### 0.1 Canonical Ensemble


**Definition 0.1.1 **  The **canonical ensemble** is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$ and particles numbers $\mathbf{N}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the thermodynamic temperature $\beta$ and the **partition function** $z$.
\[\mathscr{p}_i = \frac{1}{\Omega(T,\mathbf{V},\mathbf{N})}=\frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]
\[z = \sum_{j}{e^{-\beta E_j}} = \sum_{j}{e^{-E_j/(k_BT)}}\]

**Definition 0.1.2 **  The **canonical energy** E of a system in the canonical ensemble is the ensemble average of energy.
\[\langle E\rangle =  \sum_{i}{E_i \mathscr{p}_i} = \frac{1}{z}\sum_{i}{\frac{-\partial}{\partial \beta}e^{-\beta E_i}} = -\frac{1}{z}\frac{\partial z}{\partial \beta} = -\frac{\partial}{\partial \beta}\log z\]

**Definition 0.1.3 **  The **heat capacity** of a system $C_V$ is the derivative of canonical energy in terms of temperature.
\[C_V = \left(\frac{\partial E}{\partial T}\right)_{\mathbf{V},\mathbf{N}} = -k_B \beta^2 \frac{\partial^2}{\partial\beta^2}\log z = -k_B \beta^2 \frac{\partial^2}{\partial\beta^2}(\beta F) = k_B\beta^2(\langle E^2\rangle - \langle E\rangle^2)\]

**Result 0.1.4 **  The *Helmholtz free energy* $F$ can be written in terms of the temperature and the partition function.
\[F = -\frac{1}{\beta}\log z\]

**Result 0.1.5 **  The *pressure* $P$ of a system can be written as the ensemble average of pressure for each microstate.
\[P = -\left(\frac{\partial F}{\partial V}\right)_{T,\mathbf{N}} = \frac{\partial}{\partial V}\left(\frac{1}{\beta}\log z\right) = \frac{1}{z}\sum_i{\left(\frac{-\partial E}{\partial V}\right)e^{-\beta E_i}} = \langle P_i\rangle\]

**Result 0.1.6 **  The *entropy* $S$ of a system can be written in terms of the ensemble average of log of the probability of each microstate.
\[S = -\left(\frac{\partial F}{\partial T}\right)_{\mathbf{V},\mathbf{N}} = k_B\beta^2\left(\frac{\partial F}{\partial \beta}\right)_{\mathbf{V},\mathbf{N}} = -k_B\beta^2\left(\frac{\partial}{\partial \beta}\frac{1}{\beta}\log z\right)_{\mathbf{V},\mathbf{N}} = \frac{k_B}{z}\sum_i{e^{-\beta E_i}(\log z +\beta E_i)}\]

**Proposition 0.1.7 **&nbsp; **Geometric series convergence** states that for $|r|<1$ the following infinite series converges to $1/(1-r)$.
\[\sum_{k=0}^\infty{r^k} = \frac{1}{1-r}\]