Gas in a box
In
Using the results from either
Thomas–Fermi approximation for the degeneracy of states
For both massive and massless
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 example, a spin 1⁄2 particle would have f = 2, one for each spin state. 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.
Without using any approximation, the number of particles with energy εi is given by
where is the
Using the Thomas−Fermi approximation, the number of particles dNE with energy between E and E + dE is:
where is the number of states with energy between E and E + dE.
Energy distribution
Using the results derived from the previous sections of this article, some distributions for the gas in a box can now be determined. For a system of particles, the distribution for a variable is defined through the expression which is the fraction of particles that have values for between and
where
- , number of particles which have values for between and
- , number of states which have values for between and
- , probability that a state which has the value is occupied by a particle
- , total number of particles.
It follows that:
For a momentum distribution , the fraction of particles with magnitude of momentum between and is:
and for an energy distribution , the fraction of particles with energy between and is:
For a particle in a box (and for a free particle as well), the relationship between energy and momentum is different for massive and massless particles. For massive particles,
while for massless particles,
where is the mass of the particle and is the speed of light. Using these relationships,
- For massive particles where Λ is thethermal wavelengthof the gas.This is an important quantity, since when Λ is on the order of the inter-particle distance , quantum effects begin to dominate and the gas can no longer be considered to be a Maxwell–Boltzmann gas.
- For massless particles where Λ is now the thermal wavelength for massless particles.
Specific examples
The following sections give an example of results for some specific cases.
Massive Maxwell–Boltzmann particles
For this case:
Integrating the energy distribution function and solving for N gives
Substituting into the original energy distribution function gives
which are the same results obtained classically for the Maxwell–Boltzmann distribution. Further results can be found in the classical section of the article on the ideal gas.
Massive Bose–Einstein particles
For this case:
where
Integrating the energy distribution function and solving for N gives the particle number
where Lis(z) is the polylogarithm function. 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. As the temperature drops towards zero, Λ will become larger and larger, until finally Λ will reach a critical value Λc where z = 1 and
where denotes the Riemann zeta function. The temperature at which Λ = Λc is the critical temperature. For temperatures below this critical temperature, the above equation for the particle number has no solution. The critical temperature is the temperature at which a Bose–Einstein condensate begins to form. The problem is, as mentioned above, that the ground state has been ignored in the continuum approximation. It turns out, however, that the above equation for particle number expresses the number of bosons in excited states rather well, and thus:
where the added term is the number of particles in the ground state. The ground state energy has been ignored. This equation will hold down to zero temperature. Further results can be found in the article on the ideal Bose gas.
Massless Bose–Einstein particles (e.g. black body radiation)
For the case of massless particles, the massless energy distribution function must be used. It is convenient to convert this function to a frequency distribution function:
where Λ is the thermal wavelength for massless particles. 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. When this is the case, the number of photons is not conserved. 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. Furthermore, since photons have two spin states, the value of f is 2. The spectral energy density is then
which is just the spectral energy density for
In certain situations, the reactions involving photons will result in the conservation of the number of photons (e.g. light-emitting diodes, "white" cavities). In these cases, the photon distribution function will involve a non-zero chemical potential. (Hermann 2005)
Another massless Bose gas is given by the
Massive Fermi–Dirac particles (e.g. electrons in a metal)
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.
See also
References
- Herrmann, F.; Würfel, P. (August 2005). "Light with nonzero chemical potential". American Journal of Physics. 73 (8): 717–723. . Retrieved 2006-11-20.
- Huang, Kerson (1967). Statistical Mechanics. New York: John Wiley & Sons.
- Isihara, A. (1971). Statistical Physics. New York: Academic Press.
- Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics (3rd Edition Part 1 ed.). Oxford: Butterworth-Heinemann.
- Yan, Zijun (2000). "General thermal wavelength and its applications". Eur. J. Phys. 21 (6): 625–631. S2CID 250870934.
- Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28.