Derivation of partition function

Consider a system ^[1] S embedded into a heat bath B. The total Energy of the system S and B is E. Let p_i denote the probability that the system S is in microstate i with Energy E_i. The probability p_i will be proportional to the number of microstates in the total closed system whereby the system S is in microstate i with Energy E_i. But, this will be proportional to the number of microstates of the heat bath B with Energy E − E_i.

$\begin{align} p_i &\propto \Omega_B\left(E_B\right) \\ &= \Omega_B \left(E - E_i\right) \\ \Rightarrow k \ln \Omega_B \left( E - E_i \right) & \approx k \ln \Omega_B \left(E\right) - \frac{\partial}{\partial E} \left(k \ln \Omega_B\left(E\right)\right) E_i \\ & \approx k \ln \Omega_B \left(E\right) - \frac{\partial S_B}{\partial E} E_i \\ & \approx k \ln \Omega_B \left(E\right) - \frac{E_i}{T} \\ \Rightarrow k \ln p_i & \approx k \ln \Omega_B \left(E\right) - \frac{E_i}{T} \\ \Rightarrow p_i & \propto e^{\ln \Omega_B \left(E\right) - \frac{E_i}{kT}} \\ \Rightarrow p_i & \propto \Omega_B \left(E\right) e^{ - \frac{E_i}{kT}} \\ \Rightarrow p_i & \propto e^{- \frac{E_i}{kT}}. \end{align}$

We shall define $\beta := \frac{1}{kT}$ and so we have the result; $p_i \propto e^{- \beta E_i }$

The partition function|Partition Function is the normalisation constant that allows us to find the probability.

$\begin{align} Z & := \sum_i e^{- \beta E_i} \end{align}$

References

1. ^ http://www.dublinmathematicsschool.ie/Notes/Derivationofpartitionfunction.pdf