Casimir力与Helmholtz力：精确结果
Casimir versus Helmholtz forces: Exact results
AI总结：
本文研究涨落诱导力的系综依赖性，通过传递矩阵方法导出具有固定磁化强度的一维Ising模型中Casimir力与Helmholtz力的精确结果，并证明两系综自由能通过Legendre变换关联且正则系综的有限尺寸修正更显著。
摘要：
最近，人们的注意力转向了涨落诱导力的系综依赖性问题。作为一个值得注意的例子，在 $O(n)$ 系统中，可以证明此类力背后的统计力学在恒定 $\vec{M}$ 磁正则系综（CE）中与在广泛研究的恒定 $\vec{h}$ 巨正则系综（GCE）中是不同的。在这里，GCE中Casimir力的对应物是CE中的Helmholtz力。鉴于有限系统的两个系综之间的差异，可以合理地预期，对于相同的几何形状和边界条件，这些力通常会有不同的行为。在此，我们针对具有周期性和反周期性边界条件的一维Ising模型，给出了Casimir力和Helmholtz力的精确结果，并比较了它们的行为。我们注意到，Ising模型最近已在Phys.Rev. E {\bf 106} L042103(2022) 中被求解，该求解针对其序参数固定值 $M$ 的情况，使用的是组合方法。在这里，我们使用传递矩阵方法（TMM）导出了具有 $N$ 个自旋和固定值 $M$ 的一维Ising模型配分函数的精确结果；先前通过TMM获得的结果仅限于 $M=0$ 且 $N$ 为偶数的情况。作为副产品，我们导出了Gauss超几何函数的几个特定积分表示。利用这些结果，我们严格证明了CE和GCE的自由能通过热力学极限中的Legendre变换相互关联，并确立了正则情况下的主导有限尺寸修正，结果表明这些修正比GCE情况中相应的修正要显著得多。
英文摘要：
Recently, attention has turned to the issue of the ensemble dependence of fluctuation induced forces. As a noteworthy example, in $O(n)$ systems the statistical mechanics underlying such forces can be shown to differ in the constant $\vec{M}$ magnetic canonical ensemble (CE) from those in the widely-studied constant $\vec{h}$ grand canonical ensemble (GCE). Here, the counterpart of the Casimir force in the GCE is the \textit{Helmholtz} force in the CE. Given the difference between the two ensembles for finite systems, it is reasonable to anticipate that these forces will have, in general, different behavior for the same geometry and boundary conditions. Here we present some exact results for both the Casimir and the Helmholtz force in the case of the one-dimensional Ising model subject to periodic and antiperiodic boundary conditions and compare their behavior. We note that the Ising model has recently being solved in Phys.Rev. E {\bf 106} L042103(2022), using a combinatorial approach, for the case of fixed value $M$ of its order parameter. Here we derive exact result for the partition function of the one-dimensional Ising model of $N$ spins and fixed value $M$ using the transfer matrix method (TMM); earlier results obtained via the TMM were limited to $M=0$ and $N$ even. As a byproduct, we derive several specific integral representations of the hypergeometric function of Gauss. Using those results, we rigorously derive that the free energies of the CE and grand GCE are related to each other via Legendre transformation in the thermodynamic limit, and establish the leading finite-size corrections for the canonical case, which turn out to be much more pronounced than the corresponding ones in the case of the GCE.
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