===== Classical canonical ensemble ===
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $ \langle \mathcal M, H,{\hat\rho}\rangle \in \mathrm{it} $ |
| @#55EE55: postulate   | @#55EE55: $\langle \mathcal M, H,{\hat\rho},{\hat\rho}_0\rangle$ ... classical statistical ensemble |
| @#55EE55: postulate   | @#55EE55: $\hat\rho: \mathbb R\to(\Gamma_{\mathcal M} \to\mathbb R_+) $ |
| @#55EE55: postulate   | @#55EE55: $\hat\rho(\beta;{\bf q},{\bf p}):=\mathrm{e}^{-\beta\ H({\bf q},{\bf p})}$ |

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=== Discussion ===
== Equivalence of the microcanonical and canonical ensemble ==

For a given a [[Classical Hamiltonian system|Hamiltonian system]], the [[classical ensemble expectation value|expectation value]] for the [[classical microcanonical ensemble|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 [[classical microcanonical ensemble|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 $g_s\cdot \mathrm{e}^{-\beta\ \varepsilon_s}$.

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Microcanonical_ensemble|Canonical ensemble]], [[http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics|Maxwell–Boltzmann statistics]]

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=== Refinement of ===
[[Classical statistical ensemble]]
=== Context ===
[[Exponential function]]