# 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:

=Classic partition function=
A system subdivided into *N* sub-systems, where each sub-system (e.g. a particle) can attain any of the energies (i.e. , , ...) has the partition function given by

*T*is the temperature). The intepretation of is that the probability that the particle (sub-system) will have energy is . When the number of energies is definite (e.g. particles with spin in a crystal lattice under an external magnetic field), then the indefinite sum with is replaced with a definite sum. However, the total partition function for the system containing

*N*sub-systems is of the form

*j*:th sub-system. Another approach is to sum over all system's total energy states,

*r*is from 1 to

*N*·(degrees of freedom). Note that for a system containing

*N*

*non-interacting*sub-system (e.g. a real gas), then the system's partition function is

*N*! is the factorial and ζ is the "common" partition function for a sub-system. This equation also has the more general form

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

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

=Quantum mechanical partition function= 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):