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.
Abstract The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential U = ∑ i = 1 d c i | x i / a i | n i are studied carefully. Detailed calculation of Kim et al. (J. Phys. Condens. Matter 11 (1999) 10269) yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume C V are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition in trapped systems as well.
Thermodynamics of ideal Fermi gas trapped in an external generic power law potential
U=∑i=1dci∣aixi∣ni are investigated systematically from the grand thermodynamic potential in
d dimensional space. These properties are explored deeply in the degenerate limit (
μ>>KBT), where the thermodynamic properties are greatly dominated by Pauli exclusion principle. Pressure and energy along with the isothermal compressibilty is non zero at
T=0K, denoting trapped Fermi system is quite live even at absolute zero temperature. The nonzero value of compressibilty denotes zero point pressure is not just a constant but depends on volume.
The Casimir effect of an ideal Bose gas trapped in a generic power-law potential and confined between two slabs with Dirichlet, Neumann, and periodic boundary conditions is investigated systematically, based on the grand potential of the ideal Bose gas, the Casimir potential and force are calculated. The scaling function is obtained and discussed. The special cases of free and harmonic potentials are also discussed. It is found that when T ⩽ T c (where T c is the critical temperature of Bose-Einstein condensation), the Casimir force is a power-law decay function; when T > T c , the Casimir force is an exponential decay function; and when T ≫ T c , the Casimir force vanishes.
According to generic relation between particle's energy and momentum,we adopt the common procedure of quanta statistics which firstly work out the grand partition function,the critical temperature and producing condition of any dimensional ideal Bose system.With no external potential and in the thermodynamics condition,it is shown that the producing condition of any dimensional ideal Bose-Einstein Condensation only lies on the size relation between space dimensionality of particle and momentum's index in energy expression,which is a universal conclusion.For it,we can explain no Bose-Einstein Condensation with no external potential and under low dimensional spaces.
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number
N of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature
Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of
x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of
Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with
104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.