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The distance of closest approach of two objects is the distance between their centers when they are externally tangent. The objects may be geometric shapes or physical particles with well defined boundaries. The distance of closest approach is sometimes referred to as the contact distance. For the simplest objects, spheres, the distance of closest approach is simply the sum of their radii. For non-spherical objects, the distance of closest approach is a function of the orientation of the objects, and its calculation can be difficult. The maximum packing density of hard particles, an important problem of ongoing interest,〔S. Torquato and Y. Jiao, (Nature 460, 876-879, 2009 )〕 depends on their distance of closest approach. The interactions of particles typically depend on their separation, and the distance of closest approach plays an important role in determining the behavior of condensed matter systems. The excluded volume of particles (the volume excluded to the centers of other particles due to the presence of one) is a key parameter in such descriptions,;〔T.L. Hill, An Introduction to Statistical Thermodynamics (Addison Wesley, London, 1960)〕〔T.A. Witten, and P.A. Pincus, Structured Fluids (Oxford University Press, Oxford, 2004)〕 the distance of closest approach is required to calculate the excluded volume. The excluded volume for identical spheres is just four times the volume of one sphere. For other anisotropic objects, the excluded volume depends on orientation, and its calculation can be surprising difficult.〔Forces, Growth and Form in Soft Condensed Matter: At the Interface between Physics and Biology, ed. A.T. Skjeltrop and A.V. Belushkin, (NATO Science Series II: Mathematics, Physics and Chemistry, 2009),〕 The simplest shapes after spheres are ellipses and ellipsoids; these have received (considerable attention ),〔A. Donev, F.H. Stillinger, P.M. Chaikin and S. Torquato, Phys. Rev. Lett. 92, 255506 (2004)〕 yet their excluded volume is not known. Vieillard Baron was able to provide an overlap criterion for two ellipses. His results were useful for computer simulations of hard particle systems and for packing problems using Monte Carlo simulations. The one anisotropic shape whose excluded volume can be expressed analytically is the spherocylinder; the solution of this problem is a classic work by Onsager.〔L. Onsager, Ann NY Acad Sci, 51, 627 (1949)〕 The problem was tackled by considering the distance between two line segments, which are the center lines of the capped cylinders. Results for other shapes are not readily available. The orientation dependence of the distance of closest approach has surprising consequences. Systems of hard particles, whose interactions are only entropic, can become ordered. Hard spherocylinders form not only orientationally ordered nematic, but also positionally ordered smectic phases.〔D. Frenkel, J. Phys. Chem. 91, 4912-4916 (1987)〕 Here, the system gives up some (orientational and even positional) disorder to gain disorder and entropy elsewhere. ==Distance of closest approach of two ellipses== Vieillard Baron first investigated this problem, and although he did not obtain a result for the distance of closest approach, he derived the overlap criterion for two ellipses. His results were useful for the study of the phase behavior of hard particles and for the packing problem using Monte Carlo simulations. Although overlap criteria have been developed,〔J. Vieillard-Baron, "Phase transition of the classical hard ellipse system" J. Chem. Phys., 56(10), 4729 (1972).〕〔J. W. Perram and M. S. Wertheim, "Statistical mechanics of hard ellipsoids. I. overlap algorithm and the contact function", J. Comput. Phys., 58, 409 (1985).〕 analytic solutions for the distance of closest approach and the location of the point of contact have only recently become available.〔X. Zheng and P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipses in two dimensions", (electronic Liquid Crystal Communications ), 2007〕〔X. Zheng and P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipses in two dimensions", Phys. Rev. E, 75, 061709 (2007).〕 The details of the calculations are provided in Ref.〔X. Zheng and P. Palffy-Muhoray, (Complete version containing contact point algorithm, May 4, 2009. )〕 The Fortran 90 subroutine is provided in Ref.〔( Fortran90 subroutine for contact distance and contact point for 2D ellipses ) by X. Zheng and P. Palffy-Muhoray, May 2009.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Distance of closest approach of ellipses and ellipsoids」の詳細全文を読む スポンサード リンク
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