Canonical Ensemble with Temperature Limitation

Abstract

Ideal Bose and Fermi systems are studied on the basis of a canonical ensemble, subject to the condition that their temperature is less than a given temperature Tmax. A single new parameter (the tau-parameter, τ) is needed to keep account of the new constraint. The parameter τ is shown to be the exponential of a pseudo-chemical potential that is linearly dependent on temperature. The inclusion of the τ- parameter leads to generalizations of usual thermodynamic quantities (internal energy, heat capacity and entropy) and various particular cases are discussed. The heat capacity of a Bose system can exhibit a maximum at a temperature less than the maximum temperature Tmax. The number of micro-states in the canonical ensemble is found to increase with τ. The heat capacity cV of a Fermi system of non-interacting spins exhibits a Schottky anomaly. The peak depends on τ, and for some cases cV/k can significantly exceed unity. The influence of τ on the entropy of the Fermi system and on the number of micro-states in the canonical ensemble is significant but not spectacular.