On spectral minimal partitions II, the case of the rectangle

In continuation of \cite{HHOT}, we discuss the question of spectral minimal 3-partitions for the rectangle $]-\frac a2,\frac a2[\times ] -\frac b2,\frac b2[$, with $0< a\leq b$. It has been observed in \cite{HHOT} that when $0<\frac ab < \sqrt{\frac 38}$ the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles $]-\frac a2,\frac a2[\times ] -\frac b2,-\frac b6[$, $]-\frac a2,\frac a2[\times ] -\frac b6,\frac b6[$ and $]-\frac a2,\frac a2[\times ] \frac b6, \frac b2[$. We will describe a possible mechanism of transition for increasing $\frac ab$ between these nodal minimal 3-partitions and non nodal minimal 3-partitions at the value $\sqrt{\frac 38}$ and discuss the existence of symmetric candidates for giving minimal 3-partitions when $\sqrt{\frac 38}<\frac ab \leq 1$. Numerical analysis leads very naturally to nice questions of isospectrality which are solved by introducing Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle

Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations

In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schr\"odinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method

The Schr\"odinger operator on an infinite wedge with a tangent magnetic field

We study a model Schr\"odinger operator with constant magnetic field on an infinite wedge with Neumann boundary condition. The magnetic field is assumed to be tangent to a face. We compare the bottom of the spectrum to the model spectral quantities coming from the regular case. We are particularly motivated by the influence of the magnetic field and the opening angle of the wedge on the spectrum of the model operator and we exhibit cases where the bottom of the spectrum is smaller than in the regular case. Numerical computations enlighten the theoretical approach

On the third critical field in Ginzburg-Landau theory

Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, $H_{C_3}$, describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near $H_{C_3}$ only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables

Experimental and thermodynamic calculations results on pwr and srf corium subsystems

International audienceDuring a severe accident in a pressurized water reactor (PWR), the nuclear oxide fuel (UO2 or (U,Pu)O2) could react with Zircaloy cladding, Inconel spacer grids, the steel controls rods cladding and the neutronic absorbers (Ag-Cd-In-B4C), leading to the relocation in the lower-head of the reactor vessel of a mixture of liquid and solid phases called in-vessel corium. If the reactor vessel is lost, the molten core can pour onto the containment concrete leading to the formation of the ex-vessel corium. As a first approximation and considering the major components, the system U-Pu-Zr-Fe-Al-Ca-Si-O can be considered as representative of the ex-vessel corium. At the Laboratory of Modelling, Thermodynamics and Thermochemistry, CEA Saclay, a series of experimental results have been achieved using ATTILHA, the novel experimental setup developed at the laboratory. These data are fundamental for the development of a reliable thermodynamic database for corium. Thermodynamic calculations have been performed to better interpret our experimental results. These calculations allow also to study the evolution of corium behavior varying different parameters, as for example the nature of the atmosphere (reducing/oxidizing), to reproduce different scenarios. Using the thermodynamic database of the representative corium developed at the laboratory, it is also possible to estimate the partition of elements in each phase as a function of temperature and composition. It can be shown that at 3000 K, in presence of a miscibility gap in the liquid phase, Pu preferentially segregate in one of the two immiscible liquids, namely in the oxide liquid

Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations

In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schrödinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method