the duration
$T\sim\xi^{z}$ of these avalanches, hence
$\dot{\epsilon}=S/(T\xi^{d})\sim(\Sigma-\Sigma_{c})^{\nu(d-d_{f}+z)}$. Hence
the strain rate $\dot{\epsilon}\sim(\Sigma-\Sigma_{c})^{\beta}$ grows with an
exponent

$\beta=\nu(d-d_{f}+z)\ .$ (10)

Stress fluctuations: Continuing to view the flow as a superposition of
avalanches of size $S_{\text{max}}$ spanning boxes of size $V=\xi^{d}$, each
will lower the stress in its box by an amount $\delta\Sigma\sim
S_{\text{max}}/\xi^{d}\sim(\Sigma-\Sigma_{c})^{\nu(d-d_{f})}$. Making the
common assumption that all quantities with the same units share the same scale
near a critical point, we expect that the fluctuations of stress
$\delta\Sigma$ on the scale $\xi$ must be of order of the distance to
threshold $\Sigma-\Sigma_{c}$, leading to Salerno13 ; Lin14 :

$\nu=1/(d-d_{f})\ .$ (11)

Putting Eqs. (11) and (10) together, one obtains $\beta=1+z/(d-d_{f})\geq 1$.
Note that the scaling relations in Eqs. (9)–(11) should work in elasto-plastic
models, particle based models, and real systems, as supported by observations
Lin14 ; nicolas2018deformation . One must note that elasto-plastic models
assume that elastic interactions propagate instantly, and sometimes give an
unphysical value of $z<1$ (leading to avalanche propagation faster than the
speed of sound). In this case, we presume that $z=1$ and the value of $\beta$
in Eq. (10) should thus be larger in physical systems than in elasto-plastic
models lin2018microscopic .

These scaling arguments can be extended to non-stationary situations
popovic2021scaling  such as the slow creep flows balmforth2014yielding  that
follows a sudden increase of stress, to non-local phenomena goyon2010does
where $\xi\sim|\Sigma-\Sigma_{c}|^{-\nu}$ is found to characterize the length
scale on which an obstacle or interface perturbs flow gueudre2017scaling , and
to finite temperatures popovic2021thermally ; ferrero2021yielding  that cause
a “thermal rounding” of the flow curves, which becomes smooth near
$\Sigma_{c}$.

##  VI Theoretical Results: Jamming

###  VI.1 Infinite-dimension calculations of avalanches

Avalanches are dynamical phenomena. Yet ‘equilibrium avalanches’ can be
defined, by tilting the energy landscape continuously by adding a force or
field, and tracking the position of the global minimum of the system Le-
Doussal12 . It then displays jumps, whose statistics can be computed in
infinite dimension using replica methods. Right at the jamming transition,
this method was successfully used to compute
$\tau=(3+\theta)/(1+\theta)\approx 1.41$franz2017mean  in close agreement with
numerical experiments for nonequilibrium avalanches in three dimensions
Combe00 .777It is known in the random-field Ising model that the universality
class of the equilibrium system is different from that of the non-equilibrium,
avalanche model at zero temperature. This has been shown in dimensions smaller
than about 5.1 using remarkable non-perturbative functional renormalization-
group calculations TarjusNonPerturbativeRG5.1 , and numerically in two
dimensions HaydenRS19 . Here, however, we are above the upper critical
dimension, where non-equilibrium and equilibrium properties should be similar.
The success of the mean-field approach arguably results from the infinite-
range nature of excitations in this system During14 .

A system of well-compressed particles (packed more densely than at the jamming
threshold) should exhibit a yielding behavior as described in subsection II.3.
Replica theory of jamming predicts $\tau=1$ for the rigid phase franz2017mean
. This mean-field result may not hold in three dimensions however, as the
length scale of excitations rapidly decreases away from jamming
shimada2018spatial , and spatial correlations are then presumably important.

###  VI.2 Suspension and granular flows of frictionless particles

Much of the jamming literature has focused on frictionless particles, both
numerically Olsson07 ; Heussinger09 ; Lerner14 ; olsson2019dimensionality ;
Trulsson12  and theoretically DeGiuli15a . Friction is of course important for
most of the applications of jamming – from soil engineering to pharmaceuticals
to geophysics. It was recently realized that a class of suspensions displaying
“shear thickening” display effectively frictionless particles at low stresses,
but they become frictional at large stresses Mari14 ; Wyart14 ;
clavaud2017revealing ; comtet2017pairwise . Controlled experimental systems
with these properties have been designed perrin2021nonlocal ;
perrin2019interparticle ; RamaswamyGLSKGSCCnn ; RamaswamyGSCCnn , allowing one
to experimentally test scaling theories of flow for frictionless jamming.

