Local density fluctuations, hyperuniformity, and order metrics

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 68, Issue 4 1, 2003, Pages 411131-4111325

  Torquato, Salvatore ^a   Stillinger, Frank H ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FOURIER TRANSFORMS; INTEGRAL EQUATIONS; INVERSE PROBLEMS; MATHEMATICAL MODELS; POTENTIAL ENERGY; QUASICRYSTALS; VOLUME FRACTION;

GRANULAR PACKING; NONLINEAR TRANSFORMATIONS;

CRYSTAL LATTICES;

EID: 0346304806     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (61)

1. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, New York, 1986).
2. P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, NJ, 1993).
3. L. Pietronero, A. Gabrielli, and F.S. Labini, Physica A 306, 395 (2002).
4. S. Warr and J.-P. Hansen, Europhys. Lett. 36, 589 (1996).
5. A. Wax, C. Yang, V. Backman, K. Badizadegan, C.W. Boone, R.R. Dasari, and M.S. Field, Biophys. J. 82, 2256 (2002).
6. D.J. Vezzetti, J. Math. Phys. 16, 31 (1975).
7. R.M. Ziff, J. Math. Phys. 18, 1825 (1977).
8. J.L. Lebowitz, Phys. Rev. A 27, 1491 (1983).
9. P.M. Bleher, F.J. Dyson, and J.L. Lebowitz, Phys. Rev. Lett. 71, 3047 (1993).
10. T.M. Truskett, S. Torquato, and P.G. Debenedetti, Phys. Rev. E 58, 7369 (1998).
11. J. Beck, Acta Math. 159, 1 (1987).
12. In very special circumstances, the variance can grow more slowly than the surface area. For example, for the infinite periodic square lattice and when Ω is a rectangle with a very irrational orientation with respect to the x axis, the variance grows as ln R: see J. Beck, Discrete Math. 229, 29 (2001).
13. D.G. Kendall, Quarterly J, Math. 19, 1 (1948).
14. D.G. Kendall and R.A. Rankin, Quarterly J. Math. 4, 178 (1953).
15. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York, 2002).
16. L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, New York, 1980).
17. Ph.A. Martin and T. Yalcin, J. Stat. Phys. 22, 435 (1980).
18. B. Lu and S. Torquato, J. Chem. Phys. 93, 3452 (1990).
19. K. Huang, Statistical Mechanics (Wiley, New York, 1987).
20. S. Torquato and F.H. Stillinger, J. Phys. Chem. B 106, 8354 (2002); 106, 11406 (2002). It is interesting to note that for a related two-point correlation function that arises in the characterization of two-phase random media (i.e., binary stochastic spatial processes), it is known that the two analogous nonnegativity conditions are necessary but not sufficient for realizability [15]; see also S. Torquato, J. Chem. Phys. 111, 8832 (1999).
21. S. Torquato and F.H. Stillinger, J. Phys. Chem. B 106, 8354 (2002); 106, 11406 (2002). It is interesting to note that for a related two-point correlation function that arises in the characterization of two-phase random media (i.e., binary stochastic spatial processes), it is known that the two analogous nonnegativity conditions are necessary but not sufficient for realizability [15]; see also S. Torquato, J. Chem. Phys. 111, 8832 (1999).
22. S. Torquato and F.H. Stillinger, J. Phys. Chem. B 106, 8354 (2002); 106, 11406 (2002). It is interesting to note that for a related two-point correlation function that arises in the characterization of two-phase random media (i.e., binary stochastic spatial processes), it is known that the two analogous nonnegativity conditions are necessary but not sufficient for realizability [15]; see also S. Torquato, J. Chem. Phys. 111, 8832 (1999).
23. N. A. C. Cressie, Statistics for Spatial Data (Wiley, New York, 1993).
