Advertisement

Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors

Chapter
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 2177)

Abstract

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic processes and models. In the context of applied image analysis of structured synthetic and biological materials, this question is central to the problem of inferring information about the formation process from spatial measurements of the resulting random structure. This chapter addresses this question by a theory-based simulation study of cell shape indices derived from tensor-valued intrinsic volumes, or Minkowski tensors, for a variety of common tessellation models. We focus on the relationship between two indices: (1) the dimensionless ratio 〈V 〉2∕〈A〉3 of empirical average cell volumes to areas, and (2) the degree of cell elongation quantified by the eigenvalue ratio 〈β10,2〉 of the interface Minkowski tensors W10,2. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and cell volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, Gibbs hard-core processes of spheres, and random sequential absorption processes as well as for Laguerre tessellations of configurations of polydisperse spheres, STIT-tessellations, and Poisson hyperplane tessellations. These data are complemented by experimental 3D image data of mechanically stable ellipsoid configurations, area-minimising liquid foam models, and mechanically stable crystalline sphere configurations. We find that, not surprisingly, the indices 〈V 〉2∕〈A〉3 and 〈β10,2〉 are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of these shape indices between many of the tessellation models listed above. Therefore, given a realization of a tessellation (e.g., an experimental image), these shape indices are able to narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by considering density-resolved volume-shape correlations.

Keywords

Point Process Voronoi Diagram Shape Index Voronoi Cell Packing Fraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We thank Andy Kraynik, Markus Spanner, and Richard Schielein for their data of the monodisperse foam, the equilibrium hard-sphere liquids, and the crystalline sphere packings, respectively. We also thank Felix Ballani for his software simulating the typical cell of a Poisson hyperplane tessellation. We thank Markus Kiderlen for valuable discussions and suggestions. We also thank the German science foundation (DFG) for the grants SCHR1148/3, HU1874/3-2, LA965/6-2, and ME1361/11 awarded as part of the DFG-Forschergruppe “Geometry and Physics of Spatial Random Systems”.