Consider a frictionless packing at the critical stress anisotropy
$\mu=\Sigma/p=\mu_{c}$. The first step of the theory consists of estimating
the number of contacts $\delta z$ that open if the stress anisotropy is
increased by $\mu-\mu_{c}$. One finds, by estimating the change of forces in
the contacts induced by such a perturbation, and using that the distribution
of small force follows $P(f)\sim f^{\theta}$, that DeGiuli15a :

$\delta z\sim(\mu-\mu_{c})^{\frac{2+2\theta}{3+\theta}}\ .$ (12)

Using this, one can use mechanical considerations to compute how an
infinitesimal applied strain $\delta\epsilon$ at the boundary moves particles
relatively to each other, by some amount $\delta r$ called the non-affine
displacement. Defining ${\cal L}\equiv\delta r/\delta\epsilon$, one finds
DeGiuli15a :

${\cal L}\sim\delta z^{-\frac{2+\theta}{1+\theta}}\ .$ (13)

The overall viscosity of such a suspension is proportional to the dissipated
power, which grows as the square of the particle velocities, and is thus
proportional to ${\cal L}^{2}$. The viscous number ${\cal J}$ is inversely
proportional to this viscosity, thus ${\cal
J}\equiv\eta_{0}\dot{\epsilon}/p\sim{\cal L}^{-2}$. Together with Eqs. (12)
and (13), one obtains the flow curve:

$\mu-\mu_{c}\sim{\cal J}^{\frac{3+\theta}{8+4\theta}}\approx{\cal J}^{0.35}\
,$ (14)

in agreement with experiments, as shown in Fig. 2. Several diverging length
scales are predicted at the transition During14 ; one of these (the divergence
of non-local effects) has been favorably tested perrin2021nonlocal .

Note that this theory is mean-field in nature, as it neglects spatial
correlations that could occur e.g. in the structure. Its quantitative
agreement with finite dimensional numerical studies DeGiuli15a  and
experiments perrin2021nonlocal  again underlines the mean-field character of
the jamming transition for frictionless particles, where excitations have
infinite range interactions at threshold. This situation does not hold for
frictional particles, where other exponents are found Trulsson16 ;
ramaswamy2021universal  and spatial correlations are relevant DeGiuli17 . An
additional complexity is that in the presence of friction, the flow curve is
non-monotonic and the transition is first order close to the critical point
DeGiuli17 ; perrin2019interparticle ; Dijksman11 ; mowlavi2021interplay .

##  VII Conclusion and open questions

We have described the closely intertwined physics of avalanches, the stability
of the rigid phase, and stationary flows in a variety of disordered systems.
Scaling concepts are the key to building such a unifying description. The
renormalization group inspires these scaling descriptions, but they are useful
even in cases where a RG procedure has not (yet) been developed. We conclude
by listing a few open problems in this field:

RG description of the yielding transition: Although the jamming transition of
hard particles appears to be mean-field in character, the yielding transition
of amorphous materials appears more subtle. Currently, there is no analytical
approach to compute exponents in that case (except for avalanche exponents in
conditions where a narrow shear band appears, where a mean-field description
may apply dahmen2009micromechanical ; antonaglia2014bulk ;
wright2016experimental ; wright2018slip ).

Avalanches and first order transitions: Systems often exhibit avalanche
precursors before abrupt changes in behavior. The abrupt change in behavior is
due to a non-monotonic flow curve: the same force (stress) can lead to
different velocities (strain rate). Quasi-brittle materials exhibit power-law
fracture precursors under tension ShekhawatZS13  before they fracture into
two. This also occurs in complex fluids (such as loosely connected colloidal
gels) and earthquakes (where frictional forces acting on the fault can
decrease with sliding velocity zheng1998conditions ). Inertia can also lead to
such flow curves in depinning problems Fisher1997 ; Schwarz2001  and in
amorphous solids Salerno12 ; nicolas2016effects . Velocity (or strain-rate)
weakening can make the transition between a rigid and flowing phase first
order and hysteretic, possibly destroying avalanches. Recent work on
frictional interfaces suggests that avalanches persist in that situation
(although their statistics change), and act as nucleation centers for system-
spanning events de2022scaling . In that view, the avalanche size diverges at
some threshold stress, beyond which the rigid material is unstable to the
presence of a large nucleus of flowing material. How generic these results are
remain to be seen.

Nucleation of failure: This contribution focuses on stationary flow in
disordered materials. Another important question is how these materials break
as the loading continuously increases. For amorphous solids, this usually
occurs by forming a shear-band where plastic strain localizes. Some have
argued that the shear band nucleates in a process similar to that of the
random-field Ising model ozawa2018random ; ozawa2021rare , or fracture
Popovic2018 ; others suggest it is governed by linear stability considerations
Fielding2021 . Numerical studies richard2021finite  report very slowly-
decaying finite-size effects that remain to be understood.

Yielding at finite temperature $T$ and the glass transition: we focused on the
yielding transition at $T=0$. At finite temperature, the flow curve loses its
singular behavior, in a stress interval around $\Sigma_{c}$ that vanishes as a
power law of $T$ popovic2021thermally ; ferrero2021yielding . Such a ’thermal
rounding’ behavior can be captured by mean field calculations or scaling
arguments popovic2021thermally . Yet, a spatial description of thermal
avalanches triggered by rare activated events is still missing. It may be very
relevant to understand the collective dynamics that occurs near the glass
transition chacko2021elastoplasticity .

## Acknowledgment

JPS would like to acknowledge the support of NSF DMR-1719490. MW acknowledges
support from the Simons Foundation Grant (No. 454953 Matthieu Wyart) and from
the SNSF under Grant No. 200021-165509.

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