Fluctuation-induced forces in confined ideal and imperfect Bose gases

Abstract

Fluctuation-induced ("Casimir") forces caused by thermal and quantum fluctuations are investigated for ideal and imperfect Bose gases confined to d-dimensional films of size ∞ d−1 × D under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions, are determined for the ideal gas case with these BCs, where λ th and ξ are the thermal de-Broglie wavelength and the bulk correlation length, respectively. The associated limiting scaling functionsξ ) describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free O(2) theory for BC = P, A, DD, DN, NN. For d = 3, they are expressed in closed analytical form in terms of polylogarithms. The analogous scaling functions Υ BC d (x λ , x ξ , c1D, c2D) and Θ R d (x ξ , c1D, c2D) under the RBCs (∂z − c1)φ|z=0 = (∂z + c2)φ|z=D = 0 with c1 ≥ 0 and c2 ≥ 0 are also determined. The corresponding scaling functions Υ P ∞,d (x λ , x ξ ) and Θ P ∞,d (x ξ ) for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with n internal degrees of freedom in the limit n → ∞. Hence, for d = 3, Θ P ∞,d (x ξ ) is known exactly in closed analytic form. To account for the breakdown of translation invariance in the direction perpendicular to the boundary planes implied by free BCs such as DDBCs, a modified imperfect Bose gas model is introduced that corresponds to the limit n → ∞ of this interacting Bose gas. Numerically and analytically exact results for the scaling function Θ DD ∞,3 (x ξ ) therefore follow from those of the O(2n) φ 4 model for n → ∞.

Thermodynamic Casimir forces in strongly anisotropic systems within the $N\to \infty$ class

“…More precisely, as established in Ref. [35] for the isotropic case, the imperfect Bose gas is equivalent to the O(2N ) model in the limit N → ∞ and the corresponding scaling functions [36,37] for Casimir energy differ by a global factor of two. One may check that (at least for periodic and von Neumann boundary conditions) the form of the dispersion has no impact on the study of Ref.…”

“…The equivalence of the presently analyzed model and interacting N -component bosons was investigated in Ref. [35] for N → ∞. In particular the existence of this limit was established therein.…”

“…Different aspects of this model were studied in recent years [35,[39][40][41][42][43][44][45][46] considering both its bulk and finite-size properties. In particular, for the isotropic continuum case it was established [35,43] that the Bose-Einstein condensation in this model is a representative of bulk O(N → ∞) universality class.…”

“…One may check that (at least for periodic and von Neumann boundary conditions) the form of the dispersion has no impact on the study of Ref. [35] in the aspects exploring connections between the imperfect Bose gas, the interacting N -component Bose gas in the limit N → ∞, and the classical Landau-Ginzburg φ 4 -type theory. In consequence an analogous correspondence holds for the anisotropic situations as well.…”

Exact results for the Casimir force of a three-dimensional model of relativistic Bose gas in a film geometry

“…In the recent years the topics of the Casimir effect and the corresponding Casimir force in Bose systems that is gaining attention [18][19][20][21][22][23][24][25][26][27][28][29][30]. The Casimir force in ideal Bose gas has been studied in [19,20,22]. The case of the imperfect Bose gas has been investigated in [22][23][24][25].…”