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Review
Cite this article: Vallianatos F, Papadakis G,
Michas G. 2016 Generalized statistical
mechanics approaches to earthquakes and
tectonics. Proc. R. Soc. A 472: 20160497.
http://dx.doi.org/10.1098/rspa.2016.0497
Received: 20 June 2016
Accepted: 7 November 2016
Subject Areas:
geophysics
Keywords:
statistical mechanics, Tsallis entropy, scaling,
earthquakes, faulting, complexity
Author for correspondence:
Filippos Vallianatos
e-mail: [email protected]
Generalized statistical
mechanics approaches to
earthquakes and tectonics
Filippos Vallianatos, Giorgos Papadakis and
Georgios Michas
UNESCO Chair on Solid Earth Physics and Geohazards Risk
Reduction, Laboratory of Geophysics and Seismology,
Technological Educational Institute of Crete, Chania, Greece
FV, 0000-0002-4600-5013
Despite the extreme complexity that characterizes
the mechanism of the earthquake generation process,
simple empirical scaling relations apply to the
collective properties of earthquakes and faults in
a variety of tectonic environments and scales. The
physical characterization of those properties and
the scaling relations that describe them attract a
wide scientific interest and are incorporated in the
probabilistic forecasting of seismicity in local, regional
and planetary scales. Considerable progress has been
made in the analysis of the statistical mechanics of
earthquakes, which, based on the principle of entropy,
can provide a physical rationale to the macroscopic
properties frequently observed. The scale-invariant
properties, the (multi) fractal structures and the
long-range interactions that have been found to
characterize fault and earthquake populations have
recently led to the consideration of non-extensive
statistical mechanics (NESM) as a consistent statistical
mechanics framework for the description of seismicity.
The consistency between NESM and observations
has been demonstrated in a series of publications
on seismicity, faulting, rock physics and other
fields of geosciences. The aim of this review is
to present in a concise manner the fundamental
macroscopic properties of earthquakes and faulting
and how these can be derived by using the
notions of statistical mechanics and NESM, providing
further insights into earthquake physics and fault
growth processes.
2016 The Author(s) Published by the Royal Society. All rights reserved.
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1. Introduction
...................................................
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Fracturing processes giving rise to subsequent earthquakes is a complex phenomenon, such that
the definition of the exact physics and the forecasting of upcoming events still represent an
outstanding challenge. The complexity of the earthquake activity is manifested in the wide range
of spatial and temporal scales that are incorporated in the process. Earthquake ruptures occur on a
complex network of fractures and faults that vary from microcracks, to major fault zones and plate
boundaries, whereas the temporal scales that are incorporated in the process vary from seconds
(during dynamic rupture), to the hundreds, thousands and millions of years that characterize
the repeat times of characteristic earthquakes and the evolution of fault zones and tectonic plate
boundaries, respectively [1,2]. Although, given a large set of seismographic records, the history of
fault rupture can be reconstructed a posteriori, the physical processes that lead to the initiation
and propagation of a rupture through a fault system, giving rise to an earthquake, is really
limited.
However, despite the extreme complexity that characterizes the mechanism of the earthquake
generation process, simple general laws seem to describe the collective properties of seismicity
and faulting. The well-established Gutenberg–Richter (GR) scaling relation indicates scale
invariance in the frequency of the dissipated seismic energies, while the Omori–Utsu scaling
relation implies that the aftershock production rate decays as power law with time [3].
Furthermore, the earthquake frequency–magnitude relationship is found to be applicable over
a wide range of earthquake sizes both globally and locally and even down to the laboratory scale
[4–6]. Short- and long-term clustering effects and scale invariance have also been exhibited in the
temporal evolution of seismicity [7]. In addition, earthquakes exhibit fractal spatial distribution
of epicentres and they occur on fractal-like structure of faults [8]. All these properties are strong
observational evidences implying nonlinear dynamics in the earthquake generation process [9].
Although the collective properties of earthquakes and faulting can be reproduced by statistical
models, the fundamental challenge in earthquake physics is to understand the transition from
the microscopic scale and the laws that govern friction, rupture propagation, chemical reactions,
fluid migration and so on, to the macroscopic scale of large earthquakes, fault networks and
tectonic plate boundaries [10]. Towards this direction, considerable progress has been made in
the statistical mechanics approach, which, based on the entropy principle, can be used to estimate
the macroscopic properties of earthquakes and faulting from the specification of the relevant
microscopic constituents and their interactions. In many studies, the classic statistical mechanics
approach due to Boltzmann–Gibbs–Shannon has been used to describe the collective properties
of earthquake populations (e.g. [2,11]).
However, typical complex systems whose elements are strongly correlated, i.e. the probability
of a certain microstate occurring depends strongly on the occurrence of another microstate,
violate Boltzmann–Gibbs (BG) statistical mechanics. Such systems, instead of the Boltzmann
distribution, typically present power-law behaviour, enhanced by (multi-) fractal geometries,
long-range interactions or large fluctuations between the various possible states; properties that
seem to correspond well to the phenomenology of earthquakes. For the statistical mechanics
description of such systems, a generalized framework has been introduced by Tsallis [12], known
as non-extensive statistical mechanics (NESM). NESM generalizes BG statistical mechanics and
its main advantage is that it considers all-length scale correlations among the elements of the
system, leading to broad distributions with power-law asymptotic behaviour. The NESM concept
has found so far many applications in nonlinear dynamic systems, including earthquakes (Tsallis
[13] and references therein). In a series of publications during the last decade, it has been
demonstrated that NESM can successfully reproduce the macroscopic properties of earthquakerelated phenomena from the laboratory scale (e.g. [14,15]), to regional (e.g. [16–18]) and global
scale (e.g. [19]).
The aim of this study is to review the fundamental properties of fault and earthquake
populations and to demonstrate in a concise manner how these properties can be derived by using
aspects of statistical mechanics and NESM. We focus on a variety of tectonic environments and
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scales and discuss the insights that are gained in the analysis of fault and earthquake populations,
with specific interest in earthquake and fracture physics and seismic hazard assessment.
(a) Phenomenology of fault and earthquake populations
(i) Scaling properties of seismicity
One of the most well-known empirical scaling relations in geophysics is the GR scaling relation
[20], which describes the frequency of earthquake magnitudes in a given region and takes the
form:
log N(>M) = a − bM,
(2.1)
where M represents the earthquake magnitude, N(>M) is the number of earthquakes with
magnitude greater than M, log is the logarithm to the base of 10 and a and b are positive constants.
