Ideal Fermi Systems Research Articles

Solution of Density Profile Function of Trapped Fermi System

Starting from many-body theory, the density profile function of ideal Fermi systems in harmonic and delta traps are obtained under the quantum mechanics and thermodynamic statistical physics, respectively. The related thermodynamic properties are analyzed based on the Fermi-Dirac distribution and exact solution of the density profile function.

Thermodynamics of Generalized Fermi Systems in a Harmonic Trap

Thermodynamics of the generalized ideal Fermi systems in the two-and three-dimensional harmonic traps are respectively calculated by the Tsallis entropy in this paper. The influences of the trap and q-number on the thermodynamic parameters (epically the heat capacity) are analysed in detail. The results yield a well agreement with the classical cases.

Thermodynamic properties of ideal Fermi gases in a harmonic potential in an n-dimensional space under the generalized uncertainty principle

Using the semi-classical (Thomas–Fermi) approximation, the thermodynamic properties of ideal Fermi gases in a harmonic potential in an n-dimensional space are studied under the generalized uncertainty principle (GUP). The mean particle number, internal energy, heat capacity and other thermodynamic variables of the Fermi system are calculated analytically. Then, analytical expressions of the mean particle number, internal energy, heat capacity, chemical potential, Fermi energy, ground state energy and amendments of the GUP are obtained at low temperatures. The influence of both the GUP and the harmonic potential on the thermodynamic properties of a copper-electron gas and other systems with higher electron densities are studied numerically at low temperatures. We find: (1) When the GUP is considered, the influence of the harmonic potential is very much larger, and the amendments produced by the GUP increase by eight to nine orders of magnitude compared to when no external potential is applied to the electron gas. (2) The larger the particle density, or the smaller the particle masses, the bigger the influence of the GUP. (3) The effect of the GUP increases with the increase in the spatial dimensions. (4) The amendments of the chemical potential, Fermi energy and ground state energy increase with an increase in temperature, while the heat capacity decreases. TF0 is the Fermi temperature of the ideal Fermi system in a harmonic potential. When the temperature is lower than a certain value (0.22 times TF0 for the copper-electron gas, and this value decreases with increasing electron density), the amendment to the internal energy is positive, however, the amendment decreases with increasing temperature. When the temperature increases to the value, the amendment is zero, and when the temperature is higher than the value, the amendment to the internal energy is negative and the absolute value of the amendment increases with increasing temperature. (5) When electron density is greater than or equal to 1037 m−3, the influence of the GUP becomes the dominant factor affecting the thermodynamic properties of the system.

Research on the Thermodynamic Property of Weakly Interacting Fermi Gases Through N-E-V Distribution and Pseudopotential Method

As the experiments of quantum system are conducted in the external field where the volume, energy and particle number are variable, the thermodynamic property of weakly interacting Fermi system is studied in this paper through the statistical distribution (N-E-V distribution) of a completely open system where the volume, energy and particle number are variable. Firstly, analytical expressions of internal energy, the equation of state, entropy and heat capacity expressed by Fermi integral are worked out. On that basis, the analytical expression and results of numerical calculation of the above thermodynamic quantities in extremely low temperature conditions can be known. Then, comparison between the calculations of N-E-V distribution and pseudopotential method (the volume and particle number remain the same) is drawn. The results show that the calculation of N-E-V distribution always makes up for the deviation of that of pseudopotential method. Besides, the most special thing about the calculation derived from N-E-V distribution is that the temperature variation curve of internal energy and that in the ideal Fermi system intersect. The characteristics of energy transfer between “attractive force” and “repulsive force” at the intersection temperature tc, which indicates that there is a phase transition. The intersection temperature does not change with the scattering length a (a0 repulsive force). Moreover, the intersection temperature of all weakly interacting Fermi system is 0.184 times as large as Fermi temperature, which is within the range of temperature for BEC and BCS phase transition and BEC-BCS crossover.

Analytical contradictions of the fixed-node density matrix

Over the last decades the fixed-node method has been used for a numerical treatment of thermo-dynamic properties of strongly correlated Fermi systems. In this work correctness of the fixed-node method for ideal Fermi systems is analytically analyzed. It is shown that the fixed-node prescription of calculation of the density matrix leads to contradictions even for two ideal fermions. The main conclusion of this work is that the fixed-node method can not reproduce the fermion density matrices and should be considered as uncontrolled empirical approach in treatment of thermodynamics of Fermi systems.

Maximum trap range and equation of state for Fermi gas in potential trap

In the Thomas-Fermi semi-classical approximation, the maximal trap range and the real trap volume of ideal Fermi gas in an n-dimensional potential trap are gaven, and the relevant equations of state are derived. These results indicate that the maximal trap range and the real pressure of trapped gas are related to the potential field and the chemical potential of the free and ideal Fermi system. When the Thomas-Fermi approximate is valid and the condition ((kT)/(hω))2 ((16π2g)/(9N))2/3<<1 is satisfied, the application of the equation of state to three-dimensional spherical symmetry harmonic trap yields the result that the change of pressure is not obvious when the temperature changes, but the change of pressure is closely related to mass of particle, number of particles and the frequency of harmonic potential.

Statistical measure of complexity and correlated behavior of Fermi systems

We apply the statistical measure of complexity, introduced by López-Ruiz, Mancini, and Calbet (LMC), to uniform Fermi systems. We investigate the connection between information and complexity measures with the strongly correlated behavior of various Fermi systems as nuclear matter, electron gas, and liquid helium. We examine the possibility that LMC complexity can serve as an index quantifying correlations in the specific system and to which extent could be related with experimental quantities. Moreover, we concentrate on thermal effects on the complexity of ideal Fermi systems. We find that complexity behaves, both at low and high values of temperature, in a similar way as the specific heat.

