373 research outputs found

### Andrade, Omori and Time-to-failure Laws from Thermal Noise in Material Rupture

Using the simplest possible ingredients of a rupture model with thermal
fluctuations, we provide an analytical theory of three ubiquitous empirical
observations obtained in creep (constant applied stress) experiments: the
initial Andrade-like and Omori-like $1/t$ decay of the rate of deformation and
of fiber ruptures and the $1/(t_c-t)$ critical time-to-failure behavior of
acoustic emissions just prior to the macroscopic rupture. The lifetime of the
material is controlled by a thermally activated Arrhenius nucleation process,
describing the cross-over between these two regimes. Our results give further
credit to the idea proposed by Ciliberto et al. that the tiny thermal
fluctuations may actually play an essential role in macroscopic deformation and
rupture processes at room temperature. We discover a new re-entrant effect of
the lifetime as a function of quenched disorder amplitude.Comment: 4 pages with 1 figur

### Bath's law Derived from the Gutenberg-Richter law and from Aftershock Properties

The empirical Bath's law states that the average difference in magnitude
between a mainshock and its largest aftershock is 1.2, regardless of the
mainshock magnitude. Following Vere-Jones [1969] and Console et al. [2003], we
show that the origin of Bath's law is to be found in the selection procedure
used to define mainshocks and aftershocks rather than in any difference in the
mechanisms controlling the magnitude of the mainshock and of the aftershocks.
We use the ETAS model of seismicity, which provides a more realistic model of
aftershocks, based on (i) a universal Gutenberg-Richter (GR) law for all
earthquakes, and on (ii) the increase of the number of aftershocks with the
mainshock magnitude. Using numerical simulations of the ETAS model, we show
that this model is in good agreement with Bath's law in a certain range of the
model parameters.Comment: major revisions, in press in Geophys. Res. Let

### Correlations and invariance of seismicity under renormalization-group transformations

The effect of transformations analogous to those of the real-space
renormalization group are analyzed for the temporal occurrence of earthquakes.
The distribution of recurrence times turns out to be invariant under such
transformations, for which the role of the correlations between the magnitudes
and the recurrence times are fundamental. A general form for the distribution
is derived imposing only the self-similarity of the process, which also yields
a scaling relation between the Gutenberg-Richter b-value, the exponent
characterizing the correlations, and the recurrence-time exponent. This
approach puts the study of the structure of seismicity in the context of
critical phenomena.Comment: Short paper. I'll be grateful to get some feedbac

### Anomalous Power Law Distribution of Total Lifetimes of Branching Processes Relevant to Earthquakes

We consider a branching model of triggered seismicity, the ETAS
(epidemic-type aftershock sequence) model which assumes that each earthquake
can trigger other earthquakes (``aftershocks''). An aftershock sequence results
in this model from the cascade of aftershocks of each past earthquake. Due to
the large fluctuations of the number of aftershocks triggered directly by any
earthquake (``productivity'' or ``fertility''), there is a large variability of
the total number of aftershocks from one sequence to another, for the same
mainshock magnitude. We study the regime where the distribution of fertilities
$\mu$ is characterized by a power law $\sim 1/\mu^{1+\gamma}$ and the bare
Omori law for the memory of previous triggering mothers decays slowly as $\sim
1/t^{1+\theta}$, with $0 < \theta <1$ relevant for earthquakes. Using the tool
of generating probability functions and a quasistatic approximation which is
shown to be exact asymptotically for large durations, we show that the density
distribution of total aftershock lifetimes scales as $\sim
1/t^{1+\theta/\gamma}$ when the average branching ratio is critical ($n=1$).
The coefficient $1<\gamma = b/\alpha<2$ quantifies the interplay between the
exponent $b \approx 1$ of the Gutenberg-Richter magnitude distribution $\sim
10^{-bm}$ and the increase $\sim 10^{\alpha m}$ of the number of aftershocks
with the mainshock magnitude $m$ (productivity) with $\alpha \approx 0.8$. More
generally, our results apply to any stochastic branching process with a
power-law distribution of offsprings per mother and a long memory.Comment: 16 pages + 4 figure