References

  1. 1.
    B.J. Alder, T.E. Wainwright, Phase transition for a hard sphere system. J. Chem. Phys. 27, 1208–1209 (1957)CrossRefGoogle Scholar
  2. 2.
    S. Alesker, Continuous rotation invariant valuations on convex sets. Ann. Math. 149(3), 977 (1999)Google Scholar
  3. 3.
    S. Alesker, Description of continuous isometry covariant valuations on convex sets. Geom. Dedicata. 74(3), 241 (1999)Google Scholar
  4. 4.
    T. Aste, T. Di Matteo, Emergence of Gamma distributions in granular materials and packing models. Phys. Rev. E 77, 021309 (2008)CrossRefGoogle Scholar
  5. 5.
    T. Aste, T. Di Matteo, M. Saadatfar, T.J. Senden, M. Schröter, H.L. Swinney, An invariant distribution in static granular media. Europhys. Lett. 79, 24003 (2007)CrossRefGoogle Scholar
  6. 6.
    F. Baccelli, B. Blaszczyszyn, Stochastic geometry and wireless networks I: theory. Found. Trends Netw. 3, 249–449 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    F. Baccelli, B. Blaszczyszyn, Stochastic geometry and wireless networks II: applications. Found. Trends Netw. 4, 1–312 (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    A. Baddeley, R. Turner, spatstat: an R package for analyzing spatial point patterns. J. Stat. Softw. 12(6), 1 (2005)Google Scholar
  9. 9.
    M. Barbosa, R. Natoli, K. Valter, J. Provis, T. Maddess, Integral-geometry characterization of photobiomodulation effects on retinal vessel morphology. Biomed. Opt. Express 5(7), 2317 (2014)Google Scholar
  10. 10.
    V. Baumstark, G. Last, Some distributional results for Poisson Voronoi tessellations. Adv. Appl. Prob. 39, 16–40 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    V. Baumstark, G. Last, Gamma distributions for stationary poisson flat processes. Adv. Appl. Prob. 41, 911–939 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J. Becker, G. Grun, R. Seemann, H. Mantz, K. Jacobs, K.R. Mecke, R. Blossey, Complex dewetting scenarios captured by thin-film models. Nat. Mater. 2, 59 (2003)CrossRefGoogle Scholar
  13. 13.
    C. Beisbart, M.S. Barbosa, H. Wagner, L.d.F. Costa, Extended morphometric analysis of neuronal cells with Minkowski valuations. Eur. Phys. J. B 52(4), 531 (2006)Google Scholar
  14. 14.
    R. Berman, Determinantal point processes and Fermions on complex manifolds: large deviations and bosonization. Commun. Math. Phys. 327(1), 1 (2014)Google Scholar
  15. 15.
    K.A. Brakke, The surface evolver. Exp. Math. 1(2), 141 (1992)Google Scholar
  16. 16.
    S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications (Wiley, New York, 2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    S. Colombi, D. Pogosyan, T. Souradeep, Tree structure of a percolating Universe. Phys. Rev. Lett. 85, 5515 (2000)CrossRefGoogle Scholar
  18. 18.
    J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups, in Grundlehren der mathematischen Wissenschaften, vol. 290 (Springer, New York, 1999)Google Scholar
  19. 19.
    R. Cowan, New classes of random tessellations arising from iterative division of cells. Adv. Appl. Prob. 42, 26–47 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes. Probability and Its Applications (Springer, New York, 2003)zbMATHGoogle Scholar
  21. 21.
    N. Deng, W. Zhou, M. Haenggi, The Ginibre point process as a model for wireless networks with repulsion. IEEE Tran. Wirel. Commun. 14(1), 107 (2015)Google Scholar
  22. 22.
    M.R. Dennis, Nodal densities of planar Gaussian random waves. Eur. Phys. J.-Spec. Top. 145(1), 191 (2007)Google Scholar
  23. 23.
    A. Ducout, F.R. Bouchet, S. Colombi, D. Pogosyan, S. Prunet, Non-Gaussianity and Minkowski functionals: forecasts for Planck. Mon. Not. R. Astron. Soc. 429(3), 2104 (2013)Google Scholar
  24. 24.
    R. Erban, S.J. Chapman, Time scale of random sequential adsorption. Phys. Rev. E 75(4), 041116 (2007)Google Scholar
  25. 25.
