Finite-temperature reaction-rate formula: Finite volume system, detailed balance, $T\to 0$ limit, and cutting rules

A complete derivation, from first principles, of the reaction-rate formula for a generic process taking place in a heat bath of finite volume is given. It is shown that the formula involves no finite-volume correction. Through perturbative diagrammatic analysis of the resultant formula, the detailed-balance formula is derived. The zero-temperature limit of the formula is discussed. Thermal cutting rules, which are introduced in previous work, are compared with those introduced by other authors.