Finite size effect and ensemble inequivalence in Fermionic and bosonic systems - Tan, Wilar Z. (Institute of Physics, College of Science, University of the Philippines Diliman, 2016)
We calculate the Fermi-Dirac and Bose-Einstein Distribution Function in the Microcanonical Ensemble (MMDF) for the system of fermions and bosons. Such MMDF is important in the study of ultracold systems with finite number of particles. To solve for the MMDF, the method of Laplace transform was utilized to solve the integral equation for the MMDF, and the thermodynamic potential was utilized to make the solution to the integral equation easier. The derived MMDF is in closed form that is dependent only on the total density of states of the system. We found that MMDF and CMDF are mathematically equivalent for two-dimensional system of bosons. We found that there is an inequivalence between the MMDF and CMDF for the two-dimensional system of fermions, three dimensional system of bosons, and three-dimensional system of bosons. Finally, since ultracold systems are typically contained in small volumes the finite size effect of the container is incorporated to the MMDF. The finite size correction to the MMDF does not use the continuum approximation to the volume of the system.