    M.E. Evans, A.M. Kraynik, D.A. Reinelt, K. Mecke, G.E. Schröder-Turk, Network-like propagation of cell-level stress in sheared random foams. Phys. Rev. Lett. 111, 138301 (2013)CrossRefGoogle Scholar
  26. 26.
    E.D. Feigelson, G.J. Babu (eds.), Statistical Challenges in Modern Astronomy (Springer, New York, 1992)Google Scholar
  27. 27.
    C. Gay, C. Pichon, D. Pogosyan, Non-Gaussian statistics of critical sets in 2D and 3D: Peaks, voids, saddles, genus, and skeleton. Phys. Rev. D 85, 023011 (2012)CrossRefGoogle Scholar
  28. 28.
    D. Göring, M.A. Klatt, C. Stegmann, K. Mecke, Morphometric analysis in gamma-ray astronomy using Minkowski functionals. Astron. Astrophys. 555, A38 (2013)CrossRefGoogle Scholar
  29. 29.
    H. Hadwiger, Beweis eines Funktionalsatzes für konvexe Körper. Abh. Math. Sem. Univ. Hamburg 17(1), 69 (1951)Google Scholar
  30. 30.
    T.C. Hales, A proof of the Kepler conjecture. Ann. Math. 162, 1065 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    H. Hansen-Goos, K. Mecke, Fundamental measure theory for inhomogeneous fluids of nonspherical hard particles. Phys. Rev. Lett. 102, 018302 (2009)CrossRefGoogle Scholar
  32. 32.
    S. Hilgenfeldt, A.M. Kraynik, S.A. Koehler, H.A. Stone, An accurate von Neumann’s law for three-dimensional foams. Phys. Rev. Lett. 86(12), 2685–2688 (2001)CrossRefGoogle Scholar
  33. 33.
    J. Hörrmann, D. Hug, M.A. Klatt, K. Mecke, Minkowski tensor density formulas for Boolean models. Adv. Appl. Math. 55(0), 48 (2014)Google Scholar
  34. 34.
    J.B. Hough, M. Krishnapur, Y. Peres, B. Virág, Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    M.L. Huber, R.L. Wolpert, Likelihood-based inference for Matérn type-III repulsive point processes. Adv. Appl. Probab. 41, 958–977 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    D. Hug, R. Schneider, Typical cells in Poisson hyperplane tessellations. Discrete Comput. Geom. 38(2), 305–319 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    D. Hug, M. Reitzner, R. Schneider, Large Poisson-Voronoi cells and Crofton cells. Adv. Appl. Probab. 36(3), 667–690 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    J. Illian, A. Penttinen, H. Stoyan, D. Stoyan, Statistical Analysis and Modelling of Spatial Point Patterns: Illian/Statistical Analysis and Modelling of Spatial Point Patterns (Wiley, Chichester, 2007)Google Scholar
  39. 39.
    M. Kac, Can one hear the shape of a drum? Am. Math. Mon. 73(4), 1–23 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    S.C. Kapfer, Morphometry and physics of particulate and porous media, Ph.D. thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2011Google Scholar
  41. 41.
    S.C. Kapfer, W. Mickel, F.M. Schaller, M. Spanner, C. Goll, T. Nogawa, N. Ito, K. Mecke, G.E. Schröder-Turk, Local anisotropy of fluids using Minkowski tensors. J. Stat. Mech. Theor. Exp. 2010(11), P11010 (2010)Google Scholar
  42. 42.
    S.C. Kapfer, W. Mickel, K. Mecke, G.E. Schröder-Turk, Jammed spheres: Minkowski tensors reveal onset of local crystallinity. Phys. Rev. E 85, 030301 (2012)CrossRefGoogle Scholar
  43. 43.
    M. Kerscher, K. Mecke, J. Schmalzing, C. Beisbart, T. Buchert, H. Wagner, Morphological fluctuations of large-scale structure: the PSCz survey. Astron. Astrophys. 373, 1 (2001)CrossRefGoogle Scholar
  44. 44.
    M. Kiderlen, M. Hörig, Matérn’s hard core models of types I and II. CSGB (2013, preprint)Google Scholar
  45. 45.
    M.A. Klatt, Morphometry of random spatial structures in physics, Ph.D. thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2016Google Scholar
  46. 46.
    M.A. Klatt, S. Torquato, Characterization of maximally random jammed sphere packings: Voronoi correlation functions. Phys. Rev. E 90, 052120 (2014)CrossRefGoogle Scholar
  47. 47.
    A.M. Kraynik, Foam structure: from soap froth to solid foams. MRS Bull. 28(04), 275–278 (2003)CrossRefGoogle Scholar
  48. 48.
    A.M. Kraynik, The structure of random foam. Adv. Eng. Mater. 8(9), 900 (2006)Google Scholar
  49. 49.
