Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics, and considering the limit of a very large box, the Thomas–Fermi approximation (named after Enrico Fermi and Llewellyn Thomas) is used to express the degeneracy of the energy states as a differential, and summations over states as integrals.
The magnitude of the momentum is given by where h is the Planck constant and L is the length of a side of the box.
Each possible state of a particle can be thought of as a point on a 3-dimensional grid of positive integers.
The distance from the origin to any point will be Suppose each set of quantum numbers specify f states where f is the number of internal degrees of freedom of the particle that can be altered by collision.
For large values of n, the number of states with magnitude of momentum less than or equal to p from the above equation is approximately which is just f times the volume of a sphere of radius n divided by eight since only the octant with positive ni is considered.
Using a continuum approximation, the number of states with magnitude of momentum between p and p + dp is therefore where V = L3 is the volume of the box.
Notice that in using this continuum approximation, also known as Thomas−Fermi approximation, the ability to characterize the low-energy states is lost, including the ground state where ni = 1.
For most cases this will not be a problem, but when considering Bose–Einstein condensation, in which a large portion of the gas is in or near the ground state, the ability to deal with low energy states becomes important.
with β = 1/kBT, the Boltzmann constant kB, temperature T, and chemical potential μ.
Using the Thomas−Fermi approximation, the number of particles dNE with energy between E and E + dE is: where
Using the results derived from the previous sections of this article, some distributions for the gas in a box can now be determined.
Using these relationships, The following sections give an example of results for some specific cases.
Further results can be found in the classical section of the article on the ideal gas.
The polylogarithm term must always be positive and real, which means its value will go from 0 to ζ(3/2) as z goes from 0 to 1.
The problem is, as mentioned above, that the ground state has been ignored in the continuum approximation.
Further results can be found in the article on the ideal Bose gas.
The spectral energy density (energy per unit volume per unit frequency) is then Other thermodynamic parameters may be derived analogously to the case for massive particles.
For example, integrating the frequency distribution function and solving for N gives the number of particles: The most common massless Bose gas is a photon gas in a black body.
Taking the "box" to be a black body cavity, the photons are continually being absorbed and re-emitted by the walls.
In the derivation of Bose–Einstein statistics, when the restraint on the number of particles is removed, this is effectively the same as setting the chemical potential (μ) to zero.
Note that the Wien distribution is recovered if this procedure is carried out for massless Maxwell–Boltzmann particles, which approximates a Planck's distribution for high temperatures or low densities.
In these cases, the photon distribution function will involve a non-zero chemical potential.
(Hermann 2005) Another massless Bose gas is given by the Debye model for heat capacity.
This model considers a gas of phonons in a box and differs from the development for photons in that the speed of the phonons is less than light speed, and there is a maximum allowed wavelength for each axis of the box.
For this case: Integrating the energy distribution function gives where again, Lis(z) is the polylogarithm function and Λ is the thermal de Broglie wavelength.
Further results can be found in the article on the ideal Fermi gas.
Applications of the Fermi gas are found in the free electron model, the theory of white dwarfs and in degenerate matter in general.