The canonical partition function is defined as $$Z=\sum_{s}e^{-\beta E_s}$$ with the sum being over all states of the system. The way I saw this derived was by assuming that for each state, the probability of the system occupying that state is proportional to the Boltzmann factor: $P(E=E_s) = c \cdot e^{-\beta E_s}$. Summing the probabilities to one gives $Z=\frac{1}{c}$.
My question is : What principles were used in order to get this probability distribution for the energy?
Answer
My favorite way to obtain the canonical partition function is via quantum statistical mechanics and involves essentially only one principle: maximum entropy. The principle says that to obtain the statistical state of a system in a certain ensemble, one extremizes the entropy subject to the constraints that define the ensemble.
In the context of quantum statistical mechanics for a system in the canonical ensemble, one extremizes the so-called von-Neumann entropy $$ S_\mathrm{vn}(\rho) = -k\,\mathrm{tr}(\rho\ln\rho) $$ subject to the constraint that the ensemble average energy has some fixed value $E$; $$ \mathrm{tr}(\rho H) = E $$ Here $\rho$ denotes the density operator of the system, and $H$ is its Hamiltonian. This constraint is, in fact, one way of defining the canonical ensemble. This is a constrained optimization problem that can be solved using the method of Lagrange multipliers. The result is that the density operator of the system is. $$ \rho = \frac{1}{Z}e^{-\beta H}, \qquad Z = \mathrm{tr}(e^{-\beta H}) $$ where $\beta$ is the Lagrange multiplier corrsponding to the constraint of fixed ensemble average energy.
Important Digression. If you use the derivation above, it's not at all clear a priori why the multiplier $\beta$ is inverse temperature. The multiplier $\beta$ can be identified with inverse temperature using the following argument. Notice that for $\rho$ of the canonical ensemble, we have \begin{align} S_\mathrm{vn}(\rho) &= -k\mathrm{tr}\left(\rho\ln \frac{e^{-\beta H}}{Z}\right)\\ &= -k\mathrm{tr}\left(\rho(\ln e^{-\beta H}-\ln Z)\right)\\ &=k\mathrm{tr}\left(\rho(\beta H+\ln Z)\right)\\ &= k(\beta \mathrm{tr}(\rho H) + \ln Z)\\ &= k(\beta E + \ln Z) \end{align} Now, notice that the Lagrange multiplier is actually a function of $E$, the constrained value of the ensemble average of $H$, so we have $$ \frac{\partial S_\mathrm{vn}}{\partial E} = k\left(\beta ' E + \beta + \frac{1}{Z}\frac{\partial Z}{\partial E}\right) $$ where $\beta'$ denotes the derivative of $\beta$ with respect to $E$, but \begin{align} \frac{1}{Z}\frac{\partial Z}{\partial E} &= \frac{1}{Z}\frac{\partial}{\partial E} \mathrm{tr}(e^{-\beta H}) = \mathrm{tr}(-\beta'He^{-\beta H}/Z) = -\beta'\mathrm{tr}(\rho H) = -\beta' E \end{align} so that putting this all together, we get $$ \frac{\partial S_\mathrm{vn}}{\partial E} = k\beta $$ on the other hand, recall that the thermodynamic temperature satisfies $$ \frac{1}{T} = \frac{\partial S}{\partial E} $$ so that if we identify the von-Neumann entropy with the thermodynamic entropy ($S = S_\mathrm{vn}$) gives $$ \beta = \frac{1}{kT} $$ as desired.
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