Particle hydrodynamic moment models in biology and microelectronics: singular relaxation limits

The hydrodynamic model has become important in the simulation of semiconductor devices during the last decade, as an alternative to the more specialized drift-diffusion model. The more comprehensive hydrodynamic model includes features of general carrier heating, velocity overshoot, and various small device features, and is based upon moments of the Boltzmann equation. It employs a macroscopic relaxation time approximation, which incorporates averaged collision mechanisms. Unlike the drift-diffusion model, the hydrodynamic model contains hyperbolic modes related to the momentum subsystem. It is therefore much more complex to analyze and simulate than the corresponding driftdiffusion model. A detailed introduction to both models may be found in the book [9]. In particular, it is demonstrated how an averaging process over group velocity space, coupled with identification of particle density as one such average, and velocity as a first moment, leads to a system whose macroscopic characterization is a conservation law system, much in the tradition of classical gas dynamics, with forcing terms due to electrical, mechanical, and heating effects, as well as frictional dissipation due to collisions. When the microscopic and macroscopic derivations are correlated, the moment closure assumptions emerge more clearly. For example, isotropy leads to the ideal gas law for the carriers. Readers may wish to consult the reference [2] for an in depth survey of some of these ideas. The reader interested in a closer explanation of the engineering aspects of semiconductor devices can consult the book [15]. In recent years, there has been an attempt to understand the relationship between the two classes of models. In the biological context of the open ionic channel, it was noticed phenomenologically that small variations of the saturation velocity in the energy relaxation term led to significant variation in the computed temperature (cf. [3]). Indeed, this interpretation suggests that the temperature tends to that of the isothermal drift-diffusion regime, which is seen as a high friction regime. Some mathematical studies suggest this also (cf. [14]). Another way of viewing high friction is in the framework of the inertial approximation, which here would assert that the drift-diffusion regime (nonisothermal) is the limit of increasingly small momentum rate of change, with respect to frequency of collisions. This approach is discussed in [10]. In this paper, we shall study the reduced hydrodynamic model, sometimes referred to as the perturbed isentropic model, with adiabatic pressure density relationship. In fact, for the most general initial conditions, rigorous mathematical results exist only in this case (see [18] for the initialboundary value problem), though recent studies have extended the models to include multi-species and geometric structure, thus permitting multi-dimensional transport with symmetry to be analyzed (cf. [5]). The paper [5] includes comprehensive simulations based on the shock capturing algorithm, ENO (cf. [16] for description of the algorithm). In particular, [5] analyzes in depth the two-valley