超冷费米系统的集体激发与热力学性质研究
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摘要
围绕超冷费米原子的零温集体激发性质和有限温度下的热力学性质,本文的研究工作如下：
一,在第二章中,我们利用流体力学方程研究了在整个BCS-BEC超流区间无旋转效应的超冷费米系统在零温碟型谐振子势约束下的集体激发模式及其动力学行为,并给出其相应的线性声速、低动量下本征频率和轴向呼吸模式下有效质量的理论预言结果。我们进一步将所得到的结果与相应雪茄型谐振子势约束下无旋超冷费米系统的结果进行了比较与分析。
二,我们研究了服从Haldane分数排斥统计任意维理想任意子有限温度的统计力学行为。在第三章中,我们推导出了任意维均匀及谐振子约束下理想任意子的粒子数、化学势、内能、熵、等容比热、等压比热、等温压缩系数、等温声速、绝热压缩系数和绝热声速的解析表达式,然后基于这些解析表达式,及相应的数值计算结果。我们发现：对于二维均匀理想任意子,其单粒子熵和等容比热的数值结果都不依赖于]Haldane分数排斥统计模型中统计参数g的取值。对于一维谐振子势约束下的理想任意子系统,单粒子熵和等容比热也具有类似于二维均匀理想任意子系统的性质,即：它们都不依赖于统计参数g的取值。在第四章中,我们推导得到了任意维均匀理想任意子和谐振子势约束下理想任意子系统的焦耳—汤姆森系数的解析表达式。在此基础上,我们考察了相应的焦耳—汤姆森转换温度随统计参数g的取值变化的行为。
三,基于幺正极限下的标度不变性,我们假设Haldane分数排斥统计下三维理想任意子可以被用来模拟真实三维空间中有着强的电磁相互作用幺正费米系统的热力学行为。在第五章中,通过设定统计参数g的取值,分别计算了幺正费米系统的化学势、单粒子内能和熵,并将所得到的理论结果与文献上相应的蒙特卡洛数值模拟结果和实验数据进行比较后,发现：在误差允许的范围内,由此分数排斥统计模型所得到的结果与文献上相应的蒙特卡洛数值模拟结果和实验数据是一致的。
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.
一,在第二章中,我们利用流体力学方程研究了在整个BCS-BEC超流区间无旋转效应的超冷费米系统在零温碟型谐振子势约束下的集体激发模式及其动力学行为,并给出其相应的线性声速、低动量下本征频率和轴向呼吸模式下有效质量的理论预言结果。我们进一步将所得到的结果与相应雪茄型谐振子势约束下无旋超冷费米系统的结果进行了比较与分析。
二,我们研究了服从Haldane分数排斥统计任意维理想任意子有限温度的统计力学行为。在第三章中,我们推导出了任意维均匀及谐振子约束下理想任意子的粒子数、化学势、内能、熵、等容比热、等压比热、等温压缩系数、等温声速、绝热压缩系数和绝热声速的解析表达式,然后基于这些解析表达式,及相应的数值计算结果。我们发现：对于二维均匀理想任意子,其单粒子熵和等容比热的数值结果都不依赖于]Haldane分数排斥统计模型中统计参数g的取值。对于一维谐振子势约束下的理想任意子系统,单粒子熵和等容比热也具有类似于二维均匀理想任意子系统的性质,即：它们都不依赖于统计参数g的取值。在第四章中,我们推导得到了任意维均匀理想任意子和谐振子势约束下理想任意子系统的焦耳—汤姆森系数的解析表达式。在此基础上,我们考察了相应的焦耳—汤姆森转换温度随统计参数g的取值变化的行为。
三,基于幺正极限下的标度不变性,我们假设Haldane分数排斥统计下三维理想任意子可以被用来模拟真实三维空间中有着强的电磁相互作用幺正费米系统的热力学行为。在第五章中,通过设定统计参数g的取值,分别计算了幺正费米系统的化学势、单粒子内能和熵,并将所得到的理论结果与文献上相应的蒙特卡洛数值模拟结果和实验数据进行比较后,发现：在误差允许的范围内,由此分数排斥统计模型所得到的结果与文献上相应的蒙特卡洛数值模拟结果和实验数据是一致的。
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.
引文
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