arXiv subjects

Florian Méhats

Publications and source records attributed to Florian Méhats.

30 records · Page 2Linked to original sources

arXiv2014

Models of dark matter halos based on statistical mechanics: II. The fermionic King model

We discuss the nature of phase transitions in the fermionic King model which describes tidally truncated quantum self-gravitating systems. This distribution function takes into account the escape of high energy particles and has a finite mass. On the other hand, the Pauli exclusion principle puts an upper bound on the phase space density of the system and stabilizes it against gravitational collapse. As a result, there exists a statistical equilibrium state for any accessible values of energy and temperature. We plot the caloric curves and investigate the nature of phase transitions as a function of the degeneracy parameter in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. Phase transitions can take place between a "gaseous" phase unaffected by quantum mechanics and a "condensed" phase dominated by quantum mechanics. The phase diagram exhibits two critical points, one in each ensemble, beyond which the phase transitions disappear. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. We apply the fermionic King model to the case of dark matter halos made of massive neutrinos. The gaseous phase describes large halos and the condensed phase describes dwarf halos. Partially degenerate configurations describe intermediate size halos. We argue that large dark matter halos cannot harbor a fermion ball because these nucleus-halo configurations are thermodynamically unstable (saddle points of entropy). Large dark matter halos may rather contain a central black hole resulting from a dynamical instability of relativistic origin occurring during the gravothermal catastrophe.

astro-ph.CO↗

arXiv2013

Analysis of models for quantum transport of electrons in graphene layers

We present and analyze two mathematical models for the self consistent quantum transport of electrons in a graphene layer. We treat two situations. First, when the particles can move in all the plane $\RR^2$, the model takes the form of a system of massless Dirac equations coupled together by a selfconsistent potential, which is the trace in the plane of the graphene of the 3D Poisson potential associated to surface densities. In this case, we prove local in time existence and uniqueness of a solution in $H^s(\RR^2)$, for $s > 3/8$ which includes in particular the energy space $H^{1/2}(\RR^2)$. The main tools that enable to reach $s\in (3/8,1/2)$ are the dispersive Strichartz estimates that we generalized here for mixed quantum states. Second, we consider a situation where the particles are constrained in a regular bounded domain $Ω$. In order to take into account Dirichlet boundary conditions which are not compatible with the Dirac Hamiltonian $H_{0}$, we propose a different model built on a modified Hamiltonian displaying the same energy band diagram as $H_{0}$ near the Dirac points. The well-posedness of the system in this case is proved in $H^s_{A}$, the domain of the fractional order Dirichlet Laplacian operator, for $1/2\leq s<5/2$.

math.AP↗

arXiv2013

On the controllability of quantum transport in an electronic nanostructure

We investigate the controllability of quantum electrons trapped in a two-dimensional device, typically a MOS field-effect transistor. The problem is modeled by the Schrödinger equation in a bounded domain coupled to the Poisson equation for the electrical potential. The controller acts on the system through the boundary condition on the potential, on a part of the boundary modeling the gate. We prove that, generically with respect to the shape of the domain and boundary conditions on the gate, the device is controllable. We also consider control properties of a more realistic nonlinear version of the device, taking into account the self-consistent electrostatic Poisson potential.

math.AP↗

arXiv2013

Solving highly-oscillatory NLS with SAM: numerical efficiency and geometric properties

In this paper, we present the Stroboscopic Averaging Method (SAM), recently introduced in [7,8,10,12], which aims at numerically solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schrödinger equation on the 1-dimensional torus and on the real line with harmonic potential, with the aim of assessing its efficiency: as compared to the well-established standard splitting schemes, the stiffer the problem is, the larger the speed-up grows (up to a factor 100 in our tests). The geometric properties of SAM are also explored: on very long time intervals, symmetric implementations of the method show a very good preservation of the mass invariant and of the energy. In a second series of experiments on 2-dimensional equations, we demonstrate the ability of SAM to capture qualitatively the long-time evolution of the solution (without spurring high oscillations).

math.NA↗

arXiv2013

Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations

This work is devoted to the numerical simulation of nonlinear Schrödinger and Klein-Gordon equations. We present a general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency. This is a stronger feature than the usual so called "Asymptotic preserving" property, the last being also satisfied by our scheme in the highly oscillatory limit. Our strategy enables to simulate the oscillatory problem without using any mesh or time step refinement, and the orders of our schemes are preserved uniformly in all regimes. In other words, since our numerical method is not based on the derivation and the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, in the highly oscillatory limit regime, and in the intermediate regime with the same order of accuracy. In the same spirit as in [5], the method is based on two main ingredients. First, we embed our problem in a suitable "two-scal" reformulation with the introduction of an additional variable. Then a link is made with classical strategies based on Chapman-Enskog expansions in kinetic theory despite the dispersive context of the targeted equations, allowing to separate the fast time scale from the slow one. Uniformly accurate (UA) schemes are eventually derived from this new formulation and their properties and performances are assessed both theoretically and numerically.

math.NA↗

arXiv2012

Stable ground states and self-similar blow-up solutions for the gravitational Vlasov-Manev system

