From a569627a872311b10d48f6642a253d52a403c6f4 Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Tue, 20 Sep 2022 18:48:56 +0200 Subject: [PATCH] smooth over analytical solution (thx Kimmo) --- src/analytical_solution.tex | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/src/analytical_solution.tex b/src/analytical_solution.tex index 2041fb5..5269e1d 100644 --- a/src/analytical_solution.tex +++ b/src/analytical_solution.tex @@ -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}