Partition function (statistical mechanics)

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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

where ( is Boltzmann's constant and 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

where is the partition function for the j:th sub-system. Another approach is to sum over all system's total energy states,

where , but then the parameter 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

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

where is the sub-system's Hamiltonian operator

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 .

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):

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.