# Self-arresting earthquakes and critical sliding nucleation theory

Didier Sornette1, Xueting Wei2 and Xiaofei Chen1,2 1 Institute of Risk
Analysis, Prediction and Management (Risks-X), Academy for Advanced
Interdisciplinary Sciences, Southern University of Science and Technology,
Shenzhen, China
2 Department of Earth and Space Sciences, Southern University of Science and
Technology, Shenzhen, China

###### Abstract

We develop a statistical thermodynamic approach for understanding earthquake
nucleation on homogeneous faults, explaining the occurrence of both self-
arresting and run-away unstable ruptures (subshear and supershear events)
previously observed in numerical simulations. Our theory identifies the
conditions under which self-arresting earthquakes occur, based on critical
sliding distance and dynamical stress drop. We also derive the Gutenberg-
Richter distribution for these earthquakes, linking the fractal nature of
faulting with the nucleation physics through the critical size’s dependence on
dynamical stress drop. Furthermore, we connect our findings to the dragon-king
theory, which suggests that the largest earthquakes differ significantly in
physical and statistical properties from smaller ones, offering new insights
for earthquake prediction and risk assessment.

Earthquakes fascinate physicists with their power law size distribution,
multi-scale faulting patterns, underlying self-organized critical behavior and
still largely unpredictable behavior, while they embody the interplay between
Earth’s physical processes and their profound societal impacts. The processes
that lead to the onset and halting of an earthquake, as well as its
preparatory stages, are still not fully understood. For decades, nucleation
has been key to understanding earthquake mechanisms, referring in literature
to the preparatory stage before an earthquake occurs [7, 9]. Slip-weakening
models describe earthquakes as starting at asperities –fault patches with
higher pre-stresses– and ending at barriers with greater frictional strength.
In contrast, rate-and-state models suggest frictional strength increases with
slip, aiding in stopping and healing. Although physical models of earthquake
nucleation at asperities vary, they agree that post-nucleation, sliding
becomes irreversibly unstable, propagating at near-sonic speeds until barriers
halt the rupture.

The necessity of a barrier to halt every earthquake implies strict conditions
and suggests a higher frequency of large earthquakes, given that barriers may
not be present at all quake locations. However, the prevalence of small
earthquakes over large ones in reality hints at a tendency for earthquakes to
self-arrest rather than being stopped by barriers. This prompted detailed
elastodynamical rupture simulations using the boundary integral equation
method, revealing that, under a broad range of parameters, earthquakes on
homogeneous faults can self-arrest after initial propagation, i.e., stop
spontaneously without barriers [1, 2]. In addition to numerical
investigations, field observations further validate the existence of self-
arresting ruptures [3, 4, 5].

Despite evidence supporting the existence of these seismic events, the
underlying physical mechanisms responsible for their occurrence remain
elusive. Here, we offer a quantitative physical explanation for the existence
of self-arresting ruptures across diverse mechanical fault properties,
adapting the thermodynamic Griffith fracture theory to frictional sliding.
This approach is inspired by classical nucleation theory, traditionally used
to describe the spontaneous emergence of a new thermodynamic phase from a
metastable state. Different from the term “nucleation” commonly used in the
context of earthquakes, our derivation suggests a need to update our
understanding of earthquake nucleation through the lens of statistical
thermodynamics.

The standard picture of an earthquake is that it arises from an irreversible
process, beginning with slow creep or sliding, followed by a gradual
acceleration of sliding and/or damage. Ultimately, this leads to the
elastodynamic rupture of the earthquake, which is perceived as the outcome of
an irreversible sliding instability. In the literature, earthquake nucleation
refers to the initial process that leads to an earthquake. It may be
associated with one or several of the following processes: a gradual increase
of stress and deformation across fault networks, progressive damage and creep
processes, evolution from stable frictional sliding on the future fault
rupture surface, various possible poro-hydro-thermal-chemical-mechanical
processes in fault zones and so on. While there is a large variety of
descriptions, earthquake nucleation is systematically thought of as an
irreversible maturation development towards a new regime where the stored
elastic energy is released suddenly, causing the fault to slip along sliding
zones with boundaries propagating at velocities close to Rayleigh wave
velocities, thus generating seismic waves perceived as an earthquake.

