The entropy due to a set of states that can be either occupied with probability  or empty with probability  can be written as:
where  is Boltzmann constant.
For a continuously distributed set of states as a function of energy, such as the eigenstates in an electronic band structure, the above sum can be written as an integral over the possible energy values, rather than a sum. Switching from summing over individual states to integrating over energy levels, the entropy can be written as:
where  is the density of states of the solid. The probability of occupation of each eigenstate is given by the Fermi function, :
where  is the Fermi energy and  is the absolute temperature. One can then re-write the entropy as:
This is the general formulation of the density-of-states based electronic entropy.