The deduction of the thermal energy distributions are pretty much just Stirling approximation $\ln(x!)=x\ln(x)-x$, Lagrange-multipliers method and a lot of permutations/combinations. You can see it at the bottom.
Thermal energy distributions contain classical models such as Maxwell-Boltzmann statistics and quantum-physical models such as
Bose-Einstein statistics and Fermi-Dirac statistics.
The "classical" term means models such as Maxwell equations, partial-derivations models, which do not contain the notion about discretization -- a big
difference to QM models such as Planck's rule $E=hf$ where the energy
of EM is quantified.
Light is an example of EM radiation. Maxwell realized this by
analyzing earlier studies of Weber and Kohlrausch
here and
concluded $c_0=\frac 1 {\sqrt{\mu_0 \epsilon_0}}$. A more realistic
model of light is a non-classical model
here that cannot be
described with classical mechanism but with quantified electromagnetic
field and quantum mechanics. Photon is a boson so it obeys
Bose-Einstein statistics, not the classical approximation ie
Maxwell-Boltzmann statistics that is only realistic with extreme
temperatures such as close to absolute zero or very high temperature.
Facts such as photoelectric effect, X-emission (opposite to photoelectric effect) and Compton-scattering prove the discretization
of the EM that QM describes. Wave-particle-dualism explains events
where light acts like a wave and like a particle. This is impossible
to explain with Maxwell equations. Examples of such events are
double-slit-experiment and single slit expriment.
Now the double-slit expriment lead into the realization of
uncertainty. You cannot see the wave nature at the same time as the
particle nature. An example of this is Heisenberg's uncertainty
principle $\Delta p\Delta x \geq \frac{h}{4\pi}$ that means you cannot
know the location of physical object and and its momentum at the same
time -- if the $\Delta p$ is close to zero, you have a particle -- and
if the $\Delta p$ is close to zero, you have a wave. Bohr
generalized this concept of complementary events from mere waves and
particles in his complementarity
here where
he realized
"It is impossible to design a measuring device that demonstrates both phenomena simultaneously not because of lack of creativity on the
part of the experimenter, but simply because such a device is
literally inconceivable." (Sentence in the Wikipedia about
complementarity)
which is actually quite thought-provoking statement. For example, I
understand this so that you cannot have a camera that minimizes all
types of noises. The QM models infer a new type of noises such as
quantum noise aka shot noise that dominates low signal-to-noise-ratio
noises in certain situations.
My lecture documents
here
at the end are confusing in this point. It mentions "You cannot force
wave nature into particle nature without losing interference." after
mentioning "You will lose interference pattern on the left if you try
to find out from which hole the photon went by filling the other hole
one-by-one" (not word-to-word translation) but the meaning should be
the same.
Now back to the statistics 'why are the statistics called "thermal"
or "energy"?'
QM models such as Bose-Einstein and Fermi-Dirac describe bosons and fermions, respectively. Classical models (ambiguous term but meaning
now Maxwell equations) are energy equations in a way: you need energy
to see their working. Thermal prepending is a bit odd but perhaps it
wants to stress the association of energy and temperature. The word
"distribution" stresses the statistical connotation.
I hope someone more experienced can explain what the "thermal energy distributions" really are! I feel my explanation is not
thorough.
Mathematical formalism
Bose-Einstein
We have particles with states $N_i$ and walls $M$ where particles can have the same quantum state, a big difference to fermions where $(n,s,l,m_l)$ cannot be the same. So horizontal alignment
$$W_h^i=\frac{(N_i+M-1)!}{N_i!(M-1)!}$$
where the total alignment is the product of all horizontal alignment so the probability function $P=\Pi_i w_h^i=\Pi_i\frac{(N_i+M_i-1)!}{N_i!(M_i-1)!}$ so
$$\ln(P)=\sum_i \ln\left[(N_i+M-1)!-\ln(N_i!)-\ln((M-1)!)\right]$$
Now we use Lagrange-multiplier method so the F-function is
$$F=\ln(P)+\alpha (N-\sum_i n_i) + \beta (MU-\sum_i n_i u_i)$$
where the first $\alpha$ restriction means the amount of particles is the sum of all particles in the states $N=\sum_i n_i$ and the second $\beta$ condition means the system energy is the sum of all energies in states.
Now we derivative this one with respect to the states variable $n_i$ where we need to use the Stirling approximation $\ln(x!)\approx x\ln(x)-x$ because of large number of particles (small amount of particles requires an extra term here). So
$$\ln(M)-\ln(N_i)-\alpha-\beta E_i=0$$
$$N_i=Me^{-\alpha-\beta E_i}$$
Fermi-Dirac
Pauli's exclusion principle is the key difference. It is otherwise the same deduction as with Bose-Einstein but $W_h^i=\frac{M!}{N!(M-N_i)!}$ where $N!$ is for "miehitetty" manned states and $(M-N_i)!$ for un-manned states due to Pauli's exclusion principle -- you cannot have same two Q-states with fermions!
Maxwell Boltzmann
I use now lectures 2061 here pages 63-65. I am not sure of this because the two teachers use a slightly different notations but I understand it this way
$$W_h^i =\frac{g_i^{n_i}}{n_i!}$$
where $g_i$ is the degenerazy, $n_i$ is the amount of state so the probability $P_{MB}=\Pi_i W_h^i.$ And we will get the statistics but taking the logarithm and using Lagrange multipliers. Our conditions are $N=\sum_i n_i$ and $E=\sum_i n_i E_i$.
SUMMARY
Most states are with Maxwell-Boltzman then Bose-Eistein and least states with Fermi-Dirac because of Walls and Pauli's exclusion principle. Please, note that there are no "walls" with Maxwell-Boltzmann where systems such as ideal-gas particles can occupy the same quantum state -- perhaps related to superfluidity phenomenon. Horizontal occupation formulae for Bose-Einstein, Fermi-Dirac and Maxwell-Boltzman:
$$W_h^i(BE)=\frac{(N_i+M-1)!}{N_i!(M-1)!}$$
$$W_h^i(FD)=\frac{M!}{N_i!(M-N_i)!}$$
$$W_h^i(MB) =\frac{G_i^{N_i}}{N_i!}$$
Study Questions
Do Fermions and Bosons have degenerazy like a Maxwell-Boltzmann system?
In other words, why no $G_i$ with BE and FD formulae?