In this work, we study the orbital stability of steady states and the existence of blow-up self-similar solutions to the so-called Vlasov-Manev (VM) system. This system is a kinetic model which has a similar Vlasov structure as the classical Vlasov-Poisson system, but is coupled to a potential in $-1/r- 1/r^2$ (Manev potential) instead of the usual gravitational potential in $-1/r$, and in particular the potential field does not satisfy a Poisson equation but a fractional-Laplacian equation. We first prove the orbital stability of the ground states type solutions which are constructed as minimizers of the Hamiltonian, following the classical strategy: compactness of the minimizing sequences and the rigidity of the flow. However, in driving this analysis, there are two mathematical obstacles: the first one is related to the possible blow-up of solutions to the VM system, which we overcome by imposing a sub-critical condition on the constraints of the variational problem. The second difficulty (and the most important) is related to the nature of the Euler-Lagrange equations (fractional-Laplacian equations) to which classical results for the Poisson equation do not extend. We overcome this difficulty by proving the uniqueness of the minimizer under equimeasurabilty constraints, using only the regularity of the potential and not the fractional-Laplacian Euler-Lagrange equations itself. In the second part of this work, we prove the existence of exact self-similar blow-up solutions to the Vlasov-Manev equation, with initial data arbitrarily close to ground states. This construction is based on a suitable variational problem with equimeasurability constraint.

math.AP↗

arXiv2012

An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit

We consider the semiclassical limit for the nonlinear Schrodinger equation. We introduce a phase/amplitude representation given by a system similar to the hydrodynamical formulation, whose novelty consists in including some asymptotically vanishing viscosity. We prove that the system is always locally well-posed in a class of Sobolev spaces, and globally well-posed for a fixed positive Planck constant in the one-dimensional case. We propose a second order numerical scheme which is asymptotic preserving. Before singularities appear in the limiting Euler equation, we recover the quadratic physical observables as well as the wave function with mesh size and time step independent of the Planck constant. This approach is also well suited to the linear Schrodinger equation.

math.NA↗

arXiv2012

Asymptotic preserving schemes for highly oscillatory kinetic equation

This work is devoted to the numerical simulation of a Vlasov-Poisson model describing a charged particle beam under the action of a rapidly oscillating external electric field. We construct an Asymptotic Preserving numerical scheme for this kinetic equation in the highly oscillatory limit. This scheme enables to simulate the problem without using any time step refinement technique. Moreover, since our numerical method is not based on the derivation of the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, and in the highly oscillatory regime as well. Our method is based on a "double-scale" reformulation of the initial equation, with the introduction of an additional periodic variable.

math.NA↗

arXiv2012

Micro-macro schemes for kinetic equations including boundary layers

We introduce a new micro-macro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution function $f$ in all its domain as the sum of an equilibrium adapted to the boundary (which is not the usual equilibrium associated with $f$) and a remaining kinetic part. This equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function. A consequence of this strategy is that no artificial boundary condition is needed in the micro-macro models and the exact boundary condition on $f$ is naturally transposed to the macro part of the model. This method provides an 'Asymptotic preserving' numerical scheme which generates a very good approximation of the space boundary values at the diffusive limit, without any mesh refinement in the boundary layers. Our numerical results are in very good agreement with the exact so-called Chandrasekhar value, which is explicitely known in some simple cases.

math.NA↗

arXiv2011

A boundary matching micro/macro decomposition for kinetic equations

We introduce a new micro/macro decomposition of collisional kinetic equations which naturally incorporates the exact space boundary conditions. The idea is to write the distribution fonction $f$ in all its domain as the sum of a Maxwellian adapted to the boundary (which is not the usual Maxwellian associated with $f$) and a reminder kinetic part. This Maxwellian is defined such that its 'incoming' velocity moments coincide with the 'incoming' velocity moments of the distribution function. Important consequences of this strategy are the following. i) No artificial boundary condition is needed in the micro/macro models and the exact boundary condition on $f$ is naturally transposed to the macro part of the model. ii) It provides a new class of the so-called 'Asymptotic preserving' (AP) numerical schemes: such schemes are consistent with the original kinetic equation for all fixed positive value of the Knudsen number $\eps$, and if $\eps \to 0 $ with fixed numerical parameters then these schemes degenerate into consistent numerical schemes for the various corresponding asymptotic fluid or diffusive models. Here, the strategy provides AP schemes not only inside the physical domain but also in the space boundary layers. We provide a numerical test in the case of a diffusion limit of the one-group transport equation, and show that our AP scheme recovers the boundary layer and a good approximation of the theoretical boundary value, which is usually computed from to the so-called Chandrasekhar function.

math.NA↗

arXiv2010

The Schrödinger-Poisson system on the sphere

We study the Schrödinger-Poisson system on the unit sphere $\SS^2$ of $\RR^3$, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results concern the Cauchy problem for this system. We prove that this problem is regularly well-posed on every $H^s(\SS ^2)$ with $s>0$, and not uniformly well-posed on $L^2(\SS ^2)$. The proof of well-posedness relies on multilinear Strichartz estimates, the proof of ill-posedness relies on the construction of a counterexample which concentrates exponentially on a closed geodesic. In a second part of the paper, we prove that this model can be obtained as the limit of the three dimensional Schrödinger-Poisson system, singularly perturbed by an external potential that confines the particles in the vicinity of the sphere.

math.AP↗

arXiv2010

An inverse problem in quantum statistical physics

We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.

math.AP↗