Can $\beta$, $\mu$ or $p$ or other intensive variables be expressed as expectation values in every ensemble? If I compare for example the canonical ensemble with partition function $\tilde{Z}_{\beta}$ and the microcanonical ensemble with partition function $E_{\beta}$, $\beta$ and $E$ seem to be  some "kind of" paired variables:

*

*They appear as a pairing in the exponential factor $e^{\beta H}$ in the canonical distribution.


*$\partial_{\beta} - \ln\tilde{Z}_{\beta}= <H> = E$, and in a similar way $\partial_E - \ln Z_E = \beta$ (of course the symbols here are to see in their respective ensembles).


*In this way of calculating $E$ and $\beta$, since $E$ is just the expectation value of an observable $H$ on the phase space, it would be completely "symmetric" if the same applied for $\beta$ as well.
I can spot the same pairing property for $V$ and $\beta p$, or $N$ and $\beta \mu$.
However: $E$ isn't only a parameter, but it's also always either an expectation value of a function of the microstate, or it the sharply peaked value of that function. $\beta$ on the other hand is nothing but a parameter - or is it?
Hence the question: Can I express those quantities like $\beta$ as expectation values / sharply peaked values of a function of the microstate as well? If so, does it only work for certain ensembles / systems, or all of them? And is there some kind of framework that explains this pairing?
To give an example of the direction I'm looking for:
I know for example that there is something called "instantaneous temperature", which is the kinetic energy of the particles, divided by the degrees of freedom. However, I have seen this definition only for ideal gasses, and it is not clear to me wether that applies to the canonical ensemble as well.
 A: The canonical ensemble is also commonly called the $NVT$ ensemble, meaning $N, V, T$ are constant and together are able to pin down all thermodynamic properties of the system. All other quantities are either directly related ($\beta=\frac{1}{k_bT}$ as a trivial example) or can be extracted from the partition function ($p,\mu$, etc). The ones that are extracted from the partition function are typically only defined through expectation values: the real value would fluctuate but in the macroscopic limit it will only take on one value. So can you express $\beta$ as a sharply peaked value? In the canonical ensemble $T$ is fixed so $\beta$ is fixed as well.
Are $\beta, E$ conjugated variables? The answer is no. The reason that $\beta, E$ appear together often is that together they form a unitless quantity since $\beta$ has units (energy)$^{-1}$. Only unitless quantities can be used in expressions like $e^x$ and they can be convenient to work with.
To find conjugated variables just look at one of the thermodynamic potentials and its corresponding fundamental equation. For example for $F(N,V,T)$:
$$dF=-SdT-pdV+\mu dN$$
The conjugated variables are any variables that appear together so $S\leftrightarrow T,p\leftrightarrow V$ and $\mu \leftrightarrow N$. Note that $V$ is conjugated to $p$ and not $\beta p$, which is just another common way to make $p$ unitless to give $p$ certain units.
And sorry but I don't know anything about instantaneous temperature but I hope this clarified things at least a little bit.
