Phase-space formulation

Approaches to Quantum Transport

Published in David K. Ferry, An Introduction to Quantum Transport in Semiconductors, 2017

One problem that is always raised with respect to the density matrix is that it is a function only of time and the position variables. In classical physics, one has the Boltzmann equation, which is a function of position, momentum, and time, so that one can generate a phase–space formulation of the transport problem. None of the quantum functions we have discussed so far has this convenient phase–space formulation. But it is often very convenient to describe the system with a phase–space formulation. And, in fact, such a formulation exists in which we first introduce the center-of-mass and difference coordinates through () R=r+r′2,s=r−r′.

Multiscale modelling and splitting approaches for fluids composed of Coulomb-interacting particles

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Journal Information

Published in Mathematical and Computer Modelling of Dynamical Systems, 2018

Jürgen Geiser

Domain Decomposition (Grid-based): Here, we underlie a spatial domain. Such methods are also known as particle-mesh methods, e.g. particle-in-cell methods (PIC), see [76]. We have to deal with particle and field equations, see [38]. The results of the phase-space formulation are interpolated to the grid formulation and vice versa, see [38]. Such methods decompose the domain, i.e. parallel domain decomposition methods, and are more delicate to parallelize, see [77].

Phase-space formulation

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Approaches to Quantum Transport

Multiscale modelling and splitting approaches for fluids composed of Coulomb-interacting particles