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arXiv:1605.06633

Density shift of Bose gas due to the Casimir effect and mean field potential

G.M. Bhuiyan

Published May 21, 2016• 1 citation
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Abstract

The shift of density of Bose gas due to the mean field potential (MFP) and the Casimir effect is systematically investigated in the $d$-dimensional configuration space from the point of thermodynamic consideration. We show that, for $d=3$, the shift of density arises completely due to the Casimir effect and, the MFP remains totally ineffective regardless of the state, condensate or non-condensate. But for dimension $d>3$ the MFP plays an active role in shifting the density of Bose gas along with the Casimir interaction. The sign of density shift becomes positive in the present case. So, the corresponding critical temperature shift would be negative, because these two shifts are related as $Δn_{c}$/$n_{c}$ $\approx$ -$ΔT_{c}$/$T_{c}$. It is important to note that, the MFP causes a shift of density for $d>3$ even when the Casimir effect is not there, and the sign of shift becomes negative then; consequently, the critical temperature shift would be positive with MFP alone in this particular situation.

DOI
10.48550/arxiv.1605.06633

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The shift of density of Bose gas due to the mean field potential (MFP) and the Casimir effect is systematically investigated in the ddd-dimensional configuration space from the point of thermodynamic consideration. We show that, for d=3d=3d=3, the shift of density arises completely due to the Casimir effect and, the MFP remains totally ineffective regardless of the state, condensate or non-condensate. But for dimension d>3d>3d>3 the MFP plays an active role in shifting the density of Bose gas along with the Casimir interaction. The sign of density shift becomes positive in the present case. So, the corresponding critical temperature shift would be negative, because these two shifts are related as ΔncΔn_{c}Δnc​/ncn_{c}nc​ ≈\approx≈ -ΔTcΔT_{c}ΔTc​/TcT_{c}Tc​. It is important to note that, the MFP causes a shift of density for d>3d>3d>3 even when the Casimir effect is not there, and the sign of shift becomes negative then; consequently, the critical temperature shift would be positive with MFP alone in this particular situation.
Abstract Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Specifically, this study enabled us to predict the shift of the density of boson in the excited states due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be 1 d − 2 for the imperfect Bose gas. This leads to a value of critical exponent 1 for d = 3. Most interestingly, in the limit when the mean field potential goes to zero the resulting expressions for the above properties coincide with those of the ideal Bose gas.
The critical temperature T c of a weakly interacting Bose gas in an isotropic power-low potential is investigated in the mean-field approximation by taking into account the fact that the particle density distribution function appearing in the mean-field depends on the chemical potential. We derive the general formula of the shift of T c from that of the ideal gas to the lowest order of an interaction. In three-dimensional space, we show that the shift of T c changes its sign from a negative value for n < 3 to a positive one for n > 3, where n is the exponent of the power-low potential.
We consider the Bose gas on a ddd-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature TTT. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature Tc{T}_{c}Tc​ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for sufficiently large values of the scaling variable, and in consequence is not an analytical function. At T>TcT>{T}_{c}T>Tc​ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.
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.
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.
In this study, we have derived close form of the Casimir force for the non-interacting ideal Bose gas between two slabs in harmonic-optical lattice potential by using Ketterle and van Druten approximation. We find that Bose-Einstein condensation temperature TcT_{c}Tc​ is a critical point for different physical behavior of the Casimir force. We have shown that Casimir force of confined Bose gas in the presence of the harmonic-optical potential decays with inversely proportional to d5d^{5}d5 when T≤TcT\leq T_{c}T≤Tc​. However, in the case of T>TcT>T_{c}T>Tc​, it decays exponentially depends on separation ddd of the slabs. Additionally we have discussed temperature dependence of Casimir force and importance of the harmonic-optical lattice potential on quantum critical systems, quantum phase transition and nano-devices.