On the concept of temperature for a small isolated system

The microcanonical temperature is shown to be a useful concept in calculati ons of the decay of a small isolated system with well defined energy. A sim pler and more transparent description is obtained than in Klots' formulatio n of finite-heat-bath theory, where the system is represented by a canonica l ensemble. As a further illustration of the utility of the microcanonical temperature concept, we discuss a formula derived by Dunbar for the probabi lities for excitation of a single oscillator in a collection of harmonic os cillators with well defined total energy. This formula expresses the excita tion probabilities in terms of the temperature for a canonical ensemble wit h mean energy equal to the energy of the system. However, a much improved a ccuracy is obtained if the canonical temperature and heat capacity are repl aced 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 rho'(E), t he derivative of the level density rho (E) of the collection of oscillators . (C) 2001 American Institute of Physics.