Nonlinear Drift-diffusion Equations Research Articles

Exponential Convergence to Equilibrium for Coupled Systems of Nonlinear Degenerate Drift Diffusion Equations

We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter . The nonlinearities and potentials are chosen such that in the decoupled system for , the evolution is metrically contractive, with a global rate . The coupling is a singular perturbation in the sense that for any , contractivity of the system is lost. Our main result is that for all sufficiently small , the global attraction to a unique steady state persists, with an exponential rate for some . The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.

Stochastic simulation algorithms for solving a nonlinear system of drift–diffusion-Poisson equations of semiconductors

Stochastic simulation algorithms for solving transient nonlinear drift diffusion recombination transport equations are developed. The governing system of equations includes two drift–diffusion equations coupled with a Poisson equation for the potential whose gradient forms the drift velocity. A stochastic algorithm for solving nonlinear drift–diffusion equations is proposed here for the first time. In each time step, the method calculates the solution on a cloud of points using a new global Monte Carlo random walk and Cellular Automata algorithms. The Poisson equation is solved by a global version of the Random Walk on Spheres method which calculates both the solutions and the derivatives without using finite difference approximations. The method is also able to calculate fluxes to any desired part of the boundary, from arbitrary sources. For transient drift–diffusion equations we suggest a stochastic expansion from cell to cell algorithm for calculating the whole solution field. All new global random walk algorithms developed in this paper are validated by comparing the simulation results with exact solutions. Application of the developed method to solve a system of 2D transport equations for electrons and holes in a semiconductor is given.

A convergent Lagrangian discretization for <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Wasserstein and flux-limited diffusion equations

We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form $ \partial_t u = \partial_x(u \cdot ({\sf c}^*)^\prime[\partial_x \mathit{h}^\prime(u)+ \mathit{v}^\prime]) $, like Fokker-Plank and $ q $-Laplace equations, on an interval. This scheme will consist of a spatio-temporal discretization founded on the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense $ p $-Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost inducing equations like Rosenau's relativistic heat equation. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.

Singular limit of the porous medium equation with a drift

We study the “stiff pressure limit” of a nonlinear drift-diffusion equation, where the density is constrained to stay below the maximal value one. The challenge lies in the presence of a drift and the consequent lack of monotonicity in time. In the limit a Hele-Shaw-type free boundary problem emerges, which describes the evolution of the congested zone where density equals one. We discuss pointwise convergence of the densities as well as the BV regularity of the limiting free boundary.

Three-Dimensional Numerical Study on the Efficiency Droop in InGaN/GaN Light-Emitting Diodes

The efficiency droop characteristics of single quantum well (SQW) InGaN/GaN light-emitting diodes (LEDs) including the phase-space filling (PSF) effect are predicted by a three-dimensional (3-D) numerical simulation. The carrier transport is based on the solution of the 3-D non-linear Poisson and drift-diffusion equations for both holes and electrons. A modified formulation of the Shockley-Reed-Hall (SRH) coefficient is proposed to describe the SRH carrier lifetime behavior, which increases at a low excitation level and decreases at a higher one. The current crowding causes a non-uniform distribution of the carrier concentration in the active layer that leads to the inversion of the local internal quantum efficiency (IQE) under the n-pad region when the injection current density increases from low to high levels. To further understand the correlation of the efficiency droop with the PSF effect, we systematically investigate carrier transport in the SQW InGaN/GaN LEDs and how the different PSF effect coefficients affect the current-voltage curve and IQE. The lumped IQE found in this study agrees well with previous experimental measurements. Moreover, the PSF effect has a strong impact on the IQE behavior including its peak and droop in efficiency.

