Fakultät für Physik, Universität Duisburg-Essen, D-47058 Duisburg, Germany.
Phys Rev E. 2017 Jun;95(6-1):062112. doi: 10.1103/PhysRevE.95.062112. Epub 2017 Jun 8.
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 Υ_{d}^{BC}(x_{λ}=D/λ_{th},x_{ξ}=D/ξ) of the residual reduced grand potential per area φ_{res,d}^{BC}(T,μ,D)=D^{-(d-1)}Υ_{d}^{BC}(x_{λ},x_{ξ}) 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 functions Θ_{d}^{BC}(x_{ξ})≡Υ_{d}^{BC}(∞,x_{ξ}) 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_{d}^{BC}(x_{λ},x_{ξ},c_{1}D,c_{2}D) and Θ_{d}^{R}(x_{ξ},c_{1}D,c_{2}D) under the RBCs (∂{z}-c{1})ϕ|{z=0}=(∂{z}+c_{2})ϕ|{z=D}=0 with c{1}≥0 and c_{2}≥0 are also determined. The corresponding scaling functions Υ_{∞,d}^{P}(x_{λ},x_{ξ}) and Θ_{∞,d}^{P}(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_{∞,d}^{P}(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_{∞,3}^{DD}(x_{ξ}) therefore follow from those of the O(2n)ϕ^{4} model for n→∞.
研究了热和量子涨落引起的涨落诱导（“卡西米尔”）力，这些力是由被限制在具有周期性（P）、反周期性（A）、狄利克雷-狄利克雷（DD）、诺伊曼-诺伊曼（NN）和罗宾（R）边界条件（BC）的无穷大(^{d-1}×D)尺寸薄膜中的理想和非理想玻色气体引起的。对于这些 BC 的理想气体情况，确定了单位面积剩余还原吉布斯势(\phi_{res,d}^{BC}(T,\mu,D)=D^{-(d-1)}\Upsilon_{d}^{BC}(x_\lambda,x_\xi))的完整标度函数(\Upsilon_{d}^{BC}(x_\lambda=x_D/\lambda_{th},x_\xi=x_D/\xi))，其中(\lambda_{th})和(\xi)分别是热德布罗意波长和体相关长度。关联的极限标度函数(\Theta_{d}^{BC}(x_\xi)=\Upsilon_{d}^{BC}(\infty,x_\xi))描述了体凝聚相变的临界行为，与先前从 BC=P、A、DD、DN、NN 的大规模自由 O(2)理论确定的一致。对于(d=3)，它们以封闭的解析形式表示为多元对数的形式。在 RBCs（(\partial_{z}-c_{1}\right)\phi|{z=0}=\left(\partial{z}+c_{2}\right)\phi|{z=D}=0，其中(c{1}\geq0)和(c_{2}\geq0)下的类似标度函数(\Upsilon_{d}^{BC}(x_\lambda,x_\xi,c_{1}D,c_{2}D))和(\Theta_{d}^{R}(x_\xi,c_{1}D,c_{2}D))也被确定。对于非理想玻色气体，相应的标度函数(\Upsilon_{\infty,d}^{P}(x_\lambda,x_\xi))和(\Theta_{\infty,d}^{P}(x_\xi))与具有 n 个内部自由度的相互作用玻色气体在 n→\infty 时的一致。因此，对于(d=3)，(\Theta_{\infty,d}^{P}(x_\xi))以封闭的解析形式精确已知。为了解决由 DDBC 等自由 BC 隐含的垂直于边界平面的平移不变性的破坏，引入了一个改进的非理想玻色气体模型，该模型对应于 n→\infty 时这个相互作用玻色气体的极限。因此，从 n→\infty 时 O(2n)\phi^{4}模型的数值和解析精确结果可以得出(\Theta_{\infty,3}^{DD}(x_\xi))的标度函数。
