How do I calculate this, $\beta$ is a parameter, $H$ a matrix:
$(1-\beta \partial_\beta)\ln(Tr(e^{-\beta H})) = \ln(Tr(e^{-\beta H})) - \beta \frac{\partial_\beta Tr(e^{-\beta H})}{Tr(e^{-\beta H})}=?$
Are there identities to simplify $\ln(Tr)$ and my derivative?
$ \def\a{\alpha}\def\b{\beta}\def\l{\lambda} \def\qiq{\quad\implies\quad} \def\tr{\operatorname{Tr}} $For typing convenience, let a dot denote derivatives wrt $\b$ and define the variables $$\eqalign{ E &= \exp(-\b H) &\qiq \dot E = -HE \\ \a &= \tr(E) &\qiq \dot\a = \tr(\dot E)\\ \l &= \log(\a) &\qiq\dot\l = \frac{\dot\a}{\a} = \left(\frac{-\tr(HE)}{\tr(E)}\right) \\ }$$ Substituting this into your equation yields $$\eqalign{ \big(\l - \b\dot\l\big) = \l \;+\; \frac{\tr(\b HE)}{\tr(E)} \\ }$$