  Transition rates among different states in a system of non-interacting
quantum particles in contact with a heat reservoir include the factor $1\mp
\bar{n}_i$, with a minus sign for fermions and a plus sign for bosons, where
$\bar{n}_i$ is the average occupation number of the final state. It is shown
that this factor can be related to the difference of the chemical potential
from that of an ideal classical mixture; this difference is formally equivalent
to the excess chemical potential in a classical system of interacting
particles. Using this analogy, Widom's insertion formula is used in the
calculation of transition rates. The result allows an alternative derivation of
quantum statistics from the condition that transition rates depend only on the
number of particles in the target energy level. Instead, if transition rates
depend on the particle number only in the origin level, the statistics of
ewkons is obtained; this is an exotic statistics that can be applied to the
description of dark energy.
