Reduced and extended weak coupling limit
The main aim of our lectures is to give a pedagogical introduction to various mathematical formalisms used to describe open quantum systems: completely positive semigroups, dilations of semigroups, quantum Langevin dynamics and the so-called Pauli-Fierz Hamiltonians. We explain two kinds of the weak coupling limit. Both of them show that Hamiltonian dynamics of a small quantum system interacting with a large resevoir can be approximated by simpler dynamics. The better known reduced weak coupling...
Regularity analysis for systems of reaction-diffusion equations
This paper is devoted to the study of the regularity of solutions to some systems of reaction–diffusion equations. In particular, we show the global boundedness and regularity of the solutions in one and two dimensions. In addition, we discuss the Hausdorff dimension of the set of singularities in higher dimensions. Our approach is inspired by De Giorgi’s method for elliptic regularity with rough coefficients. The proof uses the specific structure of the system to be considered and is not a mere...
Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off.
We show that in the setting of the spatially homogeneous Boltzmann equation without cut-off, the entropy dissipation associated to a function f ∈ L1(RN) yields a control of √f in Sobolev norms as soon as f is locally bounded below. Under this additional assumption of lower bound, our result is an improvement of a recent estimate given by P.-L. Lions, and is optimal in a certain sense.
Regularity in kinetic formulations via averaging lemmas
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity...
Regularity in kinetic formulations via averaging lemmas
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to...
Regularity of local manifolds in dispersing billiards.
Regularity properties of the diffusion coefficient for a mean zero exclusion process
Reinforced walk on graphs and neural networks
A directed-edge-reinforced random walk on graphs is considered. Criteria for the walk to end up in a limit cycle are given. Asymptotic stability of some neural networks is shown.
Relativity and irreversibility.
Relaxation lengths and non-negative solutions in neutron transport
Remarques sur la formulation cinétique des lois de conservation scalaires
Renormalization analysis of catalytic Wright-Fisher diffusions.
Repulsion of an evolving surface on walls with random heights
Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit
We consider the Pn model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P1 model...
Résultats de convergence et de non-convergence de l’équation de Von-Neumann périodique vers l’équation de Boltzmann quantique.
Sampling the Fermi statistics and other conditional product measures
Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km ln k for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non-homogeneous)...
Scaling limit and cube-root fluctuations in SOS surfaces above a wall
Consider the classical -dimensional Solid-On-Solid model above a hard wall on an box of . The model describes a crystal surface by assigning a non-negative integer height to each site in the box and 0 heights to its boundary. The probability of a surface configuration is proportional to , where is the inverse-temperature and sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures....
Scaling limit of particle systems, incompressible Navier-Stokes equation and Boltzmann equation.
Scaling of a random walk on a supercritical contact process
We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk...