N-E-V分布及赝势法研究弱磁场中弱相互作用费米子气体的热力学性质

Investigation of thermodynamic properties of weakly interacting Fermi gas in weakly magnetic field by using the N-E-V distribution and pseudopotential method

基于低温下量子系统的相关实验多是在体积、能量和粒子数都可变的外场束缚下进行的事实,由体积、能量和粒子数可变的完全开放系统的统计分布（N-E-V分布）研究了弱磁场中弱相互作用费米系统的热力学性质.首先求出了一般情况下由费米积分表示的内能和热容的解析表达式.在此基础上,又给出了在低温极限条件下内能与热容的解析表达式和数值计算结果,并将N-E-V分布（粒子数密度变化）的结果与赝势法（粒子数密度不变）的结果进行了比较.结果表明：N-E-V分布方法的计算结果总是补偿赝势法计算结果的过度偏差.由N-E-V分布方法所得结果最特异之处在于：在低温条件下,弱磁场中弱相互作用费米系统存在一相变温度tc,其正处于费米系统发生玻色-爱因斯坦凝聚（BEC）和费米原子形成库珀对的超流状态（BCS）相变及BEC-BCS跨越的温度范围内,且不随反映弱相互作用大小和特征的散射长度a（a〈0引力,a〉0斥力）变化,但随弱磁场的加强而降低,即弱磁场可调节该相变温度.磁场为零时,相变温度最高,为费米温度的0.184倍.

Based on the fact that most of low temperature experiments of quantum systems are explored in an external field on condition that the particle numbers, volumes and energies of systems may be changed, the thermodynamic properties of weakly interacting Fermi systems in weak magnetic field are studied by using the statistical distribution of the completely open system with variable particle number, volume, and energy （N-E-V distribution）. Firstly, the analytical expressions of internal energy and heat capacity, which are in the Fermi integral form, are obtained in the general case, and the analytical expressions and numerical results of energy and heat capacity are given under the extreme condition of supper- low temperature. The calculation results by the N-E-V distribution （with particle number density being variable） are compared with those by a pseudopotential method （with particle number density being unchanged）. It can be found that the deviations of the internal energy and heat capacity calculated by the two different methods are very small, and the N-E-V distribution method can partially compensate for the error caused by the pseudo potential method. The most interesting point of the results obtained by the N-E-V distribution method is that there is a phase transition temperature in the weakly interacting Fermi system in weak magnetic field under the low temperature condition. The phase transition temperature is just in the range where occur the Fermi systems, Bose-Einstein condensation （BEe）, Bardeen-Cooper- Schrieffer （BCS） phase transition, and BEC-BCS crossover, and does not vary with strength nor characteristic （attraction or repulsion） of the weak interaction, but it decreases with the strengthening of the external magnetic. When there is no external magnetic, the phase transition temperature is highest （more than 0.184 times Fermi temperature）.