Attention in the literature has increasingly turned to the issue of the dependence on ensemble and boundary conditions of fluctuation-induced forces.
We have recently investigated this problem in the one-dimensional Ising model with periodic and antiperiodic boundary conditions (Annals of Physics {\bf 459}, 169533 (2023)). Significant variations of the behavior of Casimir and Helmholtz forces was observed, depending on both ensemble and boundary conditions. Here we extend our study by considering the problem in the important case of Dirichlet (also termed free, or missing neighbors) boundary conditions. The advantage of the mathematical formulation of the problem in terms of Chebyshev polynomials is demonstrated and, in this approach, expressions for the partition functions in the canonical and the grand canonical ensembles are presented. We prove analytically that the Casimir force is attractive for all values of temperature and external ordering field, while the Helmholtz force can be both attractive and repulsive.
Casimir versus Helmholtz forces in the Gaussian model: exact results for Dirichlet--Dirichlet, Neumann--Dirichlet, Neumann--Neumann, and periodic boundary conditions
Institute of Mechanics, Bulgarian Academy of Sciences（保加利亚科学院力学研究所）；Center of Competence for Mechatronics and Clean Technologies “Mechatronics, Innovation, Robotics, Automation and Clean Technologies” - MIRACle（机电一体化与清洁技术能力中心“机电一体化、创新、机器人、自动化与清洁技术”-MIRACle）；Max-Planck-Institut für Intelligente Systeme（马克斯·普朗克智能系统研究所）；Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理与天文学系）
Ensemble dependence of Critical Casimir Forces in Films with Dirichlet Boundary Conditions
在最近的一项研究[Phys. Rev. E \textbf{94}，022103（2016）]中，已经表明，对于经受临界吸附的流体膜，所得的临界卡西米尔力（CCF）可能显著取决于热力学系综。在这里，我们通过考虑所谓的普通表面普适性类内的流体膜来扩展该研究。我们专注于平均场论，其中的OP配置文件满足狄利克雷边界条件，并产生一个非平凡的CCF中存在的外部散装领域或，分别为非零的总序参量内的电影。三维伊辛模型的蒙特卡罗模拟支持了我们的分析结果。我们表明，在正则系综，即，当固定膜内所谓的总质量时，CCF通常是排斥的，而不是像在巨正则系综中那样是吸引的。在Landau-Ginzburg自由能的基础上，进一步得到了薄膜中序参量分布的解析表达式，并分析了薄膜中总质量与外部体场的关系.
