The BCS gap equation for spin-polarized fermions

We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For $\cosh(\delta_\mu/T) \leq 2$, with $T$ the temperature and $\delta_\mu$ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in \cite{FHNS,HHSS,HS}. For $\cosh(\delta_\mu/T) > 2$ the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.Comment: 23 pages, 1 figur

Detecting premonitory seismicity patterns based on critical point dynamics

We test the hypothesis that critical point dynamics precedes strong earthquakes in a region surrounding the future hypocenter. Therefore, we search systematically for regions obeying critical point dynamics in terms of a growing spatial correlation length (GCL). The question of whether or not these spatial patterns are correlated with future seismicity is crucial for the problem of predictability. The analysis is conducted for earthquakes with <i>M</i> <u>></u> 6.5 in California. As a result, we observe that GCL patterns are correlated with the distribution of future seismicity. In particular, there are clear correlations in some cases, e.g. the 1989 Loma Prieta earthquake and the 1999 Hector Mine earthquake. We claim that the critical point concept can improve the seismic hazard assessment

On Blowup for time-dependent generalized Hartree-Fock equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.Comment: 24 page

A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case

This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in details a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, S\'er\'e and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations.Comment: Final version, to appear in Comm. Math. Phy

Statistical analysis of the induced Basel 2006 earthquake sequence: Introducing a probability-based monitoring approach for Enhanced Geothermal Systems

ISSN:0956-540XISSN:1365-246

Statistical analysis of the induced Basel 2006 earthquake sequence: introducing a probability-based monitoring approach for Enhanced Geothermal Systems

Geothermal energy is becoming an important clean energy source, however, the stimulation of a reservoir for an Enhanced Geothermal System (EGS) is associated with seismic risk due to induced seismicity. Seismicity occurring due to the water injection at depth have to be well recorded and monitored. To mitigate the seismic risk of a damaging event, an appropriate alarm system needs to be in place for each individual experiment. In recent experiments, the so-called traffic-light alarm system, based on public response, local magnitude and peak ground velocity, was used. We aim to improve the pre-defined alarm system by introducing a probability-based approach; we retrospectively model the ongoing seismicity in real time with multiple statistical forecast models and then translate the forecast to seismic hazard in terms of probabilities of exceeding a ground motion intensity level. One class of models accounts for the water injection rate, the main parameter that can be controlled by the operators during an experiment. By translating the models into time-varying probabilities of exceeding various intensity levels, we provide tools which are well understood by the decision makers and can be used to determine thresholds non-exceedance during a reservoir stimulation; this, however, remains an entrepreneurial or political decision of the responsible project coordinators. We introduce forecast models based on the data set of an EGS experiment in the city of Basel. Between 2006 December 2 and 8, approximately 11 500 m3 of water was injected into a 5-km-deep well at high pressures. A six-sensor borehole array, was installed by the company Geothermal Explorers Limited (GEL) at depths between 300 and 2700 m around the well to monitor the induced seismicity. The network recorded approximately 11 200 events during the injection phase, more than 3500 of which were located. With the traffic-light system, actions where implemented after an ML 2.7 event, the water injection was reduced and then stopped after another ML 2.5 event. A few hours later, an earthquake with ML 3.4, felt within the city, occurred, which led to bleed-off of the well. A risk study was later issued with the outcome that the experiment could not be resumed. We analyse the statistical features of the sequence and show that the sequence is well modelled with the Omori-Utsu law following the termination of water injection. Based on this model, the sequence will last 31+29/−14 years to reach the background level. We introduce statistical models based on Reasenberg and Jones and Epidemic Type Aftershock Sequence (ETAS) models, commonly used to model aftershock sequences. We compare and test different model setups to simulate the sequences, varying the number of fixed and free parameters. For one class of the ETAS models, we account for the flow rate at the injection borehole. We test the models against the observed data with standard likelihood tests and find the ETAS model accounting for the on flow rate to perform best. Such a model may in future serve as a valuable tool for designing probabilistic alarm systems for EGS experiment

Modeling of Kashmir Aftershock Decay Based on Static Coulomb Stress Changes and Laboratory-Derived Rate-and-State Dependent Friction Law

We model the spatial and temporal evolution of October 8, 2005 Kashmir earthquake\u2019s aftershock activity using the rate-and-state dependent friction model incorporating uncertainties in computed coseismic stress perturbations. We estimated the best possible value for frictional resistance \u2018\u2018Arn\u2019\u2019, background seismicity rate \u2018\u2018r\u2019\u2019 and coef\ufb01cient of stress variation \u2018\u2018CV\u2019\u2019 using maximum log-likelihood method. For the whole Kashmir earthquake sequence, we measure a frictional resistance Arn * 0.0185 MPa, r * 20 M3.7? events/year and CV = 0.94 \ub1 0.01. The spatial and temporal forecasted seismicity rate of modeled aftershocks \ufb01ts well with the spatial and temporal distribution of observed aftershocks that occurred in the regions with positive static stress changes as well as in the apparent stress shadow region. To quantify the effect of secondary aftershock triggering, we have re-run the estimations for 100 stochastically declustered catalogs showing that the effect of aftershock-induced secondary stress changes is obviously minor compared to the overall uncertainties, and that the stress variability related to uncertain slip model inversions and receiver mechanisms remains the major factor to provide a reasonable data \ufb01t

Scale Dependence of the Retarded van der Waals Potential

We study the ground state energy for a system of two hydrogen atoms coupled to the quantized Maxwell field in the limit $\alpha \to 0$ together with the relative distance between the atoms increasing as $\alpha^{-\gamma} R$, $\gamma > 0$. In particular we determine explicitly the crossover function from the $R^{-6}$ van der Waals potential to the $R^{-7}$ retarded van der Waals potential, which takes place at scale $\alpha^{-2} R$.Comment: 19 page

Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation

We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the self-consistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently established by Hainzl-Lewin-Sere, we prove the existence of global-in-time solutions of the considered evolution equation.Comment: 12 pages; more explanations added, some final (minor) corrections include