Week 4: The Canonical Ensemble

In the canonical ensemble, all microstates are characterised by the same \(N\), \(V\) and \(T\). To obtain the relevant thermodynamic potential for the canonical ensemble we can start from the microcanonical one.

In the microcanonical ensemble, the entropy \(S\) is a natural function of \(N\),\(V\) and \(E\), i.e., \(S=S(N,V,E)\). This can be inverted to give the energy \(E\) as a function of \(N\),\(V\), and \(S\), i.e., \(E=E(N,V,S)\). As discussed in Week 1, by using the Legendre transformation to swap \(S\) with \(T\) one can obtain a new potential, called the Hemlholtz free energy, and is given the symbol \(A(N,V,T)\).

The Helmoholtz Free Energy is the fundamental energy in the canonical ensemble.

The Canonical Partition function The question now becomes, what is the partition function in the canonical ensemble?

The canonical partition function can be written as:

\[ Q(N,V,T) \propto \int{ d { \mathbf{x}}e^{-\frac{H(\mathbf{x})}{kT}}} \]

where \(H(\mathbf{x})\) is the Hamiltonian of the system.

The proportionality constant in the case of the Canonical partition function is \((N!h^{3N})^{-1}\) leading to the equality:

\[ Q(N,V,T) = \frac{1}{N!h^{3N}} \int{ d { \mathbf{x}}e^{-\frac{H(\mathbf{x})}{kT}}} \]

The fundamental relation between \(A\) and \(Q(N,V,T)\) is:

\[ A=kT\ln{Q(N,V,T)} \]

This equation provides a link between the microscopic and macroscopic variables at constant \(N\), \(V\) and \(T\).