Formula_757

\[\begin{gathered}\langle\ldots\rangle_{t} \equiv \operatorname{Sp}\left\{\ldots \rho\left(t_{0}\right)\right\}, \tilde{a}_{k} \equiv a_{k}\left(t_{0}\right) \exp \left\{-i \omega_{k}\left(t-t_{0}\right)\right\}, \\N_{k} \equiv \exp \left\{-\beta \omega_{k}\right\} /\left(1-\exp \left\{-\beta \omega_{k}\right\}\right),\end{gathered}\]