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The classical time-dependent drift-diffusion model for semiconductors is considered for small scaled Debye length (which is a singular perturbation parameter). The corresponding limit is carried out on both the dielectric relaxation time scale and the diffusion time scale. The latter is a quasineutral limit, and the former can be interpreted as an initial time layer problem. The main mathematical tool for the analytically rigorous singular perturbation theory of this paper is the (physical) entropy of the system.

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In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schrödinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schrödinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schrödinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.