Anton Arnold

ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS

It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view. In our unified presentation of the theory we present new results to the following topics: an elementary derivation of Bakry-Emery type conditions, results concerning perturbations of invariant measures with general admissible entropies, sharpness of convex Sobolev inequalities, applications to non-symmetric linear and certain non-linear Fokker-Planck type equations (Desai-Zwanzig model, drift-diffusion-Poisson model).

Discrete transparent boundary conditions for the Schrödinger equation

This paper is concerned with transparent boundary conditions for the one dimensional time–dependent Schrodinger equation. They are used to restrict the original PDE problem that is posed on an unbounded domain onto a finite interval in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretization of such transparent boundary conditions in conjunction with some chosen discretization of the PDE (usually Crank–Nicolson finite differences in the case of the Schrodinger equation). The presented discrete transparent boundary conditions yield an unconditionally stable numerical scheme and are completely reflection–free at the boundary.

An Analysis of Quantum Fokker-Planck Models: A Wigner Function Approach

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.

MATHEMATICAL CONCEPTS OF OPEN QUANTUM BOUNDARY CONDITIONS

This paper is concerned with the derivation and the numerical discretization of open boundary conditions for the 1D Schrödinger equation in order to simulate quantum devices. New discrete transparent boundary conditions are presented that are able to handle the situation of a continuous plane wave influx into a device. Also, we give a review of various formulations of boundary conditions that are used within the Schrödinger and the Wigner formalism of quantum mechanics, and we discuss their mathematical properties.

Self-consistent relaxation-time models in quantum mechanics

ABSTRACT. This paper is concerned with the relaxation-time von Neumann- Poisson (or quantum Liouville-Poisson) equation in three spatial dimensions which describes the self-consistent time evolution of an open quantum me- chanical system that includes some relaxation mechanism. This model and the equivalent relaxation-time Wigner-Poisson system play an important role in the simulation of quantum semiconductor devices. For initial density matrices with finite kinetic energy, we prove that this problem, formulated in the space of Hermitian trace class operators, admits a unique global strong solution. A key ingredient for our analysis is a new generalization of the Lieb-Thirring inequality for density matrix operators.

On generalized Csiszár-Kullback inequalities

Classical Csiszar-Kullback inequalities bound the L1-distance of two probability densities in terms of their relative (convex) entropies. Here we generalize such inequalities to not necessarily normalized and possibly non-positive L1 functions. Also, we analyse the optimality of the derived Csiszar-Kullback type inequalities and show that they are in many important cases significantly sharper than the classical ones (in terms of the functional dependence of the L1 bound on the relative entropy). Moreover our construction of these bounds is rather elementary.

An Operator Splitting Method for the Wigner--Poisson Problem

The Wigner--Poisson equation describes the quantum-mechanical motion of electrons in a self-consistent electrostatic field. The equation consists of a transport term and a non-linear pseudodifferential operator. In this paper we analyze an operator splitting method for the linear Wigner equation and the coupled Wigner--Poisson problem. For this semidiscretization in time, consistency and nonlinear stability are established in an L2-framework. We present a numerical example to illustrate the method.

On large time asymptotics for drift-diffusion-poisson systems

Abstract In this paper we analyze the convergence rate of solutions of certain drift-diffusion-Poisson systems to their unique steady state. These bi-polar equations model the transport of two populations of charged particles and have applications for semiconductor devices and plasmas. When prescribing a confinement potential for the particles we prove exponential convergence to the equilibrium. Without confinement the solution decays with an algebraic rate towards a self-similar state. The analysis is based on a relative entropy type functional and it uses logarithmic Sobolev inequalities.

Mathematical Properties of Quantum Evolution Equations

This chapter focuses on the mathematical analysis of nonlinear quantum transport equations that appear in the modeling of nano-scale semi-conductor devices. We start with a brief introduction on quantum devices like the resonant tunneling diode and quantum waveguides. For the mathematical analysis of quantum evolution equations we shall mostly focus on whole space problems to avoid the technicalities due to boundary conditions. We shall discuss three different quantum descriptions: Schrodinger wave functions, density matrices, and Wigner functions. For the Schrodinger–Poisson analysis (in H 1 and L 2) we present Strichartz inequalities. As for density matrices, we discuss both closed and open quantum systems (in Lindblad form). Their evolution is analyzed in the space of trace class operators and energy subspaces, employing Lieb–Thirring-type inequalities. For the analysis of the Wigner–Poisson–Fokker–Planck system we shall first derive (quantum) kinetic dispersion estimates (for Vlasov–Poisson and Wigner–Poisson). The large-time behavior of the linear Wigner–Fokker–Planck equation is based on the (parabolic) entropy method. Finally, we discuss boundary value problems in the Wigner framework.

LOW AND HIGH FIELD SCALING LIMITS FOR THE VLASOV– AND WIGNER–POISSON–FOKKER–PLANCK SYSTEMS

This paper is concerned with scaling limits in kinetic semiconductor models. For the classical Vlasov–Poisson–Fokker–Planck equation and its quantum mechanical counterpart, the Wigner–Poisson–Fokker–Planck equation, three distinguished scaling regimes are presented. Using Hilbert and Chapman–Enskog expansions, we derive two drift-diffusion type approximations. The test case of a n − – n – n + diode reveals that different scaling regimes may be present at the same time in different subregions of a semiconductor device. Numerical simulations of the stationary solution illustrate the good approximation of the kinetic solution by a drift-diffusion model and by a hybrid (adaptive domain decomposition) model.