In the Wigner picture we have for each a density . In the QTBM we have for each a mode and a corresponding hydro density
This formal analogy is a guideline which was helpful in finding a scattering model for the use with the QTBM. It should not be stretched too far.
From the modes we build the mixed state
We are not interested in Schrödinger models for relaxation time scattering per se, but it is a good test case and gives important hints on how to model scattering. We will propose a refined QTBM Schrödinger model for scattering which includes a better model for inscattering. The models proposed in this section share some properties with the Wigner relaxation time models. They are not identical in the sense of being equivalent via a Wigner transform, hence we call them ``relaxation-time-like''.
As a first proposal for a model of this type we try (using the anology above):
This model has the property that for
However, the scattering term
We will now look at the same problem in the hydrodynamical formulation which provides further insight. The Schrödinger equation with relaxation time outscattering becomes for a mode
Note that the equation for stays unchanged. We obtain a modified continuity equation. The resulting current is no longer constant.
As a hydrodynamical relaxation time model which also includes inscattering we propose
The corresponding wave function scattering term is
This model does not relax , but only relaxes towards the equilibrium values. We can relax left and right going modes separately or in parallel, in the last case we have
This model conserves mass. To compare it with the Wigner relaxation time model we look at the pure state
From the Schrödinger equation for with scattering term 7.35 we derive a von Neumann equation for where the scattering is of the form
For the outscattering this gives
The inscattering from Wigner relaxation time scattering is (writing Equation 7.24 in hydro variables):
Comparing our proposed model for inscattering and the original Wigner relaxation time scattering we see that the QTBM hydro model approximates
The equation for is the same in both cases, scattering does not enter.
Implementation of hydrodynamical models is a formidable task, see [Bit00] for related work. Our experience with the Riccati equation is also disheartening. Hence the numerical validation of all proposed scattering models is an open issue.
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