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The thermostatistic problems of a q-deformed ideal Fermi gas in any dimensional space and with a general energy spectrum are studied, based on the q-deformed Fermi-Dirac distribution. The effects of the deformation parameter q on the properties of the system are revealed. It is shown that q-deformation results in some novel characteristics different from those of an ordinary system. Besides, it is found that the effects of the q-deformation on the properties of the Fermi systems are very different for different dimensional spaces and different energy spectrums.

-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.The original Casimir effect [1] shows that zerotemperature quantum fluctuations in an electromagnetic vacuum give rise to an attractive force between two closely spaced perfectly conducting plates. It is a pure quantum effect because there is no force between the plates in classical electrodynamics. The Casimir effect for massive quantum particles is less explored than its counterpart for the photon gas [2], but the Casimir effect in a confined Bose-Einstein condensate (BEC) system caused by the quantum fluctuations of the ground state at zero temperature or thermal fluctuations at finite temperature has recently attracted considerable interest [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The Casimir effect caused by thermal fluctuations in a Bose gas confined by two slabs was studied under the Dirichlet, Neumann, and periodic boundary conditions by Martin and Zagrebnov [2]. The asymptotic expressions of the grand potential with a universal Casimir term [2], the relationship between the thermodynamic Casimir effect in the Bose gas slabs and the critical Casimir forces [3][4][5], and the scaling function [6] for an ideal Bose gas in the case of the Dirichlet boundary condition were obtained. The Casimir effect of an ideal Bose gas was also considered at finite temperatures with or without traps for the Dirichlet boundary condition [8,9].By using the field-theoretical method [6,7] for the plate geometry instead of the thermodynamic method, it became possible to consider the Casimir effect of a weakly interacting Bose gas. Recently, the Casimir force due to zero-temperature quantum fluctuations and thermal fluctuations at finite temperature of a weakly-interacting dilute BEC confined by a pair of parallel plates with Dirichlet and periodic boundary conditions was also investigated [10][11][12][13]. In addition, it has been noted [10,12] that the quasiparticle vacuum in a zerotemperature dilute weakly-interacting BEC should give rise to a measurable Casimir force.

An analytical description of the low temperature behaviour of a trapped interacting Bose gas is presented by using a simple approach that is based on the principle of the constancy of chemical potentials in equilibrium and the local-density approximation. Several thermodynamic quantities, which include the ground-state fraction, chemical potential, total energy, entropy and heat capacity, are derived analytically. It is shown that the results obtained here are in excellent agreement with the experimental data and the theoretical predictions based on the numerical calculation. Meanwhile, by selecting a suitable variable, the divergent problem existing in some papers is solved.

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