How the concept of earthquake nucleation is used in seismological literature
is exemplified by the work of Ohnaka and Kuwahara who defined the nucleation
zone as the (transition) zone where a crack grows stably with accelerating
speeds from the point where the slip failure nucleus is formed to the critical
point, beyond which the crack grows unstably at a speed close to sonic
velocities [7]. To account for their laboratory experimental data, they
envision four stages, (i) static formation of the slip failure nucleus, (ii)
subsequent quasistatic steady crack growth defined as stable rupture caused by
further increase in the applied external stress, (iii) stable, but
accelerating crack growth up to a critical crack length, (iv) beyond which
instability begins to occur. This nucleation picture is further supported by
recent experiments [8]. Beroza and Ellsworth called “seismic nucleation phase”
the initial regime of low moment rate relative to the rest of the earthquake
rupture [9]. Abercrombie and Mori analyzes the first three seconds of the
Landers earthquake for information concerning its “nucleation” and early
growth [11]. Ide suggested that the initial onsets of earthquakes may be
related to the nucleation process, believed to be initiated before the dynamic
rupture process in rocky materials [12]. One perspective is that there is a
‘pre-slip’ nucleation process before the development of the full rupture
dynamics radiating seismic waves, during which a confined portion of the
fault, referred to as the nucleation zone, undergoes a period of slow aseismic
and transiently stable sliding. This sliding progressively accelerates until
the slipping patch reaches a “critical size” [7, 13, 14, 15, 16]. In Ohnaka’s
theory, the “critical size” is the length beyond which the slip in the
nucleation zone transitions into an accelerated unstable regime leading to a
fast sub-Rayleigh or supershear rupture. Ohnaka argued that the nucleation
process is itself an immediate earthquake precursor [17] and proposed to infer
the “critical size” of the nucleation zone from the immediate foreshock
activity [14]. Similar concepts pervade the interpretation of laboratory
stick-slip experiments [8, 18], as well as in theoretical and numerical
studies examining the spatial and temporal characteristics of foreshocks and
nucleation processes [19, 20, 21, 22, 23]. A general loose view is to define
the nucleation stage as the time period during which the fault slip velocity
increases slowly from a background rate to a higher slip rate [18] leading to
a transition into an accelerated and unstable regime, ultimately resulting in
the rapid sub-Rayleigh or supershear rupture.

However, the above smooth nucleation paradigm has been suggested to be
relevant only to unnaturally homogeneous faults [16]. Alternative models of
earthquake initiation, such as cascades and accumulation of small stochastic
events may be more appropriate. In this alternative view, seismic nucleation
involves the accumulation of small stochastic events, which ultimately leads
to the largest sub-event of the earthquake. This physical picture is commonly
developed in ‘cascade’ models of sequential or interconnected patterns of
earthquake events, where the energy released by one earthquake serves as a
stress perturbation that can trigger another earthquake on a nearby fault
segment [24, 25, 26, 11, 27]. In this spirit, Brune discussed a triggering
rupture propagation model and focused on the rupture initiations [28].
Examples of cascades models include also self-organised critical models [29,
30, 31].

The standard concept is thus that aseismic creep or stochastic small
instabilities are an intrinsic part of the nucleation process of earthquakes.
In other words, earthquakes are considered to be preceded by a preparatory
stage called earthquake nucleation that leads to irreversible dynamical
ruptures.

While the above physical pictures have contributed to the understanding of
earthquakes, there are still important open questions. In particular, the
discovery of self-arresting earthquakes through numerical simulations and
empirical analyses [1, 2, 3, 4, 5] reveals that elastodynamic ruptures may
spontaneously come to a stop in the absence of barriers. This finding
challenges the longstanding belief that earthquake rupture is irreversible,
revealing the process of earthquake nucleation to be more complex than
previously understood.

We now propose a simple theory for the existence of self-arresting earthquakes
based on a reasoning analogous to that used in the classical theory of
nucleation in statistical thermodynamics. In this way, we propose an extension
of the concept of nucleation as used in seismology. Our reasoning also
parallels Griffith’s fundamental principle, which forms the foundation of
modern linear elastic fracture mechanics. Griffith’s theory is based on
comparing the decrease in potential energy that accompanies fracturing and the
increase in surface energy resulting from the formation of new free surfaces
as a crack expands [32]. Rapid fracture initiation, as described by the
Griffith criterion, happens when the bulk potential energy released by the
growth of an existing crack exceeds the energy required for fracture.

