Temperature and the Canonical Ensemble

Microcanonical ensemble only deals with isolated systems, but a canonical ensemble deals will closed (energy exchange can happen).
A canonical ensemble is a collection of replicas of a system all having fixed number of particles ( $N$ ), volume ( $V$ ), and temperature ( $T$ ).
Microcanonical ensemble has constant $N V E$ . Grand canonical ensemble $μ V T$ , where $μ$ is the chemical potential of the particle.

Ω_{i} (E) = Ω_{R} (E - E_{i})

Ω (E) = \sum_{i} Ω_{i} (E) = \sum_{i} Ω_{i} (E - E_{i})

Boltzmann definition of entropy:

S_{R} = k_{B} \ln Ω_{R} (E - E_{i}) \approx k_{B} \ln Ω_{R} (E) - \frac{\partial}{\partial E} (k_{B} \ln Ω_{R} (E)) E_{i}

where in the last step I used Taylor expansion. We know that $\frac{1}{T} = \frac{\partial S}{\partial E}$

= k_{B} \ln Ω_{R} (E) - \frac{E_{i}}{T}

Hence,

Ω_{R} (E - E_{i}) = Ω_{R} (E) e^{- E_{i} / k_{B} T}

Total number of microstates is the sum over $i$ :

Ω (E) = Ω_{R} (E) \sum_{i} e^{- E_{i} / k_{B} T}

\Rightarrow p_{i} \propto e^{- E_{i} / k_{B} T} $ $ $ $ \Rightarrow p_{i} = \frac{e^{- E_{i} / k_{B} T}}{\sum_{j} e^{- E_{j} / k_{B} T}} = \frac{e^{- E_{i} / k_{B} T}}{Z} $ $ w h e r e $ Z $ i s c a l l e d t h e p a r t i t i o n f u n c t i o n : $ Z = \sum_{j} e^{- E_{j} / k_{B} T} $