Partition function (statistical mechanics)

In statistical mechanics, the partition function Z is used for the statistical description of a system in thermodynamic equilibrium. It depends on the physical system under consideration and is a function of temperature as well as other parameters (such as volume enclosing a gas etc.). The partition function forms the basis for most calculations in statistical mechanics. It is most easily formulated in quantum mechanics:

Given the energy eigenvalues of the system's Hamiltonian operator , the partition function at temperature is defined as:

Here the sum runs over all energy eigenstates (counted by the index j) and is Boltzmann's constant.

The partition function has the following meanings:

• It is needed as the normalization denominator for Boltzmann's probability distribution which gives the probability to find the system in state j when it is in thermal equilibrium at temperature T (the sum over probabilities has to be equal to one):

• Qualitatively, Z grows when the temperature rises, because then the exponential weights increase for states of larger energy. Roughly, Z is a measure of how many different energy states are populated appreciably in thermal equilibrium (at least when we suppose the ground state energy to be zero).

• Given Z as a function of temperature, we may calculate the average energy as

The free energy of the system is basically the logarithm of Z:

• From these two relations, the entropy S may be obtained as

Alternatively, with , we have and , as well as .

More formally, the partition function Z of a quantum-mechanical system may be written as a trace over all states (which may be carried out in any basis, as the trace is basis-independent):

If the Hamiltonian contains a dependence on a parameter , as in then the statistical average over may be found from the dependence of the partition function on the parameter, by differentiation:

If one is interested in the average of an operator that does not appear in the Hamiltonian, one often adds it artificially to the Hamiltonian, calculates Z as a function of the extra new parameter and sets the parameter equal to zero after differentiation.