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:
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 .