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arXiv:1905.02985 · Symmetry

Particle Fluctuations in Mesoscopic Bose Systems

Vyacheslav I. Yukalov

Published May 1, 2019• 8 citations
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Abstract

Particle fluctuations in mesoscopic Bose systems of arbitrary spatial dimensionality are considered. Both ideal Bose gases and interacting Bose systems are studied in the regions above the Bose–Einstein condensation temperature T c , as well as below this temperature. The strength of particle fluctuations defines whether the system is stable or not. Stability conditions depend on the spatial dimensionality d and on the confining dimension D of the system. The consideration shows that mesoscopic systems, experiencing Bose–Einstein condensation, are stable when: (i) ideal Bose gas is confined in a rectangular box of spatial dimension d > 2 above T c and in a box of d > 4 below T c ; (ii) ideal Bose gas is confined in a power-law trap of a confining dimension D > 2 above T c and of a confining dimension D > 4 below T c ; (iii) the interacting Bose system is confined in a rectangular box of dimension d > 2 above T c , while below T c , particle interactions stabilize the Bose-condensed system, making it stable for d = 3 ; (iv) nonlocal interactions diminish the condensation temperature, as compared with the fluctuations in a system with contact interactions.

DOI
10.3390/sym11050603

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Particle fluctuations in mesoscopic Bose systems of arbitrary spatial dimensionality are considered. Both ideal Bose gases and interacting Bose systems are studied in the regions above the Bose–Einstein condensation temperature T c , as well as below this temperature. The strength of particle fluctuations defines whether the system is stable or not. Stability conditions depend on the spatial dimensionality d and on the confining dimension D of the system. The consideration shows that mesoscopic systems, experiencing Bose–Einstein condensation, are stable when: (i) ideal Bose gas is confined in a rectangular box of spatial dimension d > 2 above T c and in a box of d > 4 below T c ; (ii) ideal Bose gas is confined in a power-law trap of a confining dimension D > 2 above T c and of a confining dimension D > 4 below T c ; (iii) the interacting Bose system is confined in a rectangular box of dimension d > 2 above T c , while below T c , particle interactions stabilize the Bose-condensed system, making it stable for d = 3 ; (iv) nonlocal interactions diminish the condensation temperature, as compared with the fluctuations in a system with contact interactions.
The notion of fluctuation indices, characterizing thermodynamic stability of statistical systems, is advanced. These indices are especially useful for investigating the stability of nonuniform and trapped atomic assemblies. The fluctuation indices are calculated for several systems with Bose-Einstein condensate. It is shown that: the ideal uniform Bose-condensed gas is thermodynamically unstable; trapped ideal gases are stable for the confining dimension larger than two; trapped gases, under the confining dimension two, are weakly unstable; harmonically trapped gas is stable only for the spatial dimension three; one-dimensional harmonically trapped gas is unstable; two-dimensional gas in a harmonic trap represents a marginal case, being weakly unstable; interacting nonuniform three-dimensional Bose-condensed gas is stable. There are no thermodynamically anomalous particle fluctuations in stable Bose-condensed systems.
The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.
We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions.
The superfluid fraction Ns/N of 27 bosons under varying confinement is investigated at finite temperature using well-known properties of the harmonic oscillator and the microscopic path integral Monte Carlo method. We find that Ns/N (i) is essentially independent of the interaction strength for all temperatures considered, (ii) changes profoundly as the effective dimensionality is varied from three to one dimensional, (iii) is approximately equal to the condensate fraction N0/N for spherical Bose gases, and (iv) deviates dramatically from N0/N for highly elongated Bose gases.