In order to understand the possibility for self-arresting earthquake ruptures,
we take inspiration from the analogy between rupture sliding nucleation and
the classical Gibbs theory of droplet nucleation [33, 34, 35]. Our formulation
contains physical ingredients similar to those entering the calculation of the
minimum patch for a rupture to expand in the case of a 2-D in-plane fault
[36], for a circular fault [37] and for the rupture propagation along a long
strike-slip fault and along a rectangular asperity [38]. In particular,
Madariaga and Olsen [38] obtain a phase diagram similar to that shown in
figure 1 below. However, the boundary they report is between subshear
propagating ruptures and a parameter domain coined “does not rupture”, instead
of the “self-arresting” rupture domain in our case. They did not consider the
scenario of “stopping after rupture broke the asperity” as an independent
seismic event but rather considered it a failure in the simulation. This
interpretation was based on the belief that earthquakes must reach a critical
nucleation size and undergo an unstable dynamic propagation phase. In
contrast, by taking seriously the possibility of genuine rupture propagation
followed by self-arrest, Xu et al. [1] and Wei et al. [2] discovered that, in
a large domain of parameter space, an earthquake nucleation along an
homogeneous fault can lead to an earthquake that self-arrests.

Based on an analogy with classical Gibbs theory of droplet nucleation, in
which droplets continuously form, grow and evaporate when not reaching the
critical radius, our physical picture is based on the existence of asperity
patches of all possible sizes on which sliding “nucleates” but then self-
arrests when their initial size is smaller than a critical value derived
below. Hence, the critical bifurcation described by Madariaga and Olsen [38]
should be interpreted as one between transiently growing ruptures that self-
arrest (similarly to a droplet that first grows and evaporates before
nucleating the macroscopic liquid state) and the unstoppable (in the absence
of barriers) sub-Rayleigh earthquakes.

The free energy $F(R)$ associated with the nucleation of a sliding event
occurring over a disk of radius $R$ can be decomposed into two terms: (a) the
decrease of free energy $F_{s}(R)$ resulting from the stress drop induced by
the sliding event and (b) the work $F_{f}$ against friction (analogous to the
creation of interfaces in crack or droplet nucleation). $F_{s}(R)$ can be
evaluated as

$F_{s}(R)=-{1\over 2}\epsilon_{e}T_{e}\cdot\eta{4\pi\over
3}R^{3}~{}~{}~{}~{}{\rm where}~{}\epsilon_{e}={T_{e}\over\mu}~{}.$ (1)

$\epsilon_{e}$ is the deformation associated with the dynamical stress drop
$T_{e}$ in a medium of shear modulus $\mu$ and the corresponding energy
density gain is ${1\over 2}\epsilon_{e}T_{e}$ (energy per unit volume). The
term $\eta{4\pi\over 3}R^{3}$ represents an estimation of the volume over
which the elastic energy drop occurs in the nucleation sliding domain. This
volume is the volume of a sphere of radius equal to the nucleation patch times
a geometrical coefficient $\eta$ of order unit. In the case of 2-D in-plane
fault [36], $\eta=\pi(1-\nu)/2\approx 1.18$ for a typical value for the
Poisson modulus $\nu=0.25$. For a circular fault [37], $\eta=12/(7\pi)\approx
0.546$. For an anti-plane rupture [39], $\eta=\pi/2\approx 1.57$. Madariaga
and Olsen [38] report values corresponding to $\eta$ between $0.625$ and
$0.83$.

The work $F_{f}$ against friction is the integral over the sliding
displacement of the frictional force, itself obtained as the product of the
frictional stress times the surface of the nucleating domain. In the
calculations of [1, 2], the frictional stress is taken to decrease linearly as
a function of slip over the critical distance $D_{c}$ and then crosses over to
a constant level $\sigma_{r}$ for $D>D_{c}$. For $D>D_{c}$, we thus obtain

$F_{f}=\pi R^{2}\left(\sigma_{r}D+{1\over 2}T_{u}D_{c}\right)~{},$ (2)

where $T_{u}$ is the breakdown stress drop. Since we are interested in the
nucleation regime $D\approx D_{c}$, and given that $\sigma_{r}\ll T_{u}$, we
can approximate expression (2) as

$F_{f}\simeq{1\over 2}\pi R^{2}~{}T_{u}D_{c}~{},$ (3)

The total free energy of a nucleation disk of size $R$ is thus

$F(R)=-{1\over 2}{T_{e}^{2}\over\mu}\cdot\eta{4\pi\over 3}R^{3}+{1\over
2}T_{u}D_{c}\cdot\pi R^{2}~{}.$ (4)

$F(R)$ first grows as $R$ increases as the work against friction dominates and
then decreases without bounds beyond a critical radius $R^{c}$ given by

${dF(R)\over dR}_{|R=R^{c}}=0~{}\to R^{c}={1\over 2\eta}{T_{u}\over
T_{e}}{\mu\over T_{e}}D_{c}~{}.$ (5)

The total free energy (4) thus predicts that a sliding event occurring over a
disk of radius $R$ will self-arrest if its radius $R$ is smaller than the
critical value $R^{c}$ given by expression (5). On the other hand, for
$R>R^{c}$, the sliding event grows without bounds until it encounters a
sufficiently strong barrier. This is similar to the classical theory for crack
or droplet nucleation. Our derivation in terms of a free energy for the
nucleation of a sliding disk rationalises and interprets the existence of the
“self-arresting ruptures” as being the equivalent to the stable nucleating
cracks or droplets in classical nucleation theory.

