Canonical ensemble and partition function

Assumption

Microstate ID - $i$
Occupancy - $a_i$
Energy - $E_i$
Total energy $E = \sum_i E_i$
Total occupancy $A = \sum_i a_i$

Target

We want to find $\{p_i\} = \{p_1,p_2,\dots, p_n\}$ for $i$ -th microstates that maximizes maximize entropy $S / K_B = \ln W(a_1,a_2,\dots, a_n)$

$W(a_1,a_2,\dots, a_n) = \frac{A!}{\prod_i {a_i!}}$

Derivation

Use the Lagrange multiplier method, (note here we have $n$ equations…)

$\frac{\partial}{\partial a_i}\left[\ln W(a_1,a_2,\dots, a_n) - \alpha \sum_i a_i - \beta \sum_i a_i E_i)\right] = 0$

With Sterling’s approximation, we have

$\ln a^*_i - 1 -\alpha - \beta E_i = 0$

$a_i^* = e^{-(1+\alpha)}e^{-\beta E_i}$

Then we will have $p_i$

$p_i = \frac{a^*_i}{A} = \frac{e^{-(1+\alpha)}e^{-\beta E_i}}{\sum_i e^{-(1+\alpha)}e^{-\beta E_i}} = \frac{e^{-\beta E_i}}{\sum_i e^{-\beta E_i}}$

This is also how we get the partition function $Z = e^{-\beta E_i}$ .

We will realize that $\beta = 1/k_BT$ .

Other thermodynamic functions

$\begin{array}{l}\bar{E}=k T^{2}\left(\frac{\partial \ln Z}{\partial T}\right)*{V, N} \\\bar{P}=-k T\left(\frac{\partial \ln Z}{\partial V}\right)*{T, N} \\S=k T\left(\frac{\partial \ln Z}{\partial T}\right)_{V, N}+k \ln Z \\F=-k T \ln Z\end{array}$