# Ensemble average in microcanonical and canonical ensembles

In Huang's statistical mechanics, the general ensemble average of a dynamical variable $$O(p,q)$$ is defined as:

$$\langle O \rangle=\frac{\int O(p,q)f(p,q,t)dpdq}{\int f(p,q,t)dpdq}$$

where $$f(p,q,t)$$ is the distribution function obeying Liouville's theorem. How does this expression simplify for the microcanonical and canonical ensembles?

For the microcanonical ensemble, the probability of a system with Hamiltonian $$H$$ being in an energy range $$[E,E+\delta E]$$ is equal for all possible microstates (i.e., combinations of $$p$$ and $$q$$). So the distribution function $$f$$ cancels and we have:

$$\langle O \rangle=\frac{\int_{E

For the canonical ensemble, the distribution function $$f=ce^{-\beta H}$$, where $$c$$ is a constant obtained by the normalization condition of probability distributions. Substituting this into our general ensemble average definition, you get:

$$\langle O \rangle=\frac{\int O(p,q)e^{-\beta H}dpdq}{\int e^{-\beta H}dpdq}$$

where the integrals are over all possible values of $$(p,q)$$.

• Is my reasoning behind these simplifications correct?
• Is the probability distribution $$f$$ that I used for each ensemble the same one that is in Liouville's theorem?

The first equation is an expression forbthe average in respect to an arbitrary probability density $$f(p,q,t)$$! Note that it does not have to he normalized - the denominator is the normalization factor.
In the microcanonical ensemble the probability density is constant everywhere in the energy shell $$[E, E+\delta E]$$, and zero everywhere else, which is indicated here by the integration limits. For the canonical ensemble $$f(p,q,t) = e^{-\beta H(p,q,t)}.$$ As I have mentioned, the normalization factor is already taken care of, and indeed it disappears from your calculation.
• The normalization factor for distribution $f(p,q,t)$ is $c = [\int dpdq f(p,q,t)]^{-1}$ - it is already present in your equation. If I take a normalized distribution, as you suggest, then this factor is 1 and need not be included in your first equation. You either use normalized distributions or write the normalization factors explicitly, but doing both is a waste of energy. Apr 17, 2020 at 13:22