Nonextensive thermostatistical investigation of free electronic gas in metal

The relevant aspects of the q-thermostatistical treatment for the free electronic gas which is an ideal Fermi system are discussed.Thermodynamical quantities of the free electronic gas such as total number of particles,total energy,heat capacity,free energy,etc.,are calculated within the Tsallis thermo-statistics where the factorization approach is incorporated,and the ensuing statistics is derived.Special attention is paid to heat capacity.The limiting cases of interest are discussed in some detail,namely,low-temperature regimes,and the approximate results pertaining to the case q→1 (the conventional Fermi-Dirac statistics corresponds to q=1).We also compare our results with previous Tsallis' works.

Nonextensive thermostatistic properties of a q-generalized Fermi system

Based on the generalized statistic distribution derived from the Tsallis’ entropy that has been successfully used in complex systems with long-range interactions and/or long-duration memory, the thermostatistic properties of a q-generalized Fermi system are studied. The total number of particles, internal energy, and heat capacity at constant volume are calculated for two different cases of q≤1 and q≥1, respectively, where q is an important parameter to evaluate the nonextensivity of the system. The thermostatistic characteristics of the system are discussed in detail. It is found from the results obtained here that the Fermi energy of such a system is independent of the parameter q and is equal to that of an original ideal Fermi system when q≤1, while other thermostatistic properties of the system depend closely on the parameter q. For example, when q 1, some thermodynamic parameters of the system at low temperatures must be cut off.

Q-thermostatistics and the analytical treatment of the ideal Fermi gas

We discuss relevant aspects of the exact q-thermostatistical treatment for an ideal Fermi system. The grand-canonical exact generalized partition function is given for arbitrary values of the nonextensivity index q, and the ensuing statistics is derived. Special attention is paid to the mean occupation numbers of single-particle levels. Limiting instances of interest are discussed in some detail, namely, the thermodynamic limit, considering in particular both the high- and low-temperature regimes, and the approximate results pertaining to the case q∼1 (the conventional Fermi–Dirac statistics corresponds to q=1). We compare our findings with previous Tsallis’ literature.

Condensation in ideal Fermi gases

I investigate the possibility of condensation in ideal Fermi systems of general single particle density of states. For this I calculate the probability $w_{N_0}$ of having exactly $N_0$ particles in the condensate and analyze its maxima. The existence of such maxima at macroscopic values of $N_0$ indicates a condensate. An interesting situation occurs for example in 1D systems, where $w_{N_0}$ may have two maxima. One is at $N_0=0$ and another one may exist at finite $N_0$ (for temperatures bellow a certain condensation temperature). This suggests the existence of a first order phase transition. % The calculation of $w_{N_0}$ allows for the exploration of ensemble equivalence of Fermi systems from a new perspective.

Cluster Expansion for Ideal Fermi Systems in the “Fixed Node Approximation”

Results of the direct path integral Monte Carlo simulation (DPIMCS) have shown that the “fixed node approximation” (FNA) rather well describes thermodynamic properties of the strongly coupled fermions at “weak and moderate” degeneracy. To explain the increasing difference for highly degenerate fermions it is rigorously proved that the exact Fermi function with index 5/2 in grand ensemble can not be reproduced in FNA, so FNA has not the correct limit to ideal fermions.

Cluster Expansion for Ideal Fermi Systems in the “Fixed Node Approximation”

Results of the direct path integral Monte Carlo simulation (DPIMCS) have shown that the “fixed node approximation” (FNA) rather well describes thermodynamic properties of the strongly coupled fermions at “weak and moderate” degeneracy. To explain the increasing difference for highly degenerate fermions it is rigorously proved that the exact Fermi function with index 5/2 in grand ensemble can not be reproduced in FNA, so FNA has not the correct limit to ideal fermions.

Cluster expansion for ideal Fermi systems in the `fixed-node approximation'

Over the last ten years the `fixed-node approximation', well known in the literature, has been widely used for a numerical treatment of thermodynamic properties of strongly correlated Fermi systems. Results of the direct path integral Monte Carlo simulation performed here show that the `fixed-node approximation' describes the thermodynamic properties of the strongly coupling fermions rather well at `weak and moderate' degeneracy. To analyse the reasons for the increasing difference between the `fixed-node approximation' and the results of direct path integral Monte Carlo simulations for highly degenerate fermions, the correctness of the `fixed-node approximation' for ideal Fermi systems has been analysed by analytical methods. A rigorous proof has been given of the fact that the exact Fermi function with index 3/2 describing the logarithm of the grand partition function of ideal fermions cannot be reproduced in the `fixed-node approximation', which means that the `fixed-node approximation' does not give the correct ideal Fermi gas limit.

Superfluidity near solid surfaces in a Thomas-Fermi-Dirac jellinm model: Potassium and nickel

Generally an almost ideal Fermi system with attraction between fermions possesses nonNewtonian behavior. Electrons described according to the Thomas-Fermi-Dirac model (TFD) form such a system. We investigated these electrons in solids with surfaces. In this kind of non-Newtonian motions, superfluidity appeared under certain condition. This conclusion is shown theoretically when we apply this model to potassium and nickel. For K, there is a layer of thickness only 0.33 bohr near the surface in which electrons move to higher potential locations similar to the climbing effect under gravity although non-Newtonian motions occur throughout the system. For nonmagnetic 4s2 electrons of Ni, there may also be a layer of this kind of thickness 0.25 bohr near the surface after we choose the surface density properly in theory. In all these systems, electrons are reflected from surface to interior layers.