Crossover from the exact factor to the Boltzmann factor

We derive the exact physical statistics of a system in thermal equilibrium with different environments-a gas, a spin, and a Boltzmann environment. The crossover from the exact factor to the Boltzmann factor is parametrized by scaled energy variables. The precise conditions that generate the Boltzman n factor emerge naturally. Such an analytic representation of the exact fac tor, the crossover, and the Boltzmann limit are not found in the standard m odel-free derivations of the Boltzmann factor. (C) 1999 American Associatio n of Physics Teachers.