In this article we obtain a fundamental measure functional for the model ofaligned hard hexagons in the plane. Our aim is not just to provide a functionalfor a new, admittedly academic, model, but to investigate the structure offundamental measure theory. A model of aligned hard hexagons has similaritieswith the hard disk model. Both share "lost cases", i.e. admit configurations ofthree particles in which there is pairwise overlap but not triple overlap.These configurations are known to be problematic for fundamental measurefunctionals, which are not able to capture their contribution correctly. Thisfailure lies in the inability of these functionals to yield a correct lowdensity limit of the third order direct correlation function. Here we derivethe functional by projecting aligned hard cubes on the plane x+y+z=0. Thecorrect dimensional crossover behavior of these functionals permits us tofollow this strategy. The functional of aligned hard cubes, however, does nothave lost cases, so neither had the resulting functional for aligned hardhexagons. The latter exhibits, in fact, a peculiar structure as compared to theone for hard disks. It depends on a uniparametric family of weighted densitiesthrough a new term not appearing in the functional for hard disks. Apart fromstudying the freezing of this system, we discuss the implications of thefunctional structure for new developments of fundamental measure theory.
展开▼
机译：在本文中，我们获得了对平面中对齐的硬六边形模型起作用的基本度量。我们的目标不仅是为公认的新学术模型提供功能，而且还应研究基本计量理论的结构。对齐的硬六边形模型与硬盘模型具有相似性。两者都具有“丢失的情况”，即，承认三个粒子具有成对重叠但没有三重重叠的构型。已知这些构型对于基本的测量功能是有问题的，这些功能无法正确地捕获它们的作用。该故障在于这些功能无法产生三阶直接相关函数的正确的低密度极限。在这里，我们通过在平面x + y + z = 0上投影对齐的硬立方体来导出函数。这些功能的正确尺寸交叉行为使我们可以遵循此策略。但是，对齐的硬立方体的功能没有丢失的情况，因此对于对齐的硬六边形来说，生成的功能也没有。实际上，与硬盘相比，后者显示出一种特殊的结构。它取决于在硬盘功能中未出现的新术语，它取决于加权密度的单参数族。除了研究冻结该系统外，我们还将讨论功能结构对基本度量理论的新发展的影响。
展开▼
