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Transition rates of non-interacting quantum particles from the Widom insertion formula

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.