    A.M. Kraynik, D.A. Reinelt, F. van Swol, Structure of random monodisperse foam. Phys. Rev. E 67, 031403 (2003)CrossRefGoogle Scholar
  50. 50.
    A.M. Kraynik, D.A. Reinelt, F. van Swol, Structure of Random Foam. Phys. Rev. Lett. 93, 208301 (2004)CrossRefGoogle Scholar
  51. 51.
    G. Last, Stationary partitions and Palm probabilities. Adv. Appl. Prob. 38, 602–620 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    G. Last, M.D. Penrose, Lectures on the Poisson Process (Cambridge University Press, Cambridge, to appear). http://www.math.kit.edu/stoch/~last/seite/lehrbuch_poissonp/de
  53. 53.
    G. Last, G. Peccati, M. Schulte, Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Prob. Theory Related Fields. 165, 667–723 (2016)CrossRefzbMATHGoogle Scholar
  54. 54.
    C. Lautensack, Random Laguerre Tessellations, Ph.D. thesis, Universität Karlsruhe, 2007Google Scholar
  55. 55.
    F. Lavancier, J. Møller, E. Rubak, Determinantal point process models and statistical inference. J. R. Stat. Soc. B 77(4), 853–877 (2015)MathSciNetCrossRefGoogle Scholar
  56. 56.
    E.A. Lazar, J.K. Mason, R.D. MacPherson, D.J. Srolovitz, Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks. Phys. Rev. E 88, 063309 (2013)CrossRefGoogle Scholar
  57. 57.
    B.D. Lubachevsky, F.H. Stillinger, Geometric properties of random disk packings. J. Stat. Phys. 60(5–6), 561 (1990)Google Scholar
  58. 58.
    O. Macchi, The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7(1), 83–122 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    V.N. Manoharan, Colloidal matter: packing, geometry, and entropy. Science 349(6251), 1253751 (2015)Google Scholar
  60. 60.
    H. Mantz, K. Jacobs, K. Mecke, Utilising Minkowski functionals for image analysis. J. Stat. Mech. 12, P12015 (2008)CrossRefGoogle Scholar
  61. 61.
    S. Mase, J. Møller, D. Stoyan, R.P. Waagepetersen, G. Döge, Packing densities and simulated tempering for hard core Gibbs point processes. Ann. Inst. Statist. Math. 53, 661–680 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    P. McCullagh, J. Møller, The permanental process. Adv. Appl. Prob. 38, 873–888 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    P. McMullen, Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo (2) Suppl. 50, 259 (1997)Google Scholar
  64. 64.
    K. Mecke, Morphological characterization of patterns in reaction-diffusion systems. Phys. Rev. E 53(5), 4794 (1996)Google Scholar
  65. 65.
    K. Mecke, Integral geometry and statistical physics. Int. J. Mod. Phys. B 12, 861 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    K. Mecke, D. Stoyan, Statistical Physics and Spatial Statistics – The Art of Analyzing and Modeling Spatial Structures and Pattern Formation. Lecture Notes in Physics, vol. 554, 1st edn. (Springer, Berlin, 2000)Google Scholar
  67. 67.
    J. Mecke, W. Nagel, V. Weiss, Some distributions for I-segments of planar random homogeneous STIT tessellations. Math. Nachr. 284, 1483–1495 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    W. Mickel, G.E. Schröder-Turk, K. Mecke, Tensorial Minkowski functionals of triply periodic minimal surfaces. Interface Focus 2(5), 623–633 (2012)CrossRefGoogle Scholar
  69. 69.
    W. Mickel, S.C. Kapfer, G.E. Schröder-Turk, K. Mecke, Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter. J. Chem. Phys. 138(4), 044501 (2013)Google Scholar
  70. 70.
    R. Miles, Poisson flats in Euclidean spaces. Part ii: Homogeneous poisson flats and complementary theorem. Adv. Appl. Prob. 3, 1–43 (1971)zbMATHGoogle Scholar
  71. 71.
    J. Møller, Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics, vol. 87 (Springer, New York, 1994)Google Scholar
  72. 72.
    J. Møller, R. Plenge Waagepetersen, Statistical Inference and Simulation for Spatial Point Processes. C& H/CRC Monographs on Statistics & Applied Probability, vol. 100 (Chapman and Hall/CRC, Boca Raton, 2003)Google Scholar
  73. 73.