The constant a represents the regional level of seismicity and b, commonly known as the seismic
b-value, is the slope of the distribution that estimates the proportion of small to large earthquakes
(figure 1). The G-R scaling relation has been found to describe the frequency-size distribution of
earthquakes in various seismic regions, tectonic environments and scales, with b-values that are
generally close to one [22,23], although regional variations may appear (e.g. [24]).
The G-R scaling relation can alternatively be written as
N(>M) = 10a−bM ,
(2.2)
which expresses a power-law dependence between the number of earthquakes N and the
magnitude M. Furthermore, the earthquake energy E (in terms of the seismic moment Mo ) is
related to the earthquake magnitude M as
log E = cM + d,
(2.3)
with a global average of c = 1.5 [25,26]. Equation (2.2) can then be written in terms of the seismic
energies E as
N(>E) ∼ E−β−1 ,
(2.4)
with β = 23 b. The latter expression has been considered as indication of scale invariance in the
distribution of earthquake magnitudes or seismic energies. In addition, the dissipated seismic
energy E, in terms of the seismic moment Mo , is related to the surface area of a fault with the
relationship Mo = μA, where μ is the rigidity of the material, A the surface area of the fault
break and the average slip during the earthquake [25]. According to the previous definition of
Mo and equation (2.4), the number of earthquakes N(>A) with rupture areas greater than A has a
power-law dependence on the area A (e.g. [8]). The latter implies that the empirical G-R scaling
relation is equivalent to a fractal distribution [27].
In the distribution of larger earthquake magnitudes, an upper bound or a taper may appear
that is associated with the finite energy release rates [28]. This upper bound limits the number
of the maximum expected earthquake magnitudes and has been modelled by modifying the G-R
scaling relation to include an exponential tail or a taper [29]. The modified G-R (MGR) scaling
...................................................
The collective properties of fault and earthquake populations exhibit some universal
characteristics, in the sense that these are appearing in various tectonic environments and scales
that may vary from the laboratory microscale to major fault zones and tectonic plate boundaries. It
is exactly these properties that motivate the statistical mechanics approach to earthquake physics.
In the following, the main properties of earthquakes and faulting and the classic statistical
mechanics approaches that have been developed to address them are briefly reviewed.
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2. Statistical mechanics and the phenomenology of earthquakes and faulting
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4
102
10
1
4.0
4.5
5.0
5.5
6.0
magnitude (M)
6.5
7.0
Figure 1. Cumulative number of earthquakes (depths ≤ 40 km) in the area of Greece (34°N – 42°N, 19°E – 29°E) for the period
1976–2009 (earthquake catalogue after Makropoulos et al. [21]). The dashed line represents the G-R scaling relation (equation
(2.2)) for the values of a = 8.25 ± 0.15 and b = 1.13 ± 0.02.
relation then takes the form of a gamma distribution:
E
−β
,
N(>E) ∼ E exp −
Ec
(2.5)
which exhibits an exponential tail for earthquake energies (seismic moments) greater than a
characteristic or ‘corner’ seismic energy Ec .
Regarding the spatio-temporal properties of seismicity, the most prominent feature is the
presence of clustering, where correlated earthquake sequences exhibit scale-free structures
[30]. Spatial clustering is exemplified by the concentration of seismicity along the tectonic
plate boundaries and regional fault networks (e.g. [1]). Temporal clustering is best observed
in aftershock sequences, where a significant increase in the regional seismicity rate occurs
immediately after the occurrence of a strong earthquake. The aftershock production rate
n(t) = dN(t)/dt (where N(t) is the number of earthquakes in time t after the major event) has
empirically been found to decay as a power law with time t according to the modified Omori
formula [3]:
n(t) = K(t + c)−p ,
(2.6)
where K and c are constants and p is the power-law exponent. The modified Omori formula
expresses a short-term clustering effect that can be seen in almost every seismic catalogue [31].
However, a truncation of the power-law regime may appear at short time-scales following the
main shock that can be attributed either to missing recorded events immediately after the main
shock or to a real non power-law regime (e.g. [32]).
The background activity of main earthquakes in a seismic region is often considered as
uncorrelated and statistically independent in time and models that express randomness, like the
general Poisson model, have been used to describe its temporal properties [33]. A Poissonian
background activity combined with triggered aftershocks that scale according to the modified
Omori formula can be realized by the distribution of inter-event times τ , i.e. the time intervals
between the successive earthquakes in a seismic region, which frequently takes the form of a
gamma distribution (e.g. [34,35]):
−τ
,
(2.7)
f (τ ) = Cτ γ −1 exp
τ0
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cumulative number N (>M)
104
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Faults are key components in the dynamic evolution of the Earth’s lithosphere and surface and are
the sites of large earthquakes. Crustal strain is accommodated in systems of faults and fractures
that exhibit complex geometries and sizes that vary from a few millimetres to tens or hundreds
of kilometres [1]. Fault systems appear self-similar in a wide range of scales, indicating scale
invariance and the absence of a characteristic length scale in the fault growth process [11,49,50].
Scale invariance in fracturing processes is also supported by the frequency-size distribution of
earthquakes that scales according to the G-R relation (equation (2.1)), which represents a powerlaw relationship between the number of earthquakes and the rupture area (see equations (2.1)–
(2.4) and the previous section). However, even if scale invariance does hold, it is restricted to a
finite range of scales that are associated with the size of the fractured medium.
In support to scale-invariant fault growth is the power-law scaling of fault lengths L found in
numerous studies and in various tectonic environments and scales (e.g. [51–53]). In terms of the
cumulative distribution of fault trace-lengths, the power-law relationship is expressed as
N(>L) = AL−D ,
(2.8)
where N(>L) is the number of faults with lengths greater than L, A is a scaling constant and D the
power-law exponent.
In other case studies though, the exponential function has been found to best describe the
cumulative number of fault lengths N(>L). In this case, the exponential function is expressed as
N(>L) = A exp
−L
,
L0
(2.9)
where A is a scaling constant and L0 a characteristic length scale that may reflect some physical
parameter in the fault system, such as the thickness of the brittle layer [50]. This type of scaling
in fault-length distributions has been observed in mid-ocean ridges [54], in the Turcana Rift
(Northern Kenya) [55] and in high-strain zones in the Corinth Rift [56] and the Afar Rift [57].
The power-law frequency-size distribution and the self-similar structure across the wide range
of scales have been considered as strong indications of fractal geometries in fracture systems
(e.g. [8]). Fractal geometry has been used as a first-order approximation to describe the complex
patterns of fault systems and physical models on fractal fault growth have been developed
[49,58,59]. Cowie et al. [58] used a two-dimensional numerical model to show how fractal fault
patterns emerge as the result of correlations in the fault network, induced from both shortand long-range elastic interactions in a heterogeneous material (random heterogeneity) that is
deformed under constant velocity (see also [59]). Cowie et al. [60] used a similar model to show
that as deformation progresses and strain localizes on a few large faults, a multifractal structure
...................................................
