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The Energy Distributions for Various Types of Particles:
Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein and Planck Blackbody

There is a correlation between certain characteristics of particles and the statistics which describe the distribution of energy. One characteristic which is important is whether the different particles are distinguishable or indistinguishable. Another characteristic is whether the particles are fermions or bosons. But that characteristic is not definitive. Photons are bosons but have different statistics from other bosons. But first there is a minor mathematical point to take care of.

A Little Lemma

If the quantity of some variable Y is a function of a variable Z, any constant factor is irrelevant in determining the proportion of Y that is accounted for by a particular value of Z. Suppose the quantity of Y is given by

Q = αf(Z)

where α is a constant. The total quantity of Y is then

Qtotal = ∫αf(Z)dZ = α∫f(Z)dZ
and hence
Q(Z)/Qtotal = f(Z)/∫f(Z')dZ'

The Maxwell-Boltzmann Statistics and Distribution

In 1860 James Clerk Maxwell presented the formula for the distribution of the speeds of molecules in a gas. In 1877 Ludwig Boltzmann developed he physical derivation of that distribution and the statistics for other properties of a classical ideal gas. An ideal gas is presumed to be composed of non-interacting particles. The distribution of the number of molecules Nj in energy state j having energy Ej is given by

Nj = Ntotal[exp(−Ej/kT)/(Σexp(−Ei/kT)]

where k is Boltzmann's constant and T is absolute temperature.

The amount of energy accounted for by energy state j is then NjEj. This quantity is proportional to Ejexp(−Ej/kT) and hence also to (Ej/kT)exp(−Ej/kT).

To get the proportional distribution of energy it is only necessary to evaluate the aggregate of ε·exp(−ε) over the allowed states of energy, where ε is equal to E/kT.

If E and hence also ε is continuous and ranges from 0 to ∞ then

0ε·exp(−ε)dε = 1

Therefore for this case

E/Etotal = (E/kT)·exp(−E/kT)
or, allowing ε=E/kT
E/Etotal = ε·exp(−ε)

The shape of this distribution is shown below.

Bose-Einstein Statistics and Distributions

Satyendra Nath Bose in 1924 formulated a form of statistics for photons and Albert Einstein extended this to atoms in 1924–25. Such statistics became known as Bose-Einstein statistics and were considered to apply to indistinguishable particles in contrast to distinguishable particles for which Maxwell-Boltzmann statisitics apply. Such particles became known as bosons. Later it was found that particles with integral spins were bosons.

The number of particles Nj in an energy state j having energy Ej is

Nj = Ntotalα/[β·exp(Ej/kT) − 1]

where α and β are constants.

This makes the quantity of energy NjEj proportional to ε/[β·exp(ε) − 1]

The shape of this distribution is shown below.

There is a marvelous story that goes with this case. Satyendra Nath Bose was teaching physics in Dhaka, now in Bangladesh but then in British India. He had discovered that the Maxwell-Boltzmann distribution implies something that was contrary to empirical observation. He decided to show this to his physics students. In the course of deriving this implication in his lecture he made a mathematical mistake. That mistaken derivation gave a result that was consistent with the empirical observation.

Bose wrote an article based upon this alternate derivation and sent it away to a physics journal in Europe. It was rejected. He sent it to other journals and they all rejected it. In desperation he sent it to Albert Einstein. Einstein read it and found a use for Bose's discovery. Einstein wrote an article utilizing Bose's discovery and sent his article along with Bose's article and recommended that they both be published, which they were.

The story however goes on. In India it is widely believed that Einstein simply added his name to Bose's article and had it published. There was no flaw in Einstein's actions concerning Bose's article. It is perhaps unfair for the physics community to designate the new statistics of particles as Bose-Einstein statistics but that is not Einstein's fault.

Fermi-Dirac Gas

Some particles obey the Pauli Exclusion Principle that no more than one particle can occupy a particular energy state. These were found to be particles with half-integral spins. The statistics for such particles were discovered independently by Enrico Fermi and Paul Dirac and published in 1926. Pascual Jordan discovered such statistics in 1925 and named them appropriately Pauli statistics, but Jordan did not publish his discovery until after Fermi and Dirac and did not get credit for it.

The distribution function for the number of particles in energy state j is

Nj = α/[β·exp(Ej/kT) + 1]

where α and β are constants.

The formula can be put in a more thermodynamically meaningful form

Nj = 1/[exp((Ej−μ)/kT) + 1]

where μ is the chemical potential of the gas.

This makes the distribution of energy proportional to Ej/[exp((Ej−μ)/kT) + 1] or equivalently to εj/[exp((εj−ν) + 1] where ε=Ej/kT and ν=μ/kT.

The shape of this distribution is shown below.

The Planck Blackbody Energy Distribution

Max Planck was able to identify the correct formula for thermal radiation from a perfect absorber/emitter (black body) from empirical measurements. He then found a derivation based upon radiation being emitted in quantum units ofhν , where h is a constant and ν is the frequency of the radiation. Planck's formula is

p(ν) = (2hν³/c²)/(exp(hν/kT) − 1)

where p(ν) is the probability density at frequency ν, c is the speed of light in a vacuum, k is Boltzmann's constant and T is absolute temperature.

Since energy E is equal to hν the Planck formula can be converted into one giving the probability density at energy E. This requires that

pEdE = pν
which will prevail if
pE(E) = pν(E/h)/|(dν/dE)| = pν(E/h)|(dE/dν)|

Since E=hν, (dE/dν equals h.

Therefore

pE(E) = [2E³/(hc²)]/[exp(E/kT)−1]

This makes the quantity of energy in the state corresponding to E proportional to ε³/[exp(ε) − 1] where ε=E/kT.

Although photons are bosons an ensemble of photons does not have the energy distribution of bosons. This is because the number of photons is not fixed. One larger energy photon may be absorbed by an atom which subsequently emits several smaller energy photons. Energy is conserved but not the number of photons.

Photon gas differs from the gases made of other kinds of particles in that the number of particles is not fixed. This makes it a puzzle as to how to represent its distribution function. This has to be the graph of the proportion of the total energy held in the various energy states as a function of the energy of the state. The distributions for the other types of particles (fermions, bosons and neutral particles) could also be put into that form for comparison. Photons are bosons but some special kind of boson. The nature of photons in this respect is intriguing. Albert Einstein once remarked that he had spent fifty years trying to understand the nature of photons but still had not achieved that goal.

Conclusions

The scheme of things is shown below

The formulas for the statistics of particles are of the form

E/Etotal = α·eγ/[β·exp(e) + δ]

where e=E/kT. For Maxwell-Boltzmann statistics δ is zero; for Bose-Einstein and Planck photon statistics it is equal to −1 and for Fermi-Dirac statistics it is +1. For all but the Planck photon statistics γ is equal to 1. For Planck photon statistics it is equal to 3. For Maxwell-Boltzmann and Planck photon statistics β is equal to 1.


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