Figure 1: Phase diagram in the plane $[{\hat{D}}_{c};{\hat{T}}_{e}]$, defined
by expressions (6), taken from [2] on which the curve
${\hat{T}}_{e}=0.5\sqrt{{\hat{D}}_{c}}$ is superimposed (dashed red line) as
obtained from the derivation leading to equation (7) with $\eta=2$.

By systematically investigating the influence of the parameters of the slip-
weakening law and the size of nucleation asperity on the dynamic rupture of a
planar fault using the boundary integral equation method, Xu et al [1] and Wei
et al. [2] have constructed a phase diagram of the different possible types of
rupture along homogenous planes: (i) subshear (or sub-Rayleigh for strike-slip
rupture), (ii) supershear, (iii) self-arresting rupture. The phase diagram is
defined in terms of the two dimensionless parameters

${\hat{T}}_{e}:={T_{e}\over
T_{u}}~{}~{}~{},~{}~{}~{}{\hat{D}}_{c}:={D_{c}\over R}{\mu\over T_{u}}$ (6)

where $T_{u}$ is the breakdown stress drop, $T_{e}$ is the dynamic stress
drop, $D_{c}=0-3$mm ($10^{-3}m$) denotes the critical slip distance, $R$ is
the effective radius of the nucleation asperity, and $\mu\approx 30~{}GPa$ is
the shear modulus of the media. Using the variables ${\hat{T}}_{e}$ and
${\hat{D}}_{c}$ thus defined, expression (5) for the critical radius $R^{c}$
separating the self-arresting ruptures from the sub-Rayleigh events reads

${\hat{T}}_{e}=\sqrt{{\hat{D}}_{c}\over 2\eta}~{},$ (7)

and is shown as the dashed red line in figure 1 for $\eta=2$.

We now derive the Gutenberg-Richter distribution for the self-arresting
earthquakes with radius smaller than $R^{c}$ given by expression (5), by
linking the fractal nature of faulting with the nucleation physics through the
critical size’s dependence on dynamical stress drop. In the real Earth crust,
$R^{c}$ is not a constant but is expected to be inhomogeneous distributed in
space since the elastic modulus $\mu$, the breakdown stress drop $T_{u}$, the
dynamic stress drop $T_{e}$ and the critical sliding distance $D_{c}$ are
material dependent and are expected to vary from place to place. Given that
$R^{c}$ is inversely proportional to the square of $T_{e}$, the largest
contribution to the variations of $R^{c}$ at large values comes from the
variations of the dynamic stress drop $T_{e}$ at small values. Let us assume
that $T_{e}$ is distributed according to some spatial distribution $p(T_{e})$,
which is regular for small $T_{e}$. The simplest forms are either a constant
or an exponential distribution for its probability density function (pdf) for
small $T_{e}$’s and give the same asymptotic result and the same Gutenberg-
Richter distribution. Let us take $p(T_{e})={1\over T_{0}}e^{-{T_{e}\over
T_{0}}}$, which is normalised to $1$ for $T_{e}$ spanning from $0$ to infinity
(this upper boundary is of no consequence for the distribution of $R^{c}$ at
intermediate and large values). Expression (5) together with the exponential
form of $p(T_{e})$ leads to the pdf of $R^{c}$ given by

$p(R^{c})=\sqrt{{1\over 2\eta}{T_{u}\over T_{0}}{\mu\over
T_{0}}D_{c}}~{}e^{-{1\over T_{0}}\sqrt{{T_{u}\mu D_{c}\over 2\eta
R^{c}}}}~{}{1\over(R^{c})^{3\over 2}}~{}.$ (8)

This implies $p(R^{c})\propto{1\over(R^{c})^{3\over 2}}$ for
$R^{c}\geq{T_{u}\mu D_{c}\over 2\eta T_{0}^{2}}$. Consider a fault network of
span $L$. The number of independent patches of size $R$ on this fault network
is proportional to $\left({L\over R}\right)^{D_{f}}$, where $D_{f}$ is the
fractal dimension of the fault network. Thus the total number of nucleation
domains that are in the self-arresting rupture regime is the sum over all
possible critical radii $R^{c}$ weighted by its pdf:

$p(R)=\left({L\over
R}\right)^{D_{f}}\int_{R}^{+\infty}dR^{c}~{}p(R^{c})\simeq{1\over
R^{D_{f}+{1\over 2}}}~{}.$ (9)

The lower bound in the above integral expresses the fact that a given asperity
of size $R$ is in the self-arresting rupture regime if it is smaller than its
local critical radius. The pdf of the disk radius $R$ translates into the pdf
of seismic moment $M_{0}$ using the standard scaling relation $M_{0}\propto
R^{3}$. Using the conservation of probabilities under a change of variable
$p(R)dR=p(M_{0})dM_{0}$ yields

$p(M_{0})\simeq{1\over M_{0}^{1+{D_{f}-{1\over 2}\over 3}}}~{}.$ (10)

Kagan and Knopoff [40] report a value $D_{f}=2.2$, which should be taken as a
lower bound since it is obtained from the two-point correlation function of
the distribution of earthquakes. Davy et al. [41] give a value
$D_{f}=1+1.7=2.7$ from analog laboratory experiments of the India-Asia
collision. Sornette [42] reviews a series of measurements by various authors
that give $D_{f}$ in the range $2.5-2.7$. Taking $D_{f}=2.5$ recovers exactly
the $2/3$ exponent of the Gutenberg-Richter distribution of seismic moments,
equivalent to the $b$-value equal to $1$ when expressed in terms of
magnitudes. This derivation informs that the Gutenberg-Richter distribution
holds for self-arresting earthquakes. In the present framework, the Gutenberg-
Richter distribution derives from the combination of the fractal nature of
faulting and the physics of earthquake nucleation via the dependence of the
critical nucleation size as a function of the dynamical stress drop.
Variations of the $b$-value could be associated with different fault network
properties with distinct fractal dimensions.

Interpreting self-arresting ruptures as analogous to stable nucleating cracks
or droplets in classical nucleation suggests to consider them as similar to
how droplets cease growing in classical thermodynamical nucleation theory.
Conversely, if a nucleation patch exceeds a critical size, the sliding rupture
escalates uncontrollably until encountering a substantial barrier. This
mirrors the unstable nucleation of a new phase in classical theory, indicating
a coexistence of self-arresting stable ruptures and runaway unstable ruptures.
The run-away unstable ruptures (the subshear and supershear events) are
subjected to positive feedbacks, i.e., as the rupture develops, more elastic
energy is released that further propels the rupture and the frictional forces
are unable to stop the propagating front. A dynamical balance is reached via
the additional dissipation mechanism involving seismic radiation. In contrast,
for self-arresting ruptures, friction together with seismic radiation combine
as dissipation mechanisms to stop the rupture. This shows that the two types
of ruptures have different mechanisms and should be indeed in different
classes. We propose that the run-away unstable ruptures are instantiations of
the concept of dragon-kings [43, 44], special events that are both extremely
large (“king”), and born of unique origins (“dragon”) relative to its peers.
The run-away unstable ruptures (the subshear and supershear events) that are
stopped only by strong geometrical barriers are the dragon-kings while the
sub-critical self-arresting ruptures are the “normal” events, distributed
according to the standard Gutenberg-Richter distribution (10).

The run-away unstable ruptures (the subshear and supershear events) are
stopped only by strong geometrical barriers. The distribution of their
magnitudes can be expected to be different, more associated with geometric
features of faults and fault networks. This is in agreement with the theory of
dragon-kings [43, 44] that predicts that the distribution of the dragon-kings
are in general different from the distribution of the smaller normal events.
Indeed, within the nucleation picture developed above, the dragon-king
subshear and supershear earthquakes should be rarer than the self-arresting
earthquakes. They should also be larger overall because only barriers can stop
them and their sizes are thus determined by the likelihood that a barrier is
present once an unstable rupture has been nucleated.

The classification into several classes of earthquakes has implications for
the interpretation of many physical and statistical properties of earthquakes.
It remains to be seen how this novel understanding could be related to the
origin of the two-branched Gutenberg-Richter distribution [45] and the fact
that large earthquakes (typically above magnitude 6 in California) may be
preferentially triggered by large earthquakes [46].

In reviewing earthquake nucleation, we noted that the smooth nucleation model
might only apply to unnaturally uniform faults. Drawing parallels with
heterogeneous nucleation theory, our theory can in principle be extended to
heterogeneous faults, by considering that the friction work $F_{f}$, akin to
creating interfaces in nucleation, could vary greatly within irregular faults.

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