(ii) Scaling in fault populations
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where C, γ and τ 0 are positive constants. The latter expression exhibits two regions, where
short inter-event times, associated with short-term clustering effects and aftershock sequences,
scale as a power law with exponent 1 − γ , while long inter-event times that are associated
with the background activity exhibit Poissonian behaviour and exponential scaling. This type of
scaling has been found in stationary earthquake time series [35–37] and is approximately the one
predicted by the epidemic-type aftershock sequence model [35,38,39]. However, in non-stationary
earthquake time series long-term clustering effects and power-law scaling have been found to
characterize long inter-event times [7,34,40,41]. The latter indicates clustering in both short and
long inter-event times and implies memory effects in the earthquake generation process [42,43].
In addition, properties like non-stationarity and intermittency in earthquake time series are
indicative of heterogeneous clustering effects that are quantitatively consistent with multifractal
geometries (e.g. [44]). The multifractal structure of earthquake time series is evident in various
earthquake catalogues [45–47] and multifractal analysis is used as a second-order approximation
to enlighten the correlations of seismicity and the local clustering effects [44,48].
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Despite the extreme complexity that characterizes the generation process of individual
earthquakes, the relatively simple phenomenology (e.g. scale invariance) that seems to apply
in the collective properties of earthquakes and faulting has motivated the statistical mechanics
approach. By using the mathematical tools of probability theory and statistics, statistical
mechanics can be used to estimate the macroscopic properties of fault and earthquake populations
from the specification of the relevant microscopic constituents and their interactions [10].
Originally, statistical mechanics and the associated concept of entropy (S) were used in
thermodynamics and the kinetic theory of gases. The concept of entropy was later incorporated
into information theory as a measure of uncertainty [72], and since the mid-1950s, it has been
extended to other fields beyond classic thermodynamics in order to provide a general principle
for inferring the least biased probability distribution from limited information [73]. The latter
is commonly known as the maximum entropy principle (MaxEnt). During recent decades,
innovative insights into the macroscopic states of many physical, chemical and biological systems
have been accomplished by applying the MaxEnt hypothesis, since only the constraints of the
large-scale system need to be known, regardless of the complexity that may characterize the
different microscopic configurations (e.g. [74]).
...................................................
(b) The classical statistical mechanics approach
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emerges as the superposition of a fractal distribution of fault displacements onto a fractal fault
pattern.
In the previous models fault interactions were an intrinsic characteristic of fault growth (see
also [61,62]). Fault interactions are created as the fault system grows in size and faults start
to interact through their stress fields. According to Sornette et al. [63], brittle failure along a
fault causes the redistribution of strain perturbations in the medium that decay algebraically
with distance according to the equations of elasticity, producing interactions in the far field.
In the near field, stress interactions around larger faults, where strain localizes, produce a
combination of stress screening and enhancement effects [61,62]. In earthquake triggering, this
mechanism is quite important as it may accelerate or retard upcoming events on neighbouring
faults [64]. In fault growth, the stress perturbations around larger faults favour deformation in
some areas, while other areas remain relatively undeformed. Hence, optimally oriented faults in
stress increase zones continue to grow, while other faults in stress drop zones may cease activity.
Other geometrical or physical factors that may control fault growth and, respectively, the
scaling properties of fault systems are the finite thickness of the brittle layer [65,66] and the
rheological properties of the lithosphere [67,68]. These factors may lead to non-universal scaling
exponents and frequency-size distributions that deviate from the power law (e.g. exponential). In
addition, simulations on numerical models [60,69,70] and on analogue laboratory experiments
[65,71] showed that the frequency-size distribution depends on the total strain that the fault
system has been imposed on and the competition between the different stages of fault growth,
i.e. fault nucleation, growth and coalescence. The models indicate that at the very early stages
of deformation, where few new faults have nucleated, their spatial organization is random and
the trace-length distribution is best described by the exponential function [65,69]. As strain
accumulates, more faults are nucleated and grow in size and number and start to coalesce
forming larger faults. At this stage, where growth and coalescence start to dominate, faults
display a power-law size distribution with decreasing exponents as strain increases, indicating
the increased importance of large faults in accommodating strain [60,65]. At later stages of
deformation, strain starts to localize along few large faults that span the mechanical layer
and the number of active faults decreases. With increasing strain, the fault population reaches
saturation, fault interactions are suppressed and the size distribution turns to exponential
[65,69–71]. The transition from power-law scaling in lower strain regimes to exponential scaling
in higher strain regimes has been observed in the Afar Rift [57] and the Corinth Rift [56],
supporting the combination of crustal processes in single tectonic settings as a function of crustal
strain.
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where p(x) is the probability distribution of x and kB is Boltzmann’s constant. The most
probable state of p(x) is estimated by maximizing the entropy SBG , subject to the constraints of
normalization of p(x)
∞
0
and the expectation (or average) value of x
x =
p(x) dx = 1,
(2.11)
∞
xp(x) dx,
(2.12)
0
By using the Lagrange multipliers method, the probability distribution p(x) that maximizes
SBG is the well-known Boltzmann distribution:
p(x) = ∞
0
e−βx
.
e−βx dx
(2.13)
The term e−βx is often called the Boltzmann factor, where β is the Lagrange multiplier, and
the denominator of equation (2.13) is a normalization factor that is referred to as the partition
function
∞
Z=
e−βx dx.
(2.14)
0
BG statistical mechanics has been applied to earthquake physics in a series of works, which
address the theory as a variational principle for deriving the large-scale properties of earthquake
populations [11,75] or to investigate the thermodynamic state of the crust [76–78]. One of the
first studies that applied statistical mechanics to earthquakes was that of Berrill & Davis [79]. The
authors maximized entropy to derive a probability density function of earthquake magnitudes
p(M) that has the form of a truncated exponential distribution:
p(M) =
β
e−βM ,
1 − e−βm1
(2.15)
where m1 is the minimum magnitude in the dataset and β the Lagrange multiplier. Furthermore,
Main & Burton [75] maximized entropy using the constraints of the average magnitude m and
the average seismic energy release E to derive a gamma distribution of earthquake energies p(E),
consisting of a power-law distribution at small magnitudes that corresponds to the G-R scaling
relation (equation (2.4)) and an exponential (Boltzmann) tail at larger magnitudes:
p(E) ∼ E−β−1 e−E/θ ,
(2.16)
where β is a scaling exponent and θ a characteristic energy that reflects the probability of
occupancy of the different energy states E. Main & Burton [75] used earthquake data from the
Mediterranean region and from Southern California to show that this type of distribution is
more consistent with real observations. Moreover, Main & Al-Kindy [77] used equation (2.16)
to investigate the proximity of global seismicity to criticality, characterized by the dissipated
seismic energy E and the entropy S. The authors defined a subcritical state for positive θ and
a supercritical state for negative θ and concluded that global seismicity is in a near-critical state,
in which large fluctuations are dominated by fluctuations in θ rather than β and large seismic
energy fluctuations can occur for smaller entropy fluctuations [77].
