What if thermodynamics and Boze-Einstein statistic can be built from first principles: discreteness.
Suppose matter is discrete and particles exchange one quantum at a time. What equilibrium follows from these simple rules?
Take M particles and Q transferable quanta.
Every particle remains nonempty, so its size is
n_i = 1 + m_i,
where m_i >= 0 and
m_1 + ... + m_M = Q.
Let particles repeatedly transfer one quantum through a reversible, symmetric process.
Given enough time, any initial allocation with the same Q and M approaches the same equilibrium distribution.
This is exactly what thermodynamic equilibrium and maximum entropy describe:
the system spends almost all its time among the overwhelmingly numerous equilibrium configurations.
Counting all possible allocations gives
Omega(Q,M) = C(Q+M-1, M-1).
Therefore entropy is
S = k log Omega.
It is calculated from the microscopic states, not introduced as an unexplained tendency.
For one particle, the exact finite-system probability is
P(m) =
C(Q-m+M-2, M-2) /
C(Q+M-1, M-1).
When Q and M become large with Q/M fixed, this becomes geometric.
Writing nu = Q/M,
P(m) -> (1-r)r^m,
where
r = nu/(1+nu).
Using 1/T = dS/dE then gives
r = exp[-epsilon/(kT)]
and
<m> = 1/(exp[epsilon/(kT)]-1).
So the familiar Bose-Einstein-shaped occupancy law, entropy, temperature, equilibrium, energy equalization, and zero net flux all arise from one reversible discrete exchange model.
The simulation independently reproduces the analytical distribution.
The red line is analytical prediction. The bars are simulation results.
EDIT.
Prediction:
The stationary law is a long-time limit, whereas a real source supplies only a finite number 𝐿 of interactions.
Fixed random-walk endpoints have a Gaussian-like tail exp[−𝑥^2], not the stationary exponential tail. Finite
formation may therefore yield a Gaussian-like high-energy cutoff; the Sun’s spectrum is one possible test of this distinction.
EDIT2.
So we simply replace electromagnetic field modes with discrete particle energies.