Abstract 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. T F0 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 T F0 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 10 37 m −3 , the influence of the GUP becomes the dominant factor affecting the thermodynamic properties of the system.
Abstract 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. T F0 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 T F0 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 10 37 m −3 , the influence of the GUP becomes the dominant factor affecting the thermodynamic properties of the system.
When taking into account the generalized uncertainty principle in statistical physics, the density of states must make a correction, which causes all the results of traditional statistical physics to have different degrees of correction. In high-energy or high-temperature conditions, this amendment can subvert the traditional concept and there are also some certain amendments at low temperatures. In this paper we study the thermodynamic properties of the ideal and weakly interacting Fermi gas in low temperature conditions when the generalized uncertainty principle is taken into account. Firstly, analytical expressions of chemical potential, internal energy and heat capacity at constant volume of ideal or weakly interacting Fermi gas are given. Then the properties of copper electron gas are computed as an example, showing that when the generalized uncertainty principle is taken into account the chemical potential, Fermi energy and the ground state energy increase with the increase of temperature, while the heat capacity decreases. When the temperature is lower than 0.3 times TF0, the internal energy increases with the increase of temperature, but becomes decreased when temperature is high than 0.3 times TF0. These amendments are mostly dependent on particle density, which becomes bigger and bigger with particle density increasing.
Abstract Significant evidence is available to support the quantum effects of gravity that lead to the generalized uncertainty principle (GUP) and the minimum observable length. Usually in quantum mechanics, statistical physics does not take gravity into account. Thermodynamic properties of ideal Bose gases in different external power-law potentials are studied under the GUP with a statistical physical method. Critical temperature, internal energy, heat capacity, entropy, particle number of ground state and excited state are calculated analytically to ideal Bose gases in the external potentials under the GUP. Below the critical temperature, taking the rubidium and sodium atoms, ideal Bose gases whose particle densities are under standard and experimental conditions, respectively, as examples, the relations of internal energy, heat capacity and entropy with temperature are analyzed numerically. Theoretical and numerical calculations show that: (1) the GUP leads to an increase in the critical temperature. (2) When the temperature is lower than the critical temperature and slightly higher than 0 K, the GUP’s amendments to internal energy, heat capacity and entropy etc are positive. As the temperature increases to a certain value, these amendments become negative. (3) The external potentials can increase or decrease the influence of the GUP on thermodynamic properties. When ε = 1 J, ε is the quantity that reflects the external potential intensity, and atomic density n = 2.687 × 10 25 m −3 , the GUP’s amendments to the internal energy, heat capacity and entropy of the rubidium atoms ideal Bose gas first decrease and then increase with the increase of X (where X ≡ Σ i 1/ t i is sum of the reciprocal of the exponents of the power function). In three-dimensional harmonic potential, the relative correction term of the GUP is 26 orders of magnitude larger than that of a free-particle system in a fixed container. (4) When ε ≈10 −31 J and n ≈ 10 20 m −3 (which are the experimental data when BEC was first verified by sodium atomic gas), the influence of the GUP can be completely ignored. (5) Under certain conditions, GUP may become the dominant factor governing the thermodynamic properties of the system.
The properties of an ideal Fermi gas in finite gravitational field are studied,based on the semiclassical(Thomas-Fermi) approximation.The general analytical expressions of the total energy,Fermi energy,chemical potential and heat capacity are derived.Moreover,the analytical expressions of the chemical potential and heat capacity in the high-and low-temperature approximations are given,and then,the example of electron gas is used to calculate exactly.All these results are compared with those of free Fermi gas and Fermi gas in infinite gravitational field.The effects of finite gravitational field on properties of Fermi gas are discussed.
At zero-temperature and finite-temperature, the thermodynamic properties of finite unitary Fermi gas in a three-dimensional harmonic trap are investigated by using fractional exclusion statistics, and the results are compared with those of the system which satisfies the thermodynamic limit. At zero-temperature, Fermi energy and average energy of per particle increase with the increase of the number of particles for finite unitary Fermi gas, and their limits are the corresponding parameters of the system which satisfy thermodynamic limits. Fermi energy and average energy of per particle each have a maximum value changing with the boundary of the potential well. For the finite-temperature trapped unitary Fermi system, when the number of particles is certain the average energy of per particle, average entropy of per particle, average heat capacity of per particle each have a characteristic temperature, respectively, when the temperature is equal to the characteristic temperature of the physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the temperature is lower (or higher) than the characteristic temperature of parameter, the physical parameter of the finite system will be greater (or less) than the corresponding parameter of the thermodynamic limit system. The characteristic temperature has particle number effect and boundary effect. When the temperature is determined, the average energy of per particle, average entropy of per particle and average heat capacity of per particle each have a characteristic number of particles, respectively, when the number of particles is equal to the characteristic number of particles for physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the number of particles is less (or more) than the characteristic number of particles for corresponding parameter, the corresponding parameter of the finite system will be less (or larger) than the thermodynamic limit of system.
In this work, improvements are introduced to the current models of the ideal Fermi gas and the ideal Bose gas by incorporating the quantum nature of phase space, which is directly linked to the uncertainty principle. These improved models build upon the recently developed concepts of quantum phase space (QPS) and the QPS representation of quantum mechanics. The Hamiltonian operator for a gas particle and its eigenstates are first determined, and quantum statistical mechanics is used to derive the thermodynamic properties of the ideal gas. Analytic expressions for thermodynamic quantities—including the grand canonical potential, particle number, internal energy, von Neumann entropy, and pressure—are derived, along with the corresponding thermodynamic equations of state for both bosons and fermions. These corrections are particularly significant at low temperatures and in confined volumes, where quantum effects related to system geometry, such as shape and size, become significant. The results also establish a direct link between thermodynamic functions and the quantum statistical variances of momentum. Importantly, the improved models recover well-known classical relations of the ideal gas in the high-temperature and large-volume limits, ensuring consistency with classical physics. By addressing quantum corrections and their thermodynamic implications, this work provides a foundation for further applications in nanoscale systems, quantum gases, low-temperature physics, ultracold physics, and astrophysics.
For the ideal Fermi gas that fills the space inside a cylindrical tube, there are calculated the thermodynamic characteristics in general form for arbitrary temperatures, namely: the thermodynamic potential, energy, entropy, equations of state, heat capacities and compressibilities. All these quantities are expressed through the introduced standard functions and their derivatives. The radius of the tube is considered as an additional thermodynamic variable. It is shown that at low temperatures in the quasi-one-dimensional case the temperature dependencies of the entropy and heat capacities remain linear. The dependencies of the entropy and heat capacities on the chemical potential have sharp maximums at the points where the filling of a new discrete level begins. The character of dependencies of thermodynamic quantities on the tube radius proves to be qualitatively different in the cases of fixed linear and fixed total density. At the fixed linear density these dependencies are monotonous and at the fixed total density they have an oscillating character. Key words: Fermi particle, nanotube, thermodynamic functions, low-dimensional systems, equation of state, heat capacity, compressibility