The hypothesis that the Earth’s lithosphere is in a state of thermodynamically driven
maximum entropy production and SOC was tested by Main & Naylor [78,80]. The authors
explored this hypothesis using the Olami–Feder–Christensen (OFC) model [81] and real
...................................................
0
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According to the principles of statistical mechanics, the relative probability that the system
possesses a given state can be estimated by optimizing the entropy, subject to the limited
knowledge about the system that is expressed through the constraints. For a continuous variable
x that takes values in the space [0, ∞], the BG entropy SBG is expressed as
∞
p(x) ln p(x) dx,
(2.10)
SBG = −kB
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bmax
10H
≈
1.2
10H
,
(2.17)
with bmax = e log e ≈ 1.2. De Santis et al. [83] studied the variability of the b-value and H
during two earthquake sequences in Italy and identified three dynamic regimes with respect
to H variations: (i) a preparatory phase, where H increases slowly with time, (ii) the phase of
occurrence of a strong earthquake, where H exhibits an abrupt increase after the occurrence of
the main shock, and (iii) a final diffuse phase, where H recovers ‘normal’ values and seismicity
spreads all over the region.
3. A non-extensive statistical mechanics approach
(a) The Tsallis entropy Sq
BG statistical mechanics seems the correct one to be used in a large and important class of
physical systems that present strongly chaotic dynamics (positive maximal Lyapunov exponent).
However, there is an important class of weakly chaotic systems (where the maximal Lyapunov
exponent vanishes) that violate some of the essential properties of BG statistical mechanics
[13,84–86]. The macroscopic behaviour of such systems, instead of the Boltzmann distribution,
typically presents power-law distributions with heavy tails, enhanced by (multi-) fractal
geometries, long-range interactions, intermittency or large fluctuations between the various
possible states; properties that seem to correspond well to the phenomenology of earthquakes. For
such systems, where BG statistical mechanics has limited applicability, a generalized framework
has been introduced by Tsallis [12], termed as NESM. NESM generalizes BG statistical mechanics
and its main advantage is that it considers all-length scale correlations among the elements of the
system, leading to broad distributions with power-law asymptotic behaviour.
Central to NESM is the non-additive (Tsallis) entropy Sq that for the discrete case is expressed
as [12]
q
1− W
i=1 pi
, (q ∈ R),
(3.1)
Sq = k
q−1
with
W
pi = 1,
(3.2)
i=1
where k is some conventional positive constant taken to be Boltzmann’s constant in thermostatistics, pi is a set of probabilities, W is the total number of microscopic configurations, and q
is the entropic index that represents a measure of non-extensivity. Tsallis entropy (Sq ) recovers
the BG entropy (SBG ) in the limit q → 1. Although Sq shares a lot of common properties with SBG
(e.g. non-negativity, expansibility, concavity; see Tsallis [13] for the full list of these properties),
SBG is additive whereas Sq (q = 1) is non-additive. According to this property and for two
B
probabilistically independent systems A and B (i.e. pA+B
= pA
i pj (∀ (i, j)), SBG satisfies
ij
SBG (A + B) = SBG (A) + SBG (B),
(3.3)
...................................................
b=
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seismicity data and concluded that their observations were consistent with the hypothesis of
entropy production as the driving mechanism for self-organized subcriticality in natural and
model seismicity [80].
In other studies, Shannon entropy (H) was used as a measure of disorder in earthquake
sequences [82,83]. Telesca et al. [82] estimated H with time for earthquake magnitudes and interevent time series in Umbria–Marche region (central Italy) and found significant variations that
correspond to the largest event in the series. De Santis et al. [83] related H to the G-R scaling
relation and derived a relationship between H and the b-value that reads as
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whereas Sq satisfies:
9
The last term on the right-hand side of equation (3.4), which introduces the concept of nonadditivity, is the fundamental principle of NESM [13]. Moreover, the cases q < 1, q = 1 and q > 1
correspond to super-additivity (super-extensivity), additivity (extensivity) and sub-additivity
(sub-extensivity), respectively. However, if the systems A and B are correlated, then a q-value
might exist such that Sq (A + B) = Sq (A) + Sq (B). In this case, Sq is extensive for q = 1 [13].
Let us now consider a continuous variable X with probability distribution p(X) (0 ≤ p(X) ≤ 1).
The X variable in earthquake physics may represent the seismic moment (Mo ), the inter-event
times (τ ) or distances (r) between successive earthquakes or the fault trace-lengths (L) in a given
region. In this case, the non-additive entropy Sq is expressed through the integral formulation:
∞
1 − 0 pq (X) dX
.
(3.5)
Sq = k
q−1
To obtain the distribution p(X) that optimizes Sq , subject to constraints, the Lagrange
multipliers method is applied. The first constraint refers to the normalization condition of p(X):
∞
p(X) dX = 1.
(3.6)
0
The second constraint is the condition regarding the generalized expectation value (qexpectation value) Xq , which is defined as
∞
Xq = Xq =
XPq (X) dX,
(3.7)
0
where Pq (X) is the escort probability distribution [13,87] that is defined as
Pq (X) = ∞
0
pq (X)
,
pq (X)dX
∞
with
0
Pq (X) dX = 1.
(3.8)
In the limit q → 1, Xq recovers the standard average value of X, X. The concept of escort
distributions [88] refers to a set of probability distributions which scan the structure of an original
probability distribution. The later one is related to the (multi-) fractal features of nonlinear
dynamic systems [87]. Using the Lagrange multipliers method and the previous constraints, the
following functional is optimized:
∞
p(X) dX − β ∗ Xq ) = 0,
(3.9)
δ(Sq − a
0
where α and β* represent the Lagrange multipliers. Optimization of equation (3.9) yields the
optimal physical probability
p(X) =
[1 − (1 − q)βq X]1/(1−q) expq (−βq X)
=
.