    J. Møller, D. Stoyan, Stochastic geometry and random tessellations. Tech. rep., Department of Mathematical Sciences, Aalborg University, 2007Google Scholar
  74. 74.
    J. Møller, S. Zuyev, Gamma-type results and other related properties of poisson processes. Adv. Appl. Prob. 28, 662–673 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    J. Møller, M.L. Huber, R.L. Wolpert, Perfect simulation and moment properties for the Matérn type III process. Stoch. Proc. Appl. 120(11), 2142–2158 (2010)CrossRefzbMATHGoogle Scholar
  76. 76.
    W. Nagel, V. Weiss, Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Probab. 37(4), 859 (2005)Google Scholar
  77. 77.
    W. Nagel, V. Weiss, Mean values for homogeneous STIT tessellations in 3D. Image Anal. Stereol. 27, 29–37 (2008)CrossRefzbMATHGoogle Scholar
  78. 78.
    X.X. Nguyen, H. Zessin, Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105–115 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    A. Okabe, B. Boots, K. Sugihara, S.N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley Series in Probability and Statistics (Wiley, Chichester, 2009)Google Scholar
  80. 80.
    E. Pineda, P. Bruna, D. Crespo, Cell size distribution in random tessellations of space. Phys. Rev. E 70, 066119 (2004)CrossRefGoogle Scholar
  81. 81.
    C. Preston, Random Fields. Lecture Notes in Mathematics, vol. 534 (Springer, Berlin, 1976)Google Scholar
  82. 82.
    C. Redenbach, Microstructure models for cellular materials. Comput. Mater. Sci. 44(4), 1397–1407 (2009)CrossRefGoogle Scholar
  83. 83.
    D. Ruelle, Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    F.M. Schaller, S.C. Kapfer, M.E. Evans, M.J. Hoffmann, T. Aste, M. Saadatfar, K. Mecke, G.W. Delaney, G.E. Schröder-Turk, Set Voronoi diagrams of 3d assemblies of aspherical particles. Philos. Mag. 93, 3993–4017 (2013)CrossRefGoogle Scholar
  85. 85.
    F.M. Schaller, M. Neudecker, M. Saadatfar, G. Delaney, K. Mecke, G.E. Schröder-Turk, M. Schröter, Tomographic analysis of jammed ellipsoid packings. AIP Conf. Proc. 1542, 377–380 (2013)CrossRefGoogle Scholar
  86. 86.
    F.M. Schaller, S.C. Kapfer, J.E. Hilton, P.W. Cleary, K. Mecke, C. De Michele, T. Schilling, M. Saadatfar, M. Schröter, G.W. Delaney, G.E. Schröder-Turk, Non-universal Voronoi cell shapes in amorphous ellipsoid packs. Europhys. Lett. 111, 24002 (2015)CrossRefGoogle Scholar
  87. 87.
    F.M. Schaller, M. Neudecker, M. Saadatfar, G.W. Delaney, G.E. Schröder-Turk, M. Schröter, Local origin of global contact numbers in frictional ellipsoid packings. Phys. Rev. Lett. 114, 158001 (2015)CrossRefGoogle Scholar
  88. 88.
    F.M. Schaller, R.F.B. Weigel, S.C. Kapfer, Densest local structures of uniaxial ellipsoids. Phys. Rev. X 6, 041032 (2016)Google Scholar
  89. 89.
    J. Schmalzing, T. Buchert, A.L. Melott, V. Sahni, B.S. Sathyaprakash, S.F. Shandarin, Disentangling the cosmic web. I. Morphology of isodensity contours. Astrophys. J. 526(2), 568 (1999)Google Scholar
  90. 90.
    R. Schneider, W. Weil, Stochastic and Integral Geometry (Probability and Its Applications) (Springer, Berlin, 2008)CrossRefzbMATHGoogle Scholar
  91. 91.
    C. Scholz, G.E. Schröder-Turk, K. Mecke, Pattern-fluid interpretation of chemical turbulence. Phys. Rev. E 91, 042907 (2015)CrossRefGoogle Scholar
  92. 92.
    C. Scholz, F. Wirner, M.A. Klatt, D. Hirneise, G.E. Schröder-Turk, K. Mecke, C. Bechinger, Direct relations between morphology and transport in boolean models. Phys. Rev. E 92, 043023 (2015)CrossRefGoogle Scholar
  93. 93.
    T. Schreiber, C. Thäle, Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations. Stoch. Process. Appl. 121, 989–1012 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  94. 94.
    T. Schreiber, C. Thäle, Second-order theory for iteration stable tessellations. Probab. Math. Stat. 32, 281–300 (2012)MathSciNetzbMATHGoogle Scholar
  95. 95.
