The canonical partition function, denoted by \( \small Z\ \), is a fundamental concept in statistical mechanics. It is a useful tool that can tell us the properties of the system, such as the probability that the system is in a particular state, the average energy of the system, and the heat capacity of the system.
For a classical, discrete canonical ensemble at a temperature \( \small T\ \), the canonical partition function is given by:
The probability of the system having energy E_i is given by:
A few useful properties can be derived from the partition function. For example, the average energy of the system is given by:
Next, the expectation value of the energy squared is given by:
Combining these two, we can get the mean square fluctiation of the energy:
$$ \langle \Delta E^2 \rangle = \langle E^2 \rangle - \langle E \rangle^2 = - \frac<\partial \langle E \rangle> <\partial \beta>$$ k Finally, the heat capacity of the system is given by:
$$ C_v(T) = \frac \langle \Delta E^2 \rangle$$