Modeling of drift-diffusion systems

Abstract

We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or -in the case of cross diffusion -even constant. (2000). 35K55, 80A20, 82D37, 82C31, 35B50, 35G25. Mathematics Subject Classification

Positivity and time behavior of a linear reaction–diffusion system, non‐local in space and time

“…We prove that it is possible to choose such manifolds M ε and initial functions f ε such that the solution to (8)-(10) u ε (x, t) converges (in a certain sense) to the solution to (1)…”

“…Important for the positivity of the solution to our system is the absence of differential operators on the off-diagonal of the main part. This problem was investigated in [1] for general linear drift-diffusion systems without memory effects.…”

Exponential decay of the free energy for discretized electro-reaction–diffusion systems

“…2. If (3.15) would be false, then we find v n ∈ N ρ , n ∈ N, such that 16) and lim n→∞ C n = +∞. Let ζ n denote the vector of the corresponding electrochemical potentials and a k ni = e ζ k ni the electrochemical activities.…”

“…To describe the fluxes j i of the species X i we need the electrochemical potentials ζ i := v i + q i v 0 . According to [1,8,16], we assume that the driving force for the flux is the antigradient of the electrochemical potential and that the flux is proportional to the inverse Hessian. In the simplest case, with Boltzmann statistics and no anisotropies of the material, j i is proportional to −u i ∇ζ i .…”