WhatBNComputes.tex

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\newcommand{\vbTInv}{\rotatebox[origin=c]{180}{$\vb T$}}
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\newcommand{\bbTInv}{\rotatebox[origin=c]{180}{$\mathbb{T}$}}
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\begin{document}

In the FVC approach, the thermal average of 
a power, force, or torque quantity $Q$ is 
computed by integrating a spectral density over all 
frequencies:
%====================================================================%
\begin{align*}
 Q &= \int_0^\infty \big\langle Q\big\rangle_\omega \, d\omega
\\
 \big\langle Q \big \rangle_\omega
   &=\text{Tr }\Big[\vb Q\sups{PFT} \vb W \vb R \vb W^\dagger\Big]
\end{align*}
%====================================================================%
where
\begin{itemize}
  \item the $\vb Q\supt{PFT}$ matrix is the matrix one sandwiches
        between the vectors of volume-current coefficients to 
        obtain the power, force, or torque
  \item $\vb R$ is a matrix describing the Rytov source density, and
  \item $\vb W$ is a matrix describing the ``dressing'' of the 
        Rytov density by the polarization response of the material
        geometry.
\end{itemize}
The elements of the $\vb R$ matrix are
%====================================================================%
$$ R_{\alpha\beta}=\frac{2k}{\pi Z_0}
   \EXPTWO{\vb b_\alpha}{ \Delta\Theta(\vb x) (\vbeps(\vb x) - \vb 1)}{\vb b_\beta}
$$
%====================================================================%
where $\Delta\Theta(\vb x)$ is the difference between the Bose-Einstein
factor at the local temperature at $\vb x$ and the Bose-Einstein
factor at the temperature of the environment:
%====================================================================%
$$\Delta\Theta(\vb x) = \Theta\Big( T(\vb x), \omega \Big) 
                       -\Theta\Big( T\sups{env}, \omega \Big).
$$
%====================================================================%
It is now convenient to split the $\vb R$ matrix into separate
contributions from the various objects in the geometry:
%====================================================================%
$$ \vb R = \vb R_1 + \vb R_2 + \cdots + \vb R_{N} $$
%====================================================================%
and to extract from the $n$th term a scalar
prefactor $\wh{\Delta\Theta}_n$ that involves the 
volume average of the temperature in body $n$:
%====================================================================%
\begin{align*}
 \wh{\Delta\Theta}_n 
&\equiv \Theta\Big( \wh{T}_n, \omega \Big) - 
        \Theta\Big( T\sups{env}, \omega \Big)
\\
\wh{T}_n 
&\equiv 
   \frac{1}{\mc V_n} \int_{\mc V_n} T(\vb x) \, d\vb x.
\end{align*}
%====================================================================%
The $\vb R$ matrix then reads
%====================================================================%
$$ \vb R = 
     \wh{\Delta\Theta}_1 \, \wh{\vb R}_1
    +\wh{\Delta\Theta}_2 \, \wh{\vb R}_2
    +\cdots
    +\wh{\Delta\Theta}_N \, \wh{\vb R}_N
$$
%====================================================================%
where the elements of the $\wh{R}_n$ matrices are just the 
$\vb R$-matrix elements normalized by $\wh{\Delta\Theta}_n$:
%====================================================================%
\numeq{NormalizedRMatrix}
{
\wh R_{n;\alpha\beta}=
   \frac{2k}{\pi Z_0 \wh{\Delta\Theta}}
   \EXPTWO{\vb b_\alpha}
          { \Delta\Theta(\vb x) (\vbeps(\vb x) - \vb 1)}{\vb b_\beta}.
}
%====================================================================%
This allows us to resolve the quantity $Q$ into contributions
from thermal sources in each object:
%====================================================================%
\begin{align*}
 Q&=\sum Q_n
\\
\int_0^\infty \sum_n \Delta\Theta(T_n,\omega)\Phi_n(\omega) \,d\omega
\label{Qn}
\Phi_n(\omega)
  &=\text{Tr }\Big[\vb Q\sups{PFT} \vb W \wh{\vb R}_n \vb W^\dagger\Big].
\end{align*}
%====================================================================%
For bodies of uniform temperature, the $\Phi_n$ factor here is just
the same flux quantity computed by {\sc scuff-neq}. 

This normalization scheme does not work for objects whose average
temperature is equal to that of the environment, $\wh T_n=T\sups{env}$.
In this case we remove the factor $\wh{\Delta\Theta}$ from both
the denominator of (\ref{NormalizedRMatrix})
and the numerator of (\ref{Qn}).

\end{document}