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Journal of Qufu Normal University

Critical Temperature of the Bose-Einstein Condensation in Random Box

Feng Zhang

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The Bose-Einstein condensation of the Bose gas is studied in a d-dimensional random box.When the length of the box L satisfies the uniform and the Gauss distributions,the critical temperature of the Bose-Einstein condensation TC is obtained respectively.Compared with that of the rigid box TRC,we find that TC is not more than TRC,and their specific relations are depended on the given distribution functions.In the same way,the condensate problem is discussed for the Bose gas confined in a potential well of harmonic-oscillator,and harmonic trap frequency is a random variable.The similar results are obtained.

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The Bose-Einstein condensation of the Bose gas is studied in a d-dimensional random box.When the length of the box L satisfies the uniform and the Gauss distributions,the critical temperature of the Bose-Einstein condensation TC is obtained respectively.Compared with that of the rigid box TRC,we find that TC is not more than TRC,and their specific relations are depended on the given distribution functions.In the same way,the condensate problem is discussed for the Bose gas confined in a potential well of harmonic-oscillator,and harmonic trap frequency is a random variable.The similar results are obtained.
The Bose–Einstein condensation in the presence of quenched disorder due to random boundary conditions is considered. We have shown that for bosons confined in a box with volume Ld and for random L the Bose condensation temperature is lowered by the presence of disorder and the onset of condensation may even be completely prohibited. Similar results hold for bosons confined in a harmonic oscillator potential well, the geometry used in recent experiments on Bose–Einstein condensation in atomic traps. Comments on possible consequences of quenched disorder in boundary conditions for many fermion systems are also given.
The q-deformed Bose–Einstein distribution is used to study the Bose–Einstein condensation (BEC) of a q-deformed Bose gas in random box. It is shown that the BEC transition temperature is lowered due to random boundary conditions. The effects of q-deformation on the properties of the system are also discussed. We find some properties of a q-deformed Bose gas, which are different from those of an ordinary Bose gas. Similar results are also shown for q-bosons confined in a harmonic oscillator potential well.
Using semi-classical approximation method, the Bose–Einstein condensation (BEC) of a relativistic ideal boson gas (RIBG) in random box is studied. The exact BEC transition temperature Tc and Helmholtz free energy at Tc are derived. The phase diagrams in the (Δ, T) plane are obtained, where Δ is the fluctuation of the box length and T is the temperature of the system. We find that Tc is lowered by the presence of the quenched disorder caused by the random boundary conditions. The effects of antibosons on the RIBG are also studied and it is found that the Helmholtz free energy of the system with antibosons at Tc is lower than that of the system without antibosons. This implies that the omission of antibosons always leads to a metastable state.
The thermodynamic properties of an ideal q-boson gas trapped in an n-dimensional harmonic potential are studied, based on the distribution function of q-bosons. The critical temperature Tc,q of Bose-Einstein condensation (BEC) and the heat capacity C of the system are derived analytically. It is shown that for the q-boson gas trapped in a harmonic potential, BEC may occur in any dimension when q ≠ 1, the critical temperature is always higher than that of an ordinary Bose gas (q = 1), and the heat capacity is continuous at Tc,q for a one-dimensional system but discontinuous at Tc,q for a two- or multi-dimensional system.
The distribution function W0(n0) of the number n0 of particles in the condensate of an ideal Bose gas confined by a trap is found. It is shown that at the temperature below the critical temperature Tc this function has a Gaussian shape and depends on the trap potential via two parameters only. The center of this function shifts to larger values of n0 with decreasing temperature and its width tends to zero, which corresponds to the suppression of fluctuations. In the narrow vicinity of the critical temperature ∣T−Tc∣⩽Tc/N\left| {T - {T_c}} \right| \leqslant {T_c}/\sqrt N ∣T−Tc​∣⩽Tc​/N​ , where N is the number of particles in the trap, the distribution function changes and at the temperature above the critical one it takes the usual form W0(n0) = [1 − exp(μ)]exp(μn0), where μ is the chemical potential in temperature units. In the limit N→∞, this change occurs at a jump.