### Acoustic Emission Monitoring of the Syracuse Athena Temple: Scale Invariance in the Timing of Ruptures

We perform a comparative statistical analysis between the acoustic-emission time series from the ancient Greek Athena temple in Syracuse and the sequence of nearby earthquakes. We find an apparent association between acoustic-emission bursts and the earthquake occurrence. The waiting-time distributions for acoustic-emission and earthquake time series are described by a unique scaling law indicating self-similarity over a wide range of magnitude scales. This evidence suggests a correlation between the aging process of the temple and the local seismic activit

### "Universal" Distribution of Inter-Earthquake Times Explained

We propose a simple theory for the ``universal'' scaling law previously
reported for the distributions of waiting times between earthquakes. It is
based on a largely used benchmark model of seismicity, which just assumes no
difference in the physics of foreshocks, mainshocks and aftershocks. Our
theoretical calculations provide good fits to the data and show that
universality is only approximate. We conclude that the distributions of
inter-event times do not reveal more information than what is already known
from the Gutenberg-Richter and the Omori power laws. Our results reinforces the
view that triggering of earthquakes by other earthquakes is a key physical
mechanism to understand seismicity.Comment: 4 pages with two figure

### Universal mean moment rate profiles of earthquake ruptures

Earthquake phenomenology exhibits a number of power law distributions
including the Gutenberg-Richter frequency-size statistics and the Omori law for
aftershock decay rates. In search for a basic model that renders correct
predictions on long spatio-temporal scales, we discuss results associated with
a heterogeneous fault with long range stress-transfer interactions. To better
understand earthquake dynamics we focus on faults with Gutenberg-Richter like
earthquake statistics and develop two universal scaling functions as a stronger
test of the theory against observations than mere scaling exponents that have
large error bars. Universal shape profiles contain crucial information on the
underlying dynamics in a variety of systems. As in magnetic systems, we find
that our analysis for earthquakes provides a good overall agreement between
theory and observations, but with a potential discrepancy in one particular
universal scaling function for moment-rates. The results reveal interesting
connections between the physics of vastly different systems with avalanche
noise.Comment: 13 pages, 5 figure

### A model for the distribution of aftershock waiting times

In this work the distribution of inter-occurrence times between earthquakes
in aftershock sequences is analyzed and a model based on a non-homogeneous
Poisson (NHP) process is proposed to quantify the observed scaling. In this
model the generalized Omori's law for the decay of aftershocks is used as a
time-dependent rate in the NHP process. The analytically derived distribution
of inter-occurrence times is applied to several major aftershock sequences in
California to confirm the validity of the proposed hypothesis.Comment: 4 pages, 3 figure

### Market dynamics immediately before and after financial shocks: quantifying the Omori, productivity and Bath laws

We study the cascading dynamics immediately before and immediately after 219
market shocks. We define the time of a market shock T_{c} to be the time for
which the market volatility V(T_{c}) has a peak that exceeds a predetermined
threshold. The cascade of high volatility "aftershocks" triggered by the "main
shock" is quantitatively similar to earthquakes and solar flares, which have
been described by three empirical laws --- the Omori law, the productivity law,
and the Bath law. We analyze the most traded 531 stocks in U.S. markets during
the two-year period 2001-2002 at the 1-minute time resolution. We find
quantitative relations between (i) the "main shock" magnitude M \equiv \log
V(T_{c}) occurring at the time T_{c} of each of the 219 "volatility quakes"
analyzed, and (ii) the parameters quantifying the decay of volatility
aftershocks as well as the volatility preshocks. We also find that stocks with
larger trading activity react more strongly and more quickly to market shocks
than stocks with smaller trading activity. Our findings characterize the
typical volatility response conditional on M, both at the market and the
individual stock scale. We argue that there is potential utility in these three
statistical quantitative relations with applications in option pricing and
volatility trading.Comment: 16 pages, double column, 13 figures, 1 Table; Changes made in Version
2 in response to referee comment

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