Zq
Zq
The denominator of equation (3.10) is the q-partition function defined as
∞
expq (−βq X) dX.
Zq =
(3.10)
(3.11)
0
The term β q is associated with the Lagrange multiplier β* and the second constraint
(equation (3.7)) as
∞
β∗
, with Cq =
pq (X) dX.
(3.12)
βq =
Cq + (1 − q)β ∗ Xq
0
...................................................
(3.4)
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160497
Sq (A + B) Sq (A) Sq (B)
Sq (A) Sq (B)
=
+
+ (1 − q)
.
k
k
k
k
k
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(a)
(b)
10
1
1
expq(x)
10–1
q = 1.5
q = 1.0
q = 0.5
q = 0.5
10–2
1
2
3
4
5
x
6
7
8
9 10
10–2
10–2
10–1
1
q = 1.0
10
x
Figure 2. The q-exponential function (equation (3.13)) for various values of q in (a) log–linear axes and (b) log–log axes. (Online
version in colour.)
The numerator of equation (3.10) is the q-exponential function defined as [13]
⎫
expq (X) = [1 + (1 − q)X]1/(1−q) , for 1 + (1 − q) ≥ 0⎬
for 1 + (1 − q) < 0,⎭
expq (X) = 0,
(3.13)
whose inverse is the q-logarithmic function:
lnq X =
X1−q − 1
.
1−q
(3.14)
According to Abe & Suzuki [16], the q-exponential distribution is a generalization of the
Zipf–Mandelbrot distribution [89], which is obtained for q > 1. If q > 1, equation (3.10) exhibits
an asymptotic power-law behaviour, whereas for 0 < q < 1 a cut-off appears at Xc = 1/(1 − q)β q
[16,17]. In the limit q → 1, the q-exponential and q-logarithmic functions lead to the ordinary
exponential and logarithmic functions, respectively. The q-exponential function for various values
of q is shown in figure 2.
In the NESM framework, it has been proposed that the cumulative distribution function (CDF)
P(>X) should be obtained upon integration of the escort probability Pq (X) rather than p(X) [16,17].
Following this approach and by using equation (3.8), P(>X) is derived as
∞
X
.
(3.15)
Pq (X) dX = expq −
P(>X) =
X0
0
where X0 is always positive, has the units of X and is defined as X0 = (1 − q)Xq + (1/βq ) [90].
Equation (3.15) can alternatively be written as
X
[P(X)]1−q − 1
=− ,
1−q
X0
which is similar to the definition of the q-logarithmic function (equation (3.14)). The latter
equation implies that after the estimation of the appropriate q-value that describes the distribution
of X, the q-logarithmic function (equation (3.14)) is linear with X, with slope −1/X0 [91].
The other type of distributions that is deeply related to statistical mechanics is that of the
squared variable X2 . Optimization of Sq in terms of X2 using the appropriate constraints leads to
the q-Gaussian distribution that has the form [13]:
2 1/(1−q)
1
X
.
(3.16)
p(X) =
1 − (1 − q)
Zq
X0
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Given the nonlinear dynamics that control fragmentation processes and the phenomenology
that fault and earthquake populations exhibit, the application of NESM to earthquake physics
arose naturally. The NESM theory addresses properties such as (multi-) fractal geometries,
long-range interactions and criticality, properties that seem quite important for understanding
earthquake physics [85]. In a series of recent publications, it has been demonstrated that NESM is
a powerful tool for describing the macroscopic properties of earthquake-related phenomena and
other geophysical problems as diverse as plate tectonics, natural hazards, geomagnetic reversals
and rock physics (see [92] and references therein). The current review focuses on the application
of NESM to earthquakes and tectonics and the insights that can be obtained in earthquake physics
using this approach.
(i) Non-extensive statistical mechanics and the earthquake-size distribution
A longstanding distribution in geophysics that is widely used in earthquake hazard assessments
is the frequency-size distribution of earthquakes. The earthquake generation process is primarily
a mechanical phenomenon, where stick–slip frictional instability inside fault zones has a
dominant role (e.g. [1]). Consistent with this idea Sotolongo-Costa & Posadas [93] introduced
the fragment-asperity interaction model for earthquake dynamics. This model considers the
interaction of two rough profiles (fault blocks) and the fragments filling the space in between
them. Stress accumulates in the crust until a fragment is displaced or an asperity is broken,
resulting in fault plane slip and energy release. The relative displacement of the fault blocks and
consequently the earthquake energy release (see the related discussion in §2a(i)) are proportional
to the size of the hindering fragments. Sotolongo-Costa & Posadas [93] used the NESM formalism
to establish the seismic energy distribution function based on the size distribution of the
fragments. The non-additive entropy Sq, in terms of the probability p(σ ) of finding a fragment
of area σ , is expressed as
1 − pq (σ ) dσ
.
(3.17)
Sq = k
q−1
Sotolongo-Costa & Posadas [93] obtained the probability p(σ ) by maximizing Sq under the
constraints of normalization of p(σ ) (equation (3.6)) and the q-mean value of pq (σ ), whereas
Silva et al. [94] introduced the condition about the q-expectation value in the second constraint
(equation (3.7)) and maximized Sq using the Lagrange multipliers method to derive the fragment
size distribution function p(σ ). Considering the proportionality between the released seismic
energy E and the size of the fragments r (E ∼ r3 ) (e.g. [95]), the following expression for the seismic
energy distribution function is derived [94]:
p(E) =
2/3
C1 E−(1/3)
[1 + C2 E2/3 ]
1/(qM −1)
2/3
,
(3.18)
with C1 = (2/3αM ) and C2 = −((1 − qM )/(2 − qM )αM ). In equation (3.18), the probability of the
energy is p(E) = n(E)/N, where n(E) corresponds to the number of earthquakes with energy E and
N is the total number of earthquakes (figure 3b). A more viable expression can now be obtained by
...................................................
(b) Fundamental ideas of non-extensive statistical mechanics applied to earthquakes
and tectonics
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The q-Gaussian distribution generalizes the standard Gaussian distribution, which is
recovered in the limit of q → 1. For q > 1, the q-Gaussian distribution displays power-law tails
with slope −2/(q − 1), thus enhancing the probabilities of rare events.
