The partition function of system with finite number of particles is derived is derived by using partition function of a weakly interacting Fermi gas trapped in weak magnetic field.Based on the derived expressions,the analytical expressions of thermodynamic quantities of a weakly interacting Fermi gas for finite particle number are solved by using statistical average method and the thermodynamic properties under different temperature conditions are given.It is shown that each thermodynamic quantity has a correctional term due to effects of a finite number of particles.The particle number has a direct effect on correction of chemical potential but has no direct effects on corrections of total energy and heat capacity except T=0.Furthermore,effects of a finite number of particles always reduce chemical potential,thereby zero point of chemical potential shifts towards low temperatures.The increase of particle number weakens the effect,but the repulsive interactions between particles strengthen the effect.
The partition function of system with finite number of particles is derived is derived by using partition function of a weakly interacting Fermi gas trapped in weak magnetic field.Based on the derived expressions,the analytical expressions of thermodynamic quantities of a weakly interacting Fermi gas for finite particle number are solved by using statistical average method and the thermodynamic properties under different temperature conditions are given.It is shown that each thermodynamic quantity has a correctional term due to effects of a finite number of particles.The particle number has a direct effect on correction of chemical potential but has no direct effects on corrections of total energy and heat capacity except T=0.Furthermore,effects of a finite number of particles always reduce chemical potential,thereby zero point of chemical potential shifts towards low temperatures.The increase of particle number weakens the effect,but the repulsive interactions between particles strengthen the effect.
Based on the pseudopotential method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.
The analytical expressions of internal energy, chemical potential and heat capacity of the little parameter r for a weakly interacting Fermi gas in weak magnetic field are derived by using “pseudopotential" method and ensemble theory. Based on the derived expressions, the thermodynamic properties of a weakly interacting Fermi gas in weak magnetic field at both high and low temperatures are given. The effects of magnetic field and interparticle interactions on the thermodynamic properties are discussed. The difference in the effects of magnetic trap and three-dimensional harmonic trap on the properties of the system and their reasons are analyzed.
Taking the anomalous magnetic moment into consideration, the analytical expression of the free energy for a weakly interacting Fermi gas in a weak magnetic field is derived by using the pseudopotential method and thermodynamic theory, therefore the thermodynamic properties can be studied. Based on the derived expression, the thermodynamic properties of this system at both high and low temperatures are given, and the effect of anomalous magnetic moment on thermodynamic properties can be analyzed. The effect of anomalous magnetic moment on the thermodynamic properties is related to temperature, and with the rise of temperature this effect increases in the low temperature zone, but decreases in the high temperature zone. For the chemical potential and internal energy of the system, the anomalous magnetic moment strengthens the influence of the magnetic field, but weakens the influence of the interaction. Under the influence of anomalous magnetic moment, the heat capacity of the system decreases in the low temperature zone, but increases in the high temperature zone.
Based on the methods of quantum statistics, by considering relativistic correction of single particle and introducing the nonrelativistic free energy of the system of spinless and free fermions,the free energy for Fermi gas in a weak magnetic field is derived, thereout the analytical expressions of thermodynamic quantities of the system under various temperature condtions are given, the effects of relativistic correction on the properties of the nonrelativistic system are exhibited particular and concrete, and the mechanism of the effects is analysed. Compared with those of nonrelativistic situation, it is shown that the effects of relativistic correction always reduce chemical potential. The effects of relativistic correction decrease the total energy and heat capacity at low temperatures, while increase them at high temperatures. The changes of the total energy and heat capacity (not include the low-temperature heat capacity ) are proportional to the square of magnetic field.
The low-temperature properties of a weakly interacting Bose gas in weak magnetic field are studied by using the conclusion of a weakly interacting Bose gas in external potential.The expressions of total energy and heat capacity are given analytically.The effects of the interparticle interactions as well as magnetic field on the thermodynamic properties of the system are discussed.It is shown that repulsion(attraction) can make the spatial distribution of particle number and energy concentrate(disperse) and the chemical potential,total energy,heat capacity of system increase(decrease).While strengthen magnetic field,the spatial distribution of particle number and energy will tend to disperse and the effects of interaction on it will be weaken.The effects of interaction on the different eigenvalue exist different individual behaviors.
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 (BEC), 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).