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smooth over analytical solution (thx Kimmo)

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Valentin Boettcher 2022-09-20 18:48:56 +02:00
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src/analytical_solution.tex
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@ -6,8 +6,8 @@ The results of \cref{chap:flow} are promising from a numerical
perspective but remain to be verified. Previous
work~\cite{Hartmann2017Dec,Hartmann2021Aug,RichardDiss} has made it
clear that the reduced system dynamics can be efficiently treated by
the method, but it is an open question whether bath related quantities
can be calculated to a similar degree of accuracy.
the NMQSD/HOPS, but it is an open question whether bath related
quantities can be calculated to a similar degree of accuracy.
The best possible verification is the comparison with a soluble model,
ideally treated with a method completely different from the NMQSD. In
@ -79,8 +79,8 @@ Then
\end{equation}
Because we are only interested in solutions for \(t\geq 0\) and the
shape of the convolution in \cref{eq:eqmotprop} the solution may be
found by virtue of the Laplace transform.
appearance of the convolution in \cref{eq:eqmotprop} the solution may
be found by virtue of the Laplace transform.
Setting
\begin{equation}
@ -360,7 +360,9 @@ now consider a model with two baths.
The considerations of~\cref{sec:oneosc} can be straight forwardly
generalized to the case of two coupled oscillators coupled in turn to
a bath each. This construction is chosen so that the previous results
can be reused and the coupling to the baths is not trivial.
can be reused and the coupling to the baths is not trivial. It is
therefore the simplest generalization of QBM for the relevant
applications.
We will not give explicit formulas for the results in terms of sums of
exponentials, as they are quite extensive and easily obtained via the
@ -448,7 +450,7 @@ now with a more complicated matrix
\end{equation}
which we have to invert.
This can be done easily\footnote{We have use a computer algebra
This can be done easily\footnote{We have used a computer algebra
system. There is probably a pattern to the inverse matrix, so that
the solution for \(N>2\) oscillators may be found.} and yields
\begin{equation}