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(a)
(b)
1
12
10–2
p (E)
N (>M)/N
10–3
10–8
10–4
10–5
10–6
1.0
1.5
2.0
2.5
M
3.0
3.5
4.0
10–10
10
102
103
104
105
106
E
Figure 3. (a) Normalized CDF of earthquake magnitudes in the West Corinth Rift (circles) and the corresponding fit according
to the fragment-asperity interaction model (equation (3.20)) for the values of qM = 1.37 ± 0.01 and aM = 19.05 ± 6.86 (solid
line). (b) The corresponding energy distribution function p(E) (circles) and the corresponding fit according to the fragmentasperity interaction model (equation (3.18)) for similar parameter values as previously (solid line). Modified from Michas
et al. [41]. (Online version in colour.)
introducing the normalized cumulative number of earthquakes, given by the integral of equation
(3.18):
(2−qM )/(1−qM )
∞
E 2/3
1 − qM
N(E > Eth )
=
p(E) dE = 1 −
,
(3.19)
N
2 − qM
αM
Eth
where N(E > Eth ) is the number of earthquakes with energy E greater than the threshold energy
Eth , N the total number of earthquakes, qM the entropic index and aM a positive constant that
expresses the proportionality between the released energy E and the fragments of size r. If we
consider the relationship between E and M (equation (2.3)), the latter expression can be written
in terms of the earthquake magnitude M. By taking into account the minimum magnitude M0
of the earthquake catalogue, the cumulative distribution of earthquake magnitudes N(>M) takes
the form [96]:
(2−qM )/(1−qM )
2/3
1 − ((1 − qM )/( 2 − qM ) )(10M /αM )
N(>M)
=
.
(3.20)
2/3
N
1 − ((1 − qM )/( 2 − qM ) )(10M0 /α )
M
The fragment–asperity interaction model describes from the first principles of statistical
mechanics the cumulative distribution of the number of earthquakes N greater than a threshold
magnitude M, normalized by the total number of earthquakes (figure 3). This model, in the form
of equation (3.20) or in the previous forms of Sotolongo-Costa & Posadas [93] and Silva et al. [94],
has been applied to various regional earthquake catalogues [18,41,94,97–105]. The results of these
studies indicate that the model can successfully reproduce the frequency–magnitude distribution
of earthquakes in diverse tectonic environments (e.g. figure 3a) and in volcanic regions [100,106].
In comparison to the G-R scaling relation (equation (2.1)), the fragment-asperity model provides
a good description of the observed earthquake magnitudes over a wider range of scales, while
for values above some threshold magnitude, the G-R relation can be derived as a particular
case, for the value of b = (2 − qM )/(qM − 1) [96]. Moreover, the b-value estimation according to
the maximum-likelihood technique (e.g. [107]) is quite sensitive to the initial selection of the
minimum earthquake magnitude M0 in the catalogue under study, while the qM -value estimation
is quite stable irrespective of the selection of M0 [108]. The stability of qM with M0 is quite
important in earthquake hazard assessments, as reliable estimates of qM can be made even in cases
where the catalogue is incomplete or presents uncertainties due to spatio-temporal variations in
the magnitude of completeness.
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1.52
1.50
qM
1.49
1.48
1.47
1.46
2 July 1992
9 Apr 1994
2 July 1997
time
Figure 4. Temporal qM variations (black continuous line) over increasing (cumulative) event-based windows and the associated
standard deviation (black dashed lines). From Papadakis et al. [110], fig.3.
In addition, the qM variations with time have been used as an index of tectonic stability in
a seismic region and its proximity towards a larger event [18,100,101,104,109–111]. Papadakis
et al. [18] estimated qM and aM along distinct seismic zones of the Hellenic Subduction Zone
(HSZ) and found that the qM variations are strongly related with the seismic energy release in
each zone. Telesca [101] studied the qM temporal variations prior to the 6 April 2009 (ML = 5.8)
L’Aquila earthquake and found a significant increase some days before the occurrence of the
strong event. Significant qM increases were also observed prior to the 12 October 2013 (ML = 6.2)
earthquake that occurred in the southwest segment of the Hellenic arc [112] and prior to the 1995
(M = 7.2) Kobe earthquake [110] (figure 4). In these case studies, the qM increases were associated
with the occurrence of moderate-size events prior to the main shock, indicating the initiation of a
preparatory phase leading to a strong earthquake (figure 4).
The probability distribution of incremental earthquake energies (i.e. the energy differences
between successive earthquakes) in real earthquake data and in numerical models has been found
to follow the q-Gaussian distribution (equation (3.16)) [92,106,113]. The q-Gaussian distribution
exhibits power-law tails, which enhance the probabilities (for q > 1) of rare events and in the case
of seismicity the occurrence of a large earthquake immediately after the occurrence of a smaller
one. In particular, Caruso et al. [113] found that in the critical regime of the dissipative OFC model
[81] (small-world lattice), the incremental earthquake energies exhibit a q-Gaussian probability
distribution, while in the non-critical regime (regular lattice) the probability distribution is close
to Gaussian (figure 5a). Caruso et al. [113] repeated their analysis on real earthquake data and
found that the probability distribution of incremental earthquake energies in global seismicity and
Northern California follows the q-Gaussian distribution (figure 5b), providing further evidence
for SOC, intermittency and long-range interactions in seismicity. These results also imply that
although long-range temporal and spatial correlations exist in seismicity, together with a certain
degree of statistical predictability, it is not possible to predict the magnitude of the events [113].
The probability distribution of incremental earthquake energies in volcano seismicity was further
studied by Vallianatos et al. [106]. In the latter study, the analysis showed that the probability
distribution of incremental earthquake energies during the 2011–2012 unrest at the Santorini
volcanic complex follows the q-Gaussian distribution for the q-value of q = 2.24, a value that is
in agreement with the Ehrenfest dog-flea SOC model [114].
In addition, q-exponential functions have been found to characterize the acoustic emission
energy release produced during the triaxial deformation of an Etna basalt in the laboratory
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17 Jan 1995
Kobe
earthquake
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(a)
14
1
PDF
10–2
10–3
10–4
–30
–20
–10
0
10
20
30
(b)
1
Northern California 1966–2006
Gaussian
q-Gaussian q = 1.75
10–1
PDF
10–2
10–3
10–4
10–5
–35 –30 –25 –20 –15 –10 –5 0
5 10 15 20 25 30 35
(X – 〈X〉)/σ
Figure 5. (a) Probability distribution function (PDF) of incremental ‘avalanche’ sizes x, normalized to the standard deviation σ ,
for the OFC model on a small-world topology (critical state; open circles) and on a regular lattice (non-critical state; filled circles).
The first curve is fitted with the q-Gaussian distribution (solid line) for q = 2 ± 0.1. The Gaussian distribution (dashed line) is
also shown for comparison. (b) PDF of incremental earthquake energies for the Northern California earthquake catalogue (open
circles) and the corresponding fit with the q-Gaussian distribution (solid line) for q = 1.75 ± 0.15. The Gaussian distribution
(dashed line) is also shown for comparison (from Caruso et al. [113]). (Online version in colour.)
[14] and the global earthquake frequency–magnitude distribution [19]. Using the global CMT
catalogue, Vallianatos & Sammonds [19] used NESM to study the effect of the Sumatra and
Honshu mega-earthquakes (Mw ≥ 9) on the global frequency–magnitude distribution (figure 6).
The authors used a cross-over formulation of NESM [115] to interpret thermodynamically the
deviation of greater magnitudes from a pure power law (see also the discussion in §2a(i)) and
found that the distribution of greater earthquakes (Mw ≥ 7.6) changes from an exponential to
another power-law prior to the two mega-events, implying a global organization of seismicity as
the two mega-events were approached (figure 6).
(ii) Spatio-temporal properties of seismicity and non-extensive statistical mechanics
The NESM approach to the spatio-temporal properties of seismicity was first introduced by
Abe & Suzuki [16,17]. These authors studied the spatio-temporal distribution of seismicity in
California and Japan and showed that the cumulative distributions of inter-event distances
P(>r) and times P(>τ ) between successive earthquakes are well described by the q-exponential
distribution (equation (3.15)) for q-values of qr < 1 and qτ > 1, respectively. In particular, Abe &
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OFC on a small world
OFC on a regular lattice
Gaussian
q-Gaussian q = 2
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1
q-describable statistics
q = 1.6
15
CDF
0.01
r-describable statistics
0.001
0.0001
1024
Mc
1025
1026
1027
1028
1029
Mo, seismic moment in dyn cm
1030
Figure 6. Normalized CDF versus seismic moment for the centroid moment tensor (CMT) earthquake catalogue up to the end
of 1990 (before the Sumatra mega event, diamonds), the end of December 2004 (after Sumatra, squares) and within a week
after Honshu mega earthquake (till 17 March 2011, triangles), for shallow events (H < 75 km), with Mw > 5.5, since 1 January
1981. From Vallianatos & Sammonds [19]. (Online version in colour.)
Suzuki [16,17] found q-values of qr = 0.77, qτ = 1.13 for California and qr = 0.747, qτ = 1.05 for
Japan and suggested the spatio-temporal duality of earthquakes, where qτ + qr ≈ 2.
The results of Abe and Suzuki were further verified in laboratory experiments [14], in
numerical models and in regional and global seismicity (e.g. [18,19,41,105,116–118]), implying
a universal character and the existence of long-term clustering effects in the spatio-temporal
evolution of seismicity. The temporal scaling properties of volcanic seismicity and in particular
during the 2011–2012 unrest at the Santorini volcanic complex were studied by Vallianatos et al.
[106]. In the latter study, the authors showed that when the volcano-related seismicity takes
swarm-like character, complex correlations of seismicity emerge that are characterized by a
q-exponential inter-event times distribution.
Papadakis et al. [18] examined the inter-event times and inter-event distances distribution
along the seismic zones of the HSZ (figure 7) and proposed that the observed variations of
the entropic indexes signify different degrees of earthquake clustering. The latter result may
imply asymmetric patterns within the clustered events in each seismic zone (e.g. [119]). In
addition, Antonopoulos et al. [105] studied the probability distribution of inter-event times
between successive earthquakes in Greece and showed that for both the entire dataset and the
declustered one, where the aftershocks have been removed, the probability distributions are better
described by the q-exponential distribution (equation (3.10)), for the q-values of qτ = 1.24 ± 0.054
and qτ = 1.14 ± 0.057, respectively. For the declustered dataset, the corresponding q-value that best
describes the observed distribution approaches unity, implying the loss of temporal correlations
and close proximity to Poissonian (random) behaviour once the aftershocks are removed from the
dataset. Moreover, Antonopoulos et al. [105] estimated the hazard function WM (T, T), defined
as the probability of at least one earthquake with magnitude greater than M will occur in the next
time interval T if the previous one occurred before time T. The hazard function is related to p(T)
by:
T+T
p(T) dT
.
(3.21)
WM (T; T) = T ∞
p(T)
dT
T
If p(T) scales according to the q-exponential function (equation (3.13)), then by simple
integration one obtains:
β(q − 1)T (q−2)/(q−1)
.
(3.22)
WM (T; T) = 1 − 1 +
1 + β(q − 1)T
The hazard function WM (T,T)for the earthquake activity in Greece according to equation
(3.22) and for various time intervals T is shown in figure 8. The graph shows that for a fixed
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1981–1990
1981–2004
1981–2011
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18°
20°
22°
24°
28°
26°
30°
16
N
E
42°
S
40°
40°
38°
1
1.09
2
1.07
38°
1.05
36°
36°
3
Crete
1
1.16
34°
5
4
0
18°
20°
22°
34°
km
24°
26°
28°
100
200
30°
Figure 7. Modified from Papadakis et al. [18] fig. 7: The qτ variation along the five seismic zones of the HSZ. (Online version
in colour.)
10
WM(T, ΔT)
1
10–1
10–2
10–3
10–3
ΔT = 1
ΔT = 5
ΔT = 20
ΔT = 50
10–2
10–1
1
10
102
103
T
Figure 8. The hazard function WM (T, T) (equation (3.22)) versus the inter-event time T (in days) for earthquake magnitudes
M ≥ 4.1 in the area of Greece, for four different T values. From Antonopoulos et al. [105].
time interval T, the probability for at least one earthquake with magnitude greater than M to
occur in the next time intervalT, decreases with increasing T (figure 8) [105]. The latter implies
that the longer it has been since the last earthquake, the longer it will take for the next one to occur
(see also Sornette & Knopoff [120]).
...................................................
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42°
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102
17
normalized p(t¢)
10–4
10–6
10–4
Mc
entire catalogue
Mc =1.6
Mc = 2
Mc = 2.5
.
gamma
q-gamma
10–3
10–2
10–1
t ¢ = RT
1
10
10–2
Figure 9. Normalized probability density p(τ ) for the rescaled inter-event times τ and for various threshold magnitudes
(Mc ). The black solid line represents the q-gamma distribution (equation (3.23)) for parameter values of C = 0.35, γ = 0.39,
τ 0 = 1.55 and q = 1.23. The fitted dashed line represents the gamma distribution (equation (2.7)). Modified from Michas et al.
[41]. (Online version in colour.)
