Canonical ensemble partition function, Q,
for a system of N identical particles each of mass m
When the particles are distinguishable then the factor N! disappears. is the Hamiltonian
(Sir William Rowan Hamilton 1805-1865 Ireland)
corresponding to the total energy of the system.
H is a function of the 3N positions and 3N momenta of the particles in the system.
The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows
Thus we have
This separation is only possible if is independent of velocity (as is generally the case).
The momentum integral can be solved analytically:
Thus we have
The integral over positions is known as the configuration integral,
In an ideal gas there are no interactions between particles so
Thus for every gas particle.
The integral of 1 over the coordinates of each atom is equal to the volume so for N particles
the configuration integral is given by where V is the volume.
Thus we have
If we define the de Broglie thermal wavelength as
where
one arrives at
Thus one can now write the partition function for a real system can be built up from
the contribution of the ideal system (the momenta) and a contribution due to
particle interactions, i.e.