Computer Algebra Challenges in Nanotechnology: Accurate Modeling of Nanoscale Electro-optic Devices Using Finite Elements Method

The simulation of silicon-based light-emitting and photodetectors nanodevices using computer algebra became a challenge. These devices couple the hyperbolic equations of electromagnetic radiation, the parabolic equations of heat conduction, the elliptic equations describing electric potential, and the eigenvalue equations of quantum mechanics—with the nonlinear drift–diffusion equations of the semiconductor physics. These complex equations must be solved by using generally mixed Dirichlet–Neumann boundary conditions in three-dimensional geometries. Comsol Multiphysics modeling software is employed integrated with MATLAB–SIMULINK and Zemax. The physical equations are discretized on a mesh using the Galerkin finite element method (FEM) and to a lesser extent the method of finite volumes. The equations can be implemented in a variety of forms such as directly as a partial differential equation, or as a variational integral, the so-called weak form. Boundary conditions may also be imposed directly or using variational constraint and reaction forces. Both choices have implication for convergence and physicality of the solution. The mesh is assembled from triangular or quadrilateral elements in two-dimensions, and hexahedral or prismatic elements in three dimensions, using a variety of algorithms. Solution is achieved using direct or iterative linear solvers and nonlinear solvers. The former are based on conjugate gradients, the latter generally on Newton–Raphson iterations. The general framework of FEM discretization, meshing and solver algorithms will be presented together with techniques for dealing with challenges such as multiple time scales, shocks and nonconvergence; these include load ramping, segregated iterations, and adaptive meshing.

Regularity Properties of Degenerate Diffusion Equations with Drifts

This paper considers a class of nonlinear degenerate drift-diffusion equations. We study well-posedness and regularity properties of the solutions with the goal of achieving uniform Holder regulari...

High-frequency limit of non-autonomous gradient flows of functionals with time-periodic forcing

We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way. On grounds of the existence results by Ferreira and Guevara (2015) on non-autonomous gradient flows (which we also extend to a broader range of energy functionals), we prove that the associated solution curves converge to a solution of the time-averaged evolution equation in the limit of infinite frequency. Under additional assumptions on the energy, we obtain an explicit rate of convergence. Furthermore, we specifically investigate nonlinear drift-diffusion equations with time-dependent drift which formally are gradient flows with respect to the L2-Wasserstein distance. We prove that a family of weak solutions obtained as a limit of the Minimizing Movements scheme exhibits the above-mentioned behavior in the high-frequency limit.

On a Nonlinear Parabolic Problem arising in the Quantum Diffusive Description of a Degenerate Fermion Gas

This article studies, both theoretically and numerically, a nonlinear drift-diffusion equation describing a gas of fermions in the zero-temperature limit. The equation is considered on a bounded domain whose boundary is divided into an "insulating" part, where homogeneous Neumann conditions are imposed, and a "contact" part, where nonhomogeneous Dirichlet data are assigned. The existence of stationary solutions for a suitable class of Dirichlet data is proven by assuming a simple domain configuration. The long-time behavior of the time-dependent solution, for more complex domain configurations, is investigated by means of numerical experiments.

Transport of dipolar excitons in (Al,Ga)N/GaN quantum wells

We investigate the transport of dipolar indirect excitons along the growth plane of polar (Al,Ga)N/GaN quantum well structures by means of spatially- and time-resolved photoluminescence spectroscopy. The transport in these strongly disordered quantum wells is activated by dipole-dipole repulsion. The latter induces an emission blue shift that increases linearly with exciton density, whereas the radiative recombination rate increases exponentially. Under continuous, localized excitation, we measure a continuous red shift of the emission, as excitons propagate away from the excitation spot. This shift corresponds to a steady-state gradient of exciton density, measured over several tens of micrometers. Time-resolved micro-photoluminescence experiments provide information on the dynamics of recombination and transport of dipolar excitons. We account for the ensemble of experimental results by solving the nonlinear drift-diffusion equation. Quantitative analysis suggests that in such structures, exciton propagation on the scale of 10 to 20 microns is mainly driven by diffusion, rather than by drift, due to the strong disorder and the presence of nonradiative defects. Secondary exciton creation, most probably by the intense higher-energy luminescence, guided along the sample plane, is shown to contribute to the exciton emission pattern on the scale up to 100 microns. The exciton propagation length is strongly temperature dependent, the emission being quenched beyond a critical distance governed by nonradiative recombination.

Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations

We exhibit a large class of Lyapunov functionals for nonlinear drift–diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero-range process and the Ginzburg–Landau dynamics, which we describe briefly. We prove, as an application, linear inequalities between such an entropy-like functional and its entropy production functional for the boundary-driven porous medium equation in a bounded domain with positive Dirichlet conditions: this implies exponential rates of relaxation related to the first Dirichlet eigenvalue of the domain. We also derive Lyapunov functions for systems of nonlinear diffusion equations, and for nonlinear Markov processes with non-reversible stationary measures.

Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties of the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We also present results from numerical experiments.

Cell polarisation model: The 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.

Modeling the sequential electric field assisted diffusion into glass of two ion species

The sequential electric field assisted diffusion of two ion species into glass is modeled in this work. Using standard assumptions, two coupled nonlinear drift-diffusion equations are derived for the ion concentration distributions. These equations are solved numerically for the case where the two ion species each have a lower mobility than the indigenous sodium ions in the glass. It is shown that stationary state distributions form if the final species injected into the glass is the less mobile of the two. Analytical expressions are derived for the stationary state distributions. A possible application in integrated optics involving Ag+ and Cu+ ion diffusion is considered.

On the regularity of solutions to the drift-diffusion approximation equations in gas discharge theory

An initial-boundary value problem is stated for the nonlinear drift-diffusion equations, which are widely used in gas discharge simulation. The class of nonlinear boundary conditions that ensure the existence and uniqueness of strong solutions is determined.

Modelling the stochastic single particle dynamics of relativistic fermions and bosons using nonlinear drift-diffusion equations

Recently, there is a considerable interest in quantum kinetic models of fermions and bosons given by generalised Boltzmann equations and their corresponding diffusion approximations: nonlinear drift-diffusion equations. In this context, we propose a model that describes the dynamical properties of relativistic fermions and bosons consistent with the Fermi-Dirac and Bose-Einstein statistics. In particular, we discuss the first- and second-order statistics of quantum particle trajectories in energy space.

A Coupled Model for Radiative Transfer: Doppler Effects, Equilibrium, and Nonequilibrium Diffusion Asymptotics

This paper is devoted to the asymptotic analysis of a coupled model arising in radiative transfer. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled to a diffusion equation satisfied by the material temperature. The interaction terms take into account both scattering and absorption/emission phenomena, as well as Doppler corrections. Two asymptotic regimes are identified, depending on the scaling assumptions about the physical parameters and observation scales. In the equilibrium regime, the system is driven only by the material temperature which satisfies a nonlinear drift-diffusion equation. In the nonequilibrium regime, the radiation temperature and the material temperature will be coupled by a system of nonlinear drift-diffusion equations.

Preface

We review a Fokker-Planck approach to nonlinear evolution equations such as nonlinear diffusion equations and nonlinear drift-diffusion equations and extend this approach to nonlinear reaction-diffusion equations. Using this Fokker-Planck approach along with appropriately defined entropy and free energy measures, we show that transient solutions converge to stationary ones in the long time limit. Implications for bifurcation theory are also addressed.

Asymptotic properties of nonlinear diffusion, nonlinear drift-diffusion, and nonlinear reaction-diffusion equations

We review a Fokker-Planck approach to nonlinear evolution equations such as nonlinear diffusion equations and nonlinear drift-diffusion equations and extend this approach to nonlinear reaction-diffusion equations. Using this Fokker-Planck approach along with appropriately defined entropy and free energy measures, we show that transient solutions converge to stationary ones in the long time limit. Implications for bifurcation theory are also addressed.

On fluid limit for the semiconductors Boltzmann equation

This paper is devoted to the derivation of (non-linear) drift-diffusion equations from the semiconductor Boltzmann equation. Collisions are taken into account through the non-linear Pauli operator, but we do not assume relation on the cross section such as the so-called detailed balance principle. In turn, equilibrium states are implicitly defined. This article follows and completes the contribution of Mellet (Monatsh. Math. 134 (4) (2002) 305–329) where the electric field is given and does not depend on time. Here, we treat the self-consistent problem, the electric potential satisfying the Poisson equation. By means of a Hilbert expansion, we shall formally derive the asymptotic model in the general case. We shall then rigorously prove the convergence in the one-dimensional case by using a modified Hilbert expansion.