    G.E. Schröder-Turk, S. Kapfer, B. Breidenbach, C. Beisbart, K. Mecke, Tensorial Minkowski functionals and anisotropy measures for planar patterns. J. Microsc. 238(1), 57 (2010)Google Scholar
  96. 96.
    G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, M.A. Klatt, F.M. Schaller, M.J.F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, K. Mecke, Minkowski tensor shape analysis of cellular, granular and porous structures. Adv. Mater. 23(22–23), 2535 (2011)Google Scholar
  97. 97.
    G.E. Schröder-Turk, R. Schielein, S.C. Kapfer, F.M. Schaller, G.W. Delaney, T. Senden, M. Saadatfar, T. Aste, K. Mecke, Minkowski tensors and local structure metrics: amorphous and crystalline sphere packings. AIP Conf. Proc. 1542(1), 349 (2013)Google Scholar
  98. 98.
    G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, F.M. Schaller, B. Breidenbach, D. Hug, K. Mecke, Minkowski tensors of anisotropic spatial structure. New J. Phys. 15(8), 083028 (2013)Google Scholar
  99. 99.
    B. Schuetrumpf, M.A. Klatt, K. Iida, J. Maruhn, K. Mecke, P.G. Reinhard, Time-dependent Hartree-Fock approach to nuclear “pasta” at finite temperature. Phys. Rev. C 87, 055805 (2013)CrossRefGoogle Scholar
  100. 100.
    B. Schuetrumpf, M.A. Klatt, K. Iida, G.E. Schröder-Turk, J.A. Maruhn, K. Mecke, P.G. Reinhard, Appearance of the single gyroid network phase in “nuclear pasta” matter. Phys. Rev. C 91, 025801 (2015)CrossRefGoogle Scholar
  101. 101.
    V. Senthil Kumar, V. Kumaran, Voronoi cell volume distribution and configurational entropy of hard-spheres. J. Chem. Phys. 123, 114501 (2005)CrossRefGoogle Scholar
  102. 102.
    T. Shirai, Y. Takahashi, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205(2), 414–463 (2003)Google Scholar
  103. 103.
    P.J. Steinhardt, D.R. Nelson, M. Ronchetti, Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784 (1983)CrossRefGoogle Scholar
  104. 104.
    J. Stienen, Die Vergröberung von Karbiden in reinen Eisen-Kohlenstoff-Staehlen, Ph.D. thesis, RWTH Aachen, 1982Google Scholar
  105. 105.
    D. Stoyan, M. Schlather, Random sequential adsorption: relationship to dead leaves and characterization of variability. J. Stat. Phys. 100, 969–979 (2000)CrossRefzbMATHGoogle Scholar
  106. 106.
    J. Talbot, G. Tarjus, P.R. Van Tassel, P. Viot, From car parking to protein adsorption: an overview of sequential adsorption processes. Colloids Surf. A 165(1), 287–324 (2000)CrossRefGoogle Scholar
  107. 107.
    J. Teichmann, F. Ballani, K.G. van den Boogaart, Generalizations of Matern’s hard-core point processes. Spat. Stat. 9, 33–53 (2013)CrossRefGoogle Scholar
  108. 108.
    S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Interdisciplinary Applied Mathematics. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  109. 109.
    S. Torquato, F.H. Stillinger, Exactly solvable disordered sphere-packing model in arbitrary-dimensional Euclidean spaces. Phys. Rev. E 73, 031106 (2006)MathSciNetCrossRefGoogle Scholar
  110. 110.
    R. Wittmann, M. Marechal, K. Mecke, Fundamental measure theory for smectic phases: scaling behavior and higher order terms. J. Chem. Phys. 141(6), 064103 (2014)Google Scholar
  111. 111.
    W.W. Wood, J.D. Jacobson, Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres. J. Chem. Phys. 27, 1207–1208 (1957)CrossRefGoogle Scholar
  112. 112.
    G. Zhang, S. Torquato, Precise algorithm to generate random sequential addition of hard hyperspheres at saturation. Phys. Rev. E 88, 053312 (2013)CrossRefGoogle Scholar
  113. 113.
    C. Zong, Sphere Packings. Universitext (Springer, New York, 1999)zbMATHGoogle Scholar
  114. 114.
    S. Zuyev, Stopping sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Institut für Theoretische PhysikUniversität Erlangen-NürnbergErlangenGermany
  3. 3.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany
  4. 4.School of Engineering and Information TechnologyMurdoch UniversityMurdochAustralia

Personalised recommendations

Cite chapter

Buy options