Partition function

For a classical system in the canonical ensemble at temperature $T$, the partition function is

where:

• $\beta =1/(kT)$.
• $H(q,p)$ is the Hamiltonian
• $\mathrm{\Gamma }$ is the full phase space
• $f$ is the number of degrees of freedom (pairs $(q_i,p_i)$)
• The factor $h^f$ sets the phase-space cell size.

It acts as the normalizing constant for the Boltzmann distribution and contains all thermodynamic information.

From $Z(\beta )$ one can obtain:

And Helmholtz free energy, defined as

So: