On the concept of temperature for a small isolated system

The microcanonical temperature is shown to be a useful concept in calculations of the decay of a small isolated system with well defined energy. A simpler and more transparent description is obtained than in Klots’ formulation of finite-heat-bath theory, where the system is represented by a canonical ensemble. As a further illustration of the utility of the microcanonical temperature concept, we discuss a formula derived by Dunbar for the probabilities for excitation of a single oscillator in a collection of harmonic oscillators with well defined total energy. This formula expresses the excitation probabilities in terms of the temperature for a canonical ensemble with mean energy equal to the energy of the system. However, a much improved accuracy is obtained if the canonical temperature and heat capacity are replaced by their microcanonical values. We justify this replacement through a modified derivation, in which the microcanonical temperature appears as the canonical temperature of a fictitious system with level density ρ′(E), the derivative of the level density ρ(E) of the collection of oscillators.