In addition, Michas et al. [41] studied the PDF for the inter-event times of earthquakes in the
West Corinth Rift, Greece. In this case, the inter-event times τ were normalized according to the
mean seismic rate as τ = R × τ , or equivalently to the mean inter-event time τ̄ as τ = τ /τ̄ . The
seismic rate acts as a weighting factor, as a higher rate in a given region corresponds to a higher
number of earthquakes that are produced and contribute to the distribution [34]. By rescaling
τ with R, the normalized probability distributions p(τ ) for various threshold magnitudes fall
approximately into a unique curve (figure 9; see also Corral [36]). For stationary periods, where
the mean seismic rate does not fluctuate, this unique curve can well be approximated by the
gamma distribution (equation (2.7)). However, if the entire non-stationary earthquake time series
are considered, an additional power-law regime at long inter-event times appears. This type of
scaling can be described by a q-gamma distribution (figure 9), where the last exponential term has
been substituted by the q-exponential function (equation (3.13)) [41]. The q-gamma distribution
then takes the form of [121]:
−τ
.
(3.23)
f (τ ) = Cτ γ −1 expq
τ0
In the limit of q → 1, the q-gamma distribution (equation (3.23)) recovers the ordinary gamma
(equation (2.7)). According to equation (3.23), for short inter-event times, p(τ ) scales as a power
law, p(τ ) ∼ τ γ −1 , whereas for longer inter-event times, p(τ ) scales as another power law, p(τ ) ∼
τ (1−γ )/(1−q) , indicating clustering effects at both short and long inter-event times.
(iii) Fault-size distribution and Tsallis entropy
The NESM formalism, as it applies to a fault system with various fault sizes (lengths) L was
introduced by Vallianatos et al. [122] and Vallianatos & Sammonds [91]. In this case, the Tsallis
entropy Sq is expressed as
1 − (dL/σ )[σ p(L)]q
Sq = k
,
(3.24)
q−1
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(a)
(b)
1
18
1
200
250
CPF
150
–4
0
500
fault length L, km
1000
1500
2000
2500
linked faults
–20
0.01
–6
–8
0
–10
independent faults
–30
lnq P (>L)
0.001
100
–2
lnq P (>L)
CPF
0.01
50
–40
–50
–60
–10
–70
–12
0.0001
1
–80
0.001
10
100
1000
1
10
fault length (km)
100
1000
fault length (km)
10 000
Figure 10. The normalized cumulative probability distribution function P(>L) for (a) independent and (b) strongly interacting
(linked) fault segments in the Valles Marineris extensional province, Mars. In the insets, the semi-q-log graphs are shown, which
indicate the q-values of q = 1.10 (correlation coefficient −0.992) for the independent faults and q = 1.75 (correlation coefficient
−0.996) for the linked ones (from Vallianatos & Sammonds [91]).
where σ is a positive scaling factor, k a positive constant, q the entropic index and p(L)dL the
probability of finding L in the interval [L, L + dL]. Optimizing Sq using the appropriate constraints
and the Lagrange multipliers method, as described in the previous sections, yields a q-exponential
type distribution, which takes the form (see also [123]):
P(>L) =
expq (−L/L0 )
expq (−Lmin /L0 )
,
(3.25)
where P(>L) is the cumulative PDF of L, L0 is a positive scaling parameter (q > 1) and Lmin is the
minimum fault-length in the dataset.
Vallianatos et al. [122] used the NESM approach to study the scaling properties of the fault
network in Crete, in the front of the Hellenic Arc, and found that this scales according to
the q-exponential distribution for the q-value of q = 1.16. Vallianatos & Sammonds [91] used
a similar approach to study the scaling properties of linked and independent faults in an
extraterrestrial fault system; the Valles Marineris extensional province, Mars, and showed that
the fault-length distributions follow a q-exponential, with q = 1.75 for linked faults and q = 1.10
for the independent ones (figure 10). These results indicate the strong mechanical correlations of
linked faults and the weaker ones for the independent faults, which exhibit a q-value close to
unity and BG statistical mechanics. Furthermore, Vallianatos [123] studied the scaling properties
of thrust (compressional) and normal (extensional) faults in Mars and found that the former are
described by the q-exponential distribution for the value of q = 1.114 and the latter for q = 1.277.
In addition, Michas et al. [56] studied the scaling properties of the fault network in the Corinth
Rift (Greece) that is one of the most tectonically active continental rifts on the Earth. By using the
NESM approach, the analysis indicated the transition from q-exponential scaling and asymptotic
power-law behaviour in the lower strain eastern zone, to exponential scaling and Poissonian
behaviour in the higher strain central and western zones. When the current strain rates were
considered, the analysis showed a similar transition from q-exponential scaling in the lower strain
rate zone to exponential scaling in the higher strain rate zone, indicating the maturity of the
fault network and fracture saturation in the currently active rift zone. The observed properties
in the various strain regimes are shown in figure 11 and are described in detail by Michas et al.
[56]. These properties provide evidence for a combination of crustal processes in natural fault
systems and further suggest that fault growth processes in the upper crust control the fault
network evolution and the localization of strain in the Corinth Rift. Furthermore, Michas et al.
[56] studied the sensitivity of the observed distributions to missing faults from the dataset by
generating synthetic fault datasets that scale according to the q-exponential-type distribution of
equation (3.20). The analysis showed that even in the case of more than 90% missing faults from
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0
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0.1
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low strain zones
high strain zones
q-exponential scaling
cumulative number
cumulative number
exponential scaling
fault length
fault length
fault nucleation
fault growth
fault linkage
Figure 11. Summary of the observed properties in the various strain regimes in the Corinth Rift [56]. (Online version in colour.)
the dataset, the observed scaling remains quite stable, with q-value variations that increase as
more data are removed from the original dataset.
4. Concluding remarks
Although significant research has been conducted on the complex properties of seismicity,
including scaling relations, temporal and spatial correlations, critical phenomena and nucleation,
many questions regarding earthquake physics remain to be answered in the future. Most of
the current research approaches rely on empirical laws rather than on a solid underlying
theoretical concept. However, statistical seismology should evolve into a genuine physically
based statistical physics of earthquakes. This review paper presents the concept of NESM, which
offers a consistent theoretical framework for the macroscopic description of fault and earthquake
populations. The NESM approach, based on the first principles of statistical mechanics, provides
a unified framework that produces a range of asymptotic power law to exponential-like
distributions that are both ubiquitous in nature.
The first section of this study reviews the scaling properties of seismicity and fault populations,
and the classical statistical mechanics approaches that have been developed to describe them.
Following, the theoretical framework of NESM and its application to earthquakes and tectonics
is extensively presented. More particularly, we present various applications on the earthquake
frequency–magnitude distribution, the spatio-temporal properties of seismicity and the fault-size
distribution.
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no. faults decreases - average fault-length increases
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increasing strain
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