Christian Vanneste

Statistical physics model for the spatiotemporal evolution of faults

A statistical physics model is used to simulate antiplane shear deformation and rupture of a tectonic plate with heterogeneous material properties. Rupture occurs when the chosen state variable reaches a threshold value. After rupture, broken elements are instantaneously healed and retain the original material properties. We document the spatiotemporal evolution of the rupture pattern in response to a constant velocity boundary condition. A fundamental feature of this model is that ruptures become strongly correlated in space and time, leading to the development of complex fractal structures. These structures, or “faults,” are simply defined by the loci where deformation accumulates. Repeated rupture of a fault occurs in events (“earthquakes”) which themselves exhibit both spatial and temporal clustering. Furthermore, we observe that a fault may be active for long periods of time until the locus of activity spontaneously switches to a different fault. The formation of the faults and the temporal variation of rupture activity is due to a complex interplay between the random small-scale structure, long-range elastic interactions, and the threshold nature of rupture physics. The characteristics of this scalar model suggest that spontaneous self-organization of active tectonics does not result solely from the tensorial nature of crustal deformation; that is, kinematic compatibility is not required for complex fault pattern formation. Furthermore, the localization of the deformation is a dynamical effect rather than a consequence of preexisting structure or preferential weakening of faults compared to the surrounding medium. We present an analysis of scaling relationships exhibited by the fault pattern and the earthquakes in this model.

Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes

Rank-ordering statistics provide a perspective on the rare, largest elements of a population, whereas the statistics of cumulative distributions are dominated by the more numerous small events. The exponent of a power law distribution can be determined with good accuracy by rank-ordering statistics from the observation of only a few tens of the largest events. Using analytical results and synthetic tests, we quantify the systematic and the random errors. We also study the case of a distribution defined by two branches, each having a power law distribution, one defined for the largest events and the other for smaller events, with application to the worldwide (Harvard) and southern California earthquake catalogs. In the case of the Harvard moment catalog, we make more precise earlier claims of the existence of a transition of the earthquake magnitude distribution between small and large earthquakes; the b values are b2 = 2.3 ± 0.3 for large shallow earthquakes and b1 = 1.00 ± 0.02 for smaller shallow earthquakes. However, the crossover magnitude between the two distributions is ill defined. The data available at present do not provide a strong constraint on the crossover which has a 50% probability of being between magnitudes 7.1 and 7.6 for shallow earthquakes; this interval may be too conservatively estimated. Thus any influence of a universal geometry of rupture on the distribution of earthquakes worldwide is ill defined at best. We caution that there is no direct evidence to confirm the hypothesis that the large-moment branch is indeed a power law. In fact, a gamma distribution fits the entire suite of earthquake moments from the smallest to the largest satisfactorily. There is no evidence that the earthquakes of the southern California catalog have a distribution with two branches or that a rolloff in the distribution is needed; for this catalog, b = 1.00 ± 0.02 up to the largest magnitude observed, MW ≃ 7.5; hence we conclude that the thickness of the seismogenic layer has no observable influence whatsoever on the frequency distribution in this region.

Modes of random lasers

In conventional lasers, the optical cavity that confines the photons also determines essential characteristics of the lasing modes such as wavelength, emission pattern, directivity, and polarization. In random lasers, which do not have mirrors or a well-defined cavity, light is confined within the gain medium by means of multiple scattering. The sharp peaks in the emission spectra of semiconductor powders, first observed in 1999, has therefore lead to an intense debate about the nature of the lasing modes in these so-called lasers with resonant feedback. We review numerical and theoretical studies aimed at clarifying the nature of the lasing modes in disordered scattering systems with gain. The past decade has witnessed the emergence of the idea that even the low-Q resonances of such open systems could play a role similar to the cavity modes of a conventional laser and produce sharp lasing peaks. We focus here on the near-threshold single-mode lasing regime where nonlinear effects associated with gain saturation and mode competition can be neglected. We discuss in particular the link between random laser modes near threshold and the resonances or quasi-bound (QB) states of the passive system without gain. For random lasers in the localized (strong scattering) regime, QB states and threshold lasing modes were found to be nearly identical within the scattering medium. These studies were later extended to the case of more lossy systems such as random systems in the diffusive regime, where it was observed that increasing the openness of such systems eventually resulted in measurable and increasing differences between quasi-bound states and lasing modes. Very recently, a theory able to treat lasers with arbitrarily complex and open cavities such as random lasers established that the threshold lasing modes are in fact distinct from QB states of the passive system and are better described in terms of a new class of states, the so-called constant-flux states. The correspondence between QB states and lasing modes is found to improve in the strong scattering limit, confirming the validity of initial work in the strong scattering limit.

Statistical physics of fault patterns self-organized by repeated earthquakes

This work presents at attempt to model brittle ruptures and slips in a continental plate and its spontaneous organization by repeated earthquakes in terms of coarse-grained properties of the mechanical plate. A statistical physics model, which simulates anti-plane shear deformation of a thin plate with inhomogeneous elastic properties, is thus analyzed theoretically and numerically in order to study the spatio-temporal evolution of rupture patterns in response to a constant applied strain rate at its borders, mimicking the effect of neighboring plates. Rupture occurs when the local stress reaches a threshold value. Broken elements are instantaneously healed and retain the original material properties, enabling the occurrence of recurrent earthquakes. Extending previous works (Cowieet al., 1993;Miltenbergeret al., 1993), we present a study of the most startling feature of this model which is that ruptures become strongly correlated in space and time leading to the spontaneous development of multifractal structures and gradually accumulate large displacements. The formation of the structures and the temporal variation of rupture activity is due to a complex interplay between the random structure, long-range elastic interactions and the threshold nature of rupture physics. The spontaneous formation of fractal fault structures by repeated earthquakes is mirrored at short times by the spatio-temporal chaotic dynamics of earthquakes, well-described by a Gutenberg-Richter power law. We also show that the fault structures can be understood as pure geometrical objects, namely minimal manifolds, which in two dimensions correspond to the random directed polymer (RDP) problem. This mapping allows us to use the results of many studies on the RDP in the field of statistical physics, where it is an exact result that the minimal random manifolds in 2D systems are self-affine with a roughness exponent 2/3. We also present results pertaining to the influence of the degree β of stress release per earthquake on the competition between faults. Our results provide a rigorous framework from which to initiate rationalization of many, reported fractal fault studies.

Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems

A study on the effects of optical gain nonuniformly distributed in one-dimensional random systems is presented. It is demonstrated numerically that even without gain saturation and mode competition, the spatial nonuniformity of gain can cause dramatic and complicated changes to lasing modes. Lasing modes are decomposed in terms of the quasimodes of the passive system to monitor the changes. As the gain distribution changes gradually from uniform to nonuniform, the amount of mode mixing increases. Furthermore, we investigate new lasing modes created by nonuniform gain distributions. We find that new lasing modes may disappear together with existing lasing modes, thereby causing fluctuations in the local density of lasing states.

Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization

Quasimodes of an open finite-size two-dimensional (2D) random system are computed and systematically characterized in terms of their spatial extension

Dispersion of B-values in Gutemberg-Richter Law as a consequence of a proposed fractal nature of continental faulting

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A Wave Automaton for Time-Dependent Wave Propagation in Random Media

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PARTIALLY PUMPED RANDOM LASERS

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Laser Raman à fibre émettant dans un très large domaine spectral.

, and phase distribution for a collection of random systems ranging from weakly scattering to localized systems. A rapid change is seen in