Social Justice and the City

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory

The one-dimensional two-species quantum hydrodynamic model is considered in the limit of small mass ratio of the charge carriers. Closure is obtained by adopting an equation of state pertaining to a zero-temperature Fermi gas for the electrons and by disregarding pressure effects for the ions. By an appropriate rescaling of the variables, a nondimensional parameter H, proportional to quantum diffraction effects, is identified. The system is then shown to support linear waves, which in the limit of small H resemble the classical ion-acoustic waves. In the weakly nonlinear limit, the quantum plasma is shown to support waves described by a deformed Korteweg–de Vries equation which depends in a nontrivial way on the quantum parameter H. In the fully nonlinear regime, the system also admits traveling waves which can exhibit periodic patterns. The quasineutral limit of the system is also discussed.

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Quantum ion-acoustic waves

Statistical Mechanics. — A Set of Lectures

The quantum hydrodynamic model for charged particle systems is extended to the cases of nonzero magnetic fields. In this way, quantum corrections to magnetohydrodynamics are obtained starting from the quantum hydrodynamical model with magnetic fields. The importance of the quantum corrections is described by a parameter H which can be significant in dense astrophysical plasmas. The quantum magnetohydrodynamic model is analyzed in the infinite conductivity limit. The conditions for equilibrium in ideal quantum magnetohydrodynamics are established. Translationally invariant exact equilibrium solutions are obtained in the case of the ideal quantum magnetohydrodynamic model.

A magnetohydrodynamic model for quantum plasmas

We analyse the classical limit of the quantum hydrodynamic equations as the Planck constant tends to zero The equations have the form of an Euler system with a constant pressure and a dispersive regularisation term, which (formally) tends to zero in the classical limit The main tool of the analysis is the exploitation of a kinetic equation, which lies behind the quantum hydrodynamic system The presented analysis can also be interpreted as an alternative approach to the geometrical optics (WKB)-analysis of the Schrequation

Quantum hydrodynamics, Wigner transforms, the classical limit

By using asymptotic analysis we derive most of the known and also new nonisothermal pipeline models starting from transient gas equations. We introduce proper scalings to identify valid regimes for the derived models and extend them to networks. Finally, we perform numerical simulations on a single pipe as well-posed as on a small network. We compare both isothermal and nonisothermal flow and pressure predictions with results obtained from the literature.

Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks

Bifurcation analysis of a class of ‘car following’ traffic models

In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schr¨odinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schr¨odinger 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 Schr¨odinger 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.

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Ingenuin Gasser

Papers

Quantum hydrodynamics, Wigner transforms, the classical limit

Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks

Bifurcation analysis of a class of ‘car following’ traffic models

A review of dispersive limits of (non)linear schr¨odinger-type equations

Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors