From Osher Doctorow

Some people are still under the impression that large areas of physics
escape either exponention functions or Probable Causation/Influence or
ratios of exponential functions. From what we have seen in recent
posts in this thread, only essentially statistically independent
events can generate such "other" scenarios

For examples of the ratio type of exponential functions, look at
Hyperphysics' "The energy distribution function(s)" on the internet,
including "Bose-Einstein" (boson) and "Fermi-Dirac" (fermion)
statistics, while the classical Maxwell-Boltzmann statistics are just
exponential. All involve basically exponential decay or contraction
rather than expansion or growth.

The probability density functions (pdfs), which is roughly speaking
probabilities of various energy levels E, are:

1) f(E) = 1/[A exp(E/kT)] (Maxwell-Boltzmann)
2 f(E) = 1/[A exp(E/kT) - 1] (Bose-Einstein)
3) f(E) = 1/[A exp(E/kT) + 1] (Fermi-Dirac)

where for example the Fermi-Dirac case (3) can be written:

4) f(E) = 1/[exp{(E - E_F)/kT} + 1]

with E_F the Fermi energy.

Osher Doctorow