Classical canonical ensemble

Set

definiendum	$ \langle \mathcal M, H,{\hat\rho}\rangle \in \mathrm{it} $
postulate	$\langle \mathcal M, H,{\hat\rho},{\hat\rho}_0\rangle$ … classical statistical ensemble
postulate	$\hat\rho: \mathbb R\to(\Gamma_{\mathcal M} \to\mathbb R_+) $
postulate	$\hat\rho(\beta;{\bf q},{\bf p}):=\mathrm{e}^{-\beta\ H({\bf q},{\bf p})}$

Discussion

Equivalence of the microcanonical and canonical ensemble

For a given a Hamiltonian system, the expectation value for the microcanonical ensemble for a given energy $E$ essentially coincide with the ones from the classical canonical ensemble if you take parameter $\beta$ above to take the value of the microcanonical inverse temperature $\beta(E)$. In fact the density $\hat\rho(\beta;{\bf q},{\bf p})$ in the definition above arises from a first order approximation for the density of a subsystem of a larger system governed by a microcanonical ensemble.

Theorems

Maxwell-Boltzmann distribution

this deserves it's own entry

Partitions the state parameters, ${\bf q},{\bf p}$ here, into bunches indexed by $s$ and compute the multiplicities $g(s)$. Then the statistical interpretation of $\hat\rho$ implies that the different systems are partitioned with weights $g_s\cdot \mathrm{e}^{-\beta\ H(s)}$.

Sometimes the Maxwell-Boltzmann distribution is derived without considering a phase space at all, just for particle species $s$, with individually are taken to carry an energy $\varepsilon_s$. Then the multiplicity is derived by straight forward counting and the exponential appears in the many particle limit a term $\lim_{m\to\infty}(1-\varepsilon_s/m)^m$ to give $g_s\cdot \mathrm{e}^{-\beta\ \varepsilon_s}$.