We present a new analysis of Kerr-Newman-deSitter black holes in terms of thermodynamic quantities that are defined in the observable portion of the universe; between the black hole and cosmological horizons. In particular, we replace the mass m with a new 'area product' parameter X. The physical region of parameter space is found analytically and thermodynamic quantities are given by simple algebraic functions of these parameters. We find that different geometrical properties of the black holes are usefully distinguished by the sum of the black hole and cosmological entropies. The physical parameter space breaks into a region in which the total entropy, together with Λ, a and q uniquely specifies the black hole, and a region in which there is a two-fold degeneracy. In this latter region, there are isentropic pairs of black holes, having the same Λ, a, and q, but different X. The thermodynamic volumes and masses differ in such that there are high and low density branches. The partner spacetimes are related by a simple inversion of X, which has a fixed point at the state of maximal total entropy. We compute the compressibility at fixed total entropy and find that it diverges at the maximal entropy point. Hence a picture emerges of high and low density phases merging at this critical point.
We confine a nematic liquid crystal with homeotropic anchoring to stable toroidal droplets and study how geometry affects the equilibrium director configuration. In contrast to the case of cylindrical confinement, we find that the equilibrium state is chiral -a twisted and escaped radial director configuration. Furthermore, we find that the magnitude of the twist distortion increases as the ratio of the ring radius to the tube radius decreases; we confirm this with computer simulations of optically polarized microscopy textures. In addition, numerical calculations also indicate that the local geometry indeed affects the magnitude of the twist distortion. We further confirm this curvature-induced twisting using bent cylindrical capillaries.
We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi,Origami6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.
In a recent series of experiments, we have shown that 5CB confined to tori and bent cylindrical capillaries with homeotropic boundary conditions also adopts a twisted escaped radial texture resulting from the curved nature of the confining boundaries. As the torus becomes more curved, the ideal location for the escaped core approaches the inner radius of the torus.
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