Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.

Forward modelling of petrological crust-forming processes on the early Earth

Tonalite-trondhjemite-granodiorite (TTG) gneisses form up to two-thirds of the preserved Archean continental crust and there is considerable debate regarding the primary magmatic processes of the generation of these rocks. The popular theories indicate that these rocks were formed by partial melting of basaltic oceanic crust which was previously metamorphosed to garnet-amphibolite and/or eclogite facies conditions either at the base of thick oceanic crust or by subduction processes.rnThis study investigates a new aspect regarding the source rock for Archean continental crust which is inferred to have had a bulk compostion richer in magnesium (picrite) than present-day basaltic oceanic crust. This difference is supposed to originate from a higher geothermal gradient in the early Archean which may have induced higher degrees of partial melting in the mantle, which resulted in a thicker and more magnesian oceanic crust. rnThe methods used to investigate the role of a more MgO-rich source rock in the formation of TTG-like melts in the context of this new approach are mineral equilibria calculations with the software THERMOCALC and high-pressure experiments conducted from 10–20 kbar and 900–1100 °C, both combined in a forward modelling approach. Initially, P–T pseudosections for natural rock compositions with increasing MgO contents were calculated in the system NCFMASHTO (Na2O–CaO–FeO–MgO–Al2O3–SiO2–H2O–TiO2) to ascertain the metamorphic products from rocks with increasing MgO contents from a MORB up to a komatiite. A small number of previous experiments on komatiites showed the development of pyroxenite instead of eclogite and garnet-amphibolite during metamorphism and established that melts of these pyroxenites are of basaltic composition, thus again building oceanic crust instead of continental crust.rnThe P–T pseudosections calculated represent a continuous development of their metamorphic products from amphibolites and eclogites towards pyroxenites. On the basis of these calculations and the changes within the range of compositions, three picritic Models of Archean Oceanic Crust (MAOC) were established with different MgO contents (11, 13 and 15 wt%) ranging between basalt and komatiite. The thermodynamic modelling for MAOC 11, 13 and 15 at supersolidus conditions is imprecise since no appropriate melt model for metabasic rocks is currently available and the melt model for metapelitic rocks resulted in unsatisfactory calculations. The partially molten region is therfore covered by high-pressure experiments. The results of the experiments show a transition from predominantly tonalitic melts in MAOC 11 to basaltic melts in MAOC 15 and a solidus moving towards higher temperatures with increasing magnesium in the bulk composition. Tonalitic melts were generated in MAOC 11 and 13 at pressures up to 12.5 kbar in the presence of garnet, clinopyroxene, plagioclase plus/minus quartz (plus/minus orthopyroxene in the presence of quartz and at lower pressures) in the absence of amphibole but it could not be explicitly indicated whether the tonalitic melts coexisting with an eclogitic residue and rutile at 20 kbar do belong to the Archean TTG suite. Basaltic melts were generated predominantly in the presence of granulite facies residues such as amphibole plus/minus garnet, plagioclase, orthopyroxene that lack quartz in all MAOC compositions at pressures up to 15 kbar. rnThe tonalitic melts generated in MAOC 11 and 13 indicate that thicker oceanic crust with more magnesium than that of a modern basalt is also a viable source for the generation of TTG-like melts and therefore continental crust in the Archean. The experimental results are related to different geologic settings as a function of pressure. The favoured setting for the generation of early TTG-like melts at 15 kbar is the base of an oceanic crust thicker than existing today or by melting of slabs in shallow subduction zones, both without interaction of tonalic melts with the mantle. Tonalitic melts at 20 kbar may have been generated below the plagioclase stability by slab melting in deeper subduction zones that have developed with time during the progressive cooling of the Earth, but it is unlikely that those melts reached lower pressure levels without further mantle interaction.rn

Measurement of the ZZ production cross section and limits to the anomalous triple gauge couplings with forward electrons with the ATLAS detector

Measurements of the self coupling between bosons are important to test the electroweak sector of the Standard Model (SM). The production of pairs of Z bosons through the s-channel is forbidden in the SM. The presence of physics, beyond the SM, could lead to a deviation of the expected production cross section of pairs of Z bosons due to the so called anomalous Triple Gauge Couplings (aTGC). Proton-proton data collisions at the Large Hadron Collider (LHC) recorded by the ATLAS detector at a center of mass energy of 8 TeV were analyzed corresponding to an integrated luminosity of 20.3 fb-1. Pairs of Z bosons decaying into two electron-positron pairs are searched for in the data sample. The effect of the inclusion of detector regions corresponding to high values of the pseudorapidity was studied to enlarge the phase space available for the measurement of the ZZ production. The number of ZZ candidates was determined and the ZZ production cross section was measured to be: rn7.3±1.0(Stat.)±0.4(Sys.)±0.2(lumi.)pb, which is consistent with the SM expectation value of 7.2±0.3pb. Limits on the aTGCs were derived using the observed yield, which are twice as stringent as previous limits obtained by ATLAS at a center of mass energy of 7 TeV.