AMaSiS 2018 Workshop: Abstracts

Energy-transport systems for optical lattices: Derivation, analysis, simulation

Marcel Braukhoff and Ansgar Jüngel

Technische Universität Wien, Institute for Analysis and Scientific Computing

In the study of semiconductors, adequate simplifying models are of big interest. A fermionic gas in an optical lattice is a physical model for a semiconductor, which can mathematically be described by a Boltzmann-type equation as well as energy-transport systems.

1. Semiconductor Boltzmann-Dirac-Benney equation

On the kinetic level, we consider

\partial_{t}f+\nabla\epsilon(p)\cdot\nabla_{x}f-U\nabla\int_{\mathbb{T}^{d}}f(% x,p,t)dp\cdot\nabla_{p}f=n_{f}(\mathcal{F}_{f}-f)\,,

(1)

where $\mathcal{F}_{f}$ denotes the Fermi-Dirac distribution associated to $f$ . Comparing an optical lattice to a semiconductor which Coulomb potential, the interaction potential here is significantly more singular. This leads to the following theorem.

Theorem 1. Eq. (1) is locally ill-posed.

2. Diffusive Limit

Using a diffusive scaling and a high temperature limit, we can formally derive an energy-transport model from (1), namely

\begin{split}\displaystyle\partial_{t}n&\displaystyle=\nabla\cdot\Big{(}\frac{% 1-E}{n}\nabla n\Big{)}\,,\\ \displaystyle\partial_{t}E&\displaystyle=\nabla\cdot\frac{\nabla E}{n}+\frac{1% -E}{n}|\nabla n|^{2}.\end{split}

(2)

In the talk we treat Eq. (2) rigorously as well as numerically and discuss its degeneracies in connection to the ill-posedness of (1).

Acknowledgments: The authors were partially funded by the Austrian Science Fund (FWF) project F 65.

References

1 Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with BGK-type collision operator: existence of solutions vs. ill-posedness, 2017; arXiv:1711.06015.
2 Marcel Braukhoff and Ansgar JÃ¼ngel. Energy-transport systems for optical lattices: derivation, analysis, simulation. Math. Models Methods Appl. Sci., 28(3):579â614, 2018.