Study Of The Collective Excitations And Thermodynamics Of An Ultra-cold Fermi System

The main work of this thesis is based on our investigations of the zero-temperature collective excitation properties and finite-temperature thermodynamics of the ultra-cold Fermi system. There are three aspects of our work in the thesis as follows.At the first aspect, by using the hydrodynamic equations, the zero-temperature col-lective excitation modes and dynamical properties of the disk-shaped harmonically trapped three-dimensional nonrotating or irrotational Fermi system along the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover have been studied in the second chapter of the thesis theoretically. The corresponding theoretical predictions for the zero-temperature linear sound velocity and low-energy eigenfrequencies and the ef-fective mass of the axial breathing mode have also been obtained. Furthermore, we make a comparison of the results of the disk-shaped nonrotating Fermi gas with the correspond-ing results of the cigar-shaped nonrotating Fermi gas.At the second aspect, we investigated the finite-temperature statistical behaviors of the anyon gas within Haldane fractional exclusion statistics. In the third chapter of the thesis, the analytical expressions for the particle number, chemical potential, internal en-ergy, entropy, isochore heat capacity, isobar heat capacity, isothermal compressibility, isothermal sound velocity, adiabatic compressibility, and adiabatic sound velocity of the homogenous and harmonically trapped ideal anyon gases in arbitrary dimensions have been derived. Based on these analytical expressions, the corresponding numerical result-s for these thermodynamic quantities have also been calculated. With careful study, it is found that the entropy and isochore heat capacity per particle are both independen-t of the statistical parameter g in Haldane fractional exclusion statistics model for the two-dimensional homogenous ideal anyon gas. The entropy and isochore heat capaci- ty per particle of the one-dimensional trapped ideal anyon gas have the similar features to the ones of the two-dimensional uniform ideal anyon gas, which means that they do not depend on g. In the next chapter, the analytical expressions and numerical results of the Joule-Thomson coefficients for both homogeneous and harmonically trapped any-dimensional ideal anyons which obey Haldane fractional exclusion statistics are derived in the fourth chapter of the thesis. Furthermore, on the basis of the study of the Joule-Thomson coefficient, the relations between the Joule-Thomson inversion temperature and the statistical parameter g have been obtained.At the last aspect, due to the scale invariance at unitarity, it is assumed that the real strongly interacting unitary Fermi gas can be modeled by the three-dimensional non-interacting anyon gas as a hypothesis. On the basis of this hypothesis, by fixing the statis-tical parameter g, we calculate the finite-temperature chemical potential, internal energy and entropy per particle of the homogeneous and harmonically trapped unitary Fermi gas. Further, we compare the theoretical results with the corresponding Monte Carlo calcula-tions and experimental data. Within the allowed range of permissible experimental error, the obtained theoretical results are reasonably consistent with the data of the experiment.