System-Size Dependence in Grand Canonical and Canonical Ensembles

The thermodynamics for a system with given temperature, density, and volume is described by the Canonical ensemble. The thermodynamics for a corresponding system with the same temperature, volume, and average density is described by the Grand Canonical ensemble. In general, a chosen thermodynamic potential (e.g., free energy) is different in the two cases. Their relationship is considered here as a function of the system size. Exact expressions relating the fundamental potential for each (free energy and pressure, respectively) are identified for arbitrary system size. A formal asymptotic analysis for large system size gives the expected equivalence, but without any characterization of the intermediate size dependence. More detailed evaluation is provided for the simple case of a homogeneous, non-interacting Fermi gas. In this case, the origin of size dependence arises from only two length scales, the average interparticle distance and quantum length scale (thermal deBroglie or Fermi length). The free energies per particle calculated from each ensemble are compared for particle numbers 2 � N � 64 for a range of temperatures above and below the Fermi temperature. The relevance of these results for applications of density functional theory is discussed briefly. Advances in Quantum Chemistry, Volume 71 � 2015 Elsevier Inc. ISSN 0065-3276 All rights reserved. http://dx.doi.org/10.1016/bs.aiq.2015.03.002 11