Exact solution of the two-level system and the Einstein solid in the microcanonical formalism

Abstract

The two-level system and the Einstein model of a crystalline solid are taught in every course of statistical mechanics and they are solved in the microcanonical formalism because the number of accessible microstates can be easily evaluated. However, their solutions are usually presented using the Stirling approximation to deal with factorials. In this paper, those two models are solved without any approximation, using the gamma function and its derivatives. Exact values are calculated for the entropy, temperature and specific heat, and the relative error between our exact solution and the approximate one using the Stirling approximation. This error is significant for small systems, with a number of particles N ∼ 100, as in studies of atomic clusters or nanoscale structures.