24. I. N. Sneddon, Fourier Transforms (Dover Publications, New York, 1995).
25. Of course, there may be situations in which A = 0, but B and/or higher-order moments of h are unbounded. In such instances, the Fourier transform of h will be a nonanalytic function of k at k=0.
26. M. Aizenman, S. Goldstein, and J.L. Lebowitz, J. Stat. Phys. 103, 601 (2001) have proved that any translation invariant measure in one dimension with bounded variance must be a superposition of periodic measures.
27. The critical exponent η associated with h(r) for thermal systems belonging to the standard Ising universality class is given exactly by 1/4 for d=2 and approximately by 0.05 for d = 3; see Ref. [19].
28. M.D. Rintoul and S. Torquato, J. Colloid Interface Sci. 186, 467 (1997); C.L.Y. Yeong and S. Torquato, Phys. Rev. E 57, 495 (1998); S. Hyun and S. Torquato, J. Mater. Res. 16, 280 (2001). These papers are concerned with "target" optimization problems in a variety of different contexts.
29. M.D. Rintoul and S. Torquato, J. Colloid Interface Sci. 186, 467 (1997); C.L.Y. Yeong and S. Torquato, Phys. Rev. E 57, 495 (1998); S. Hyun and S. Torquato, J. Mater. Res. 16, 280 (2001). These papers are concerned with "target" optimization problems in a variety of different contexts.
30. M.D. Rintoul and S. Torquato, J. Colloid Interface Sci. 186, 467 (1997); C.L.Y. Yeong and S. Torquato, Phys. Rev. E 57, 495 (1998); S. Hyun and S. Torquato, J. Mater. Res. 16, 280 (2001). These papers are concerned with "target" optimization problems in a variety of different contexts.
31. F.H. Stillinger, P.G. Debenedetti, and S. Sastry, J. Chem. Phys. 109, 3983 (1998).
32. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover Publications, New York, 1972).
33. Indeed, this is true for other periodic lattices and for statistically homogeneous point patterns in two and three dimensions. The fact that the asymptotic surface-area coefficient λ̄ can be estimated from the average of λ(R) over a small range of R near R = 0 can be exploited to speed up significantly optimization algorithms to find extremal or targeted values of A.
34. R.A. Rankin, Proc. Glasgow Math. Assoc. 1, 149 (1953).
35. E.A. Jagla, J. Chem. Phys. 110, 451 (1998); Phys. Rev. E 58, 1478 (1999).
36. E.A. Jagla, J. Chem. Phys. 110, 451 (1998); Phys. Rev. E 58, 1478 (1999).
37. A proof that the extremal lattice for d=3 is the bcc lattice currently does not exist. In this connection, it should be noted that the surface-area coefficient for the bcc lattice is related to the Epstein zeta function for the fcc lattice, which was shown [V. Ennola, Proc. Cambridge Philos. Soc. 60, 855 (1964)] to provide a local minimum value among all lattices.
38. Of course, the extremal point pattern depends on the shape of the window.
39. S. Goldstein, E. Speer, and J. L. Lebowitz (unpublished).
40. A. Gabrielli, B. Jancovici, M. Joyce, J.L. Lebowitz, L. Pietronero, and F.S. Labini, Phys. Rev. D 67, 043506 (2003).
41. C. Radin, Ann. Math. 139, 661 (1994).
42. D. Levesque, J-J. Weis, and J.L. Lebowitz, J. Stat. Phys. 100, 209 (2000).
43. B. Jancovici, Phys. Rev. Lett. 46, 386 (1980).
44. E.R. Harrison, Phys. Rev. D 1, 2726 (1970); Ya.B. Zeldovich, Mon. Not. R. Astron. Soc. 160, 1 (1972).
45. E.R. Harrison, Phys. Rev. D 1, 2726 (1970); Ya.B. Zeldovich, Mon. Not. R. Astron. Soc. 160, 1 (1972).
46. F.H. Stillinger, S. Torquato, J.M. Eroles, and T.M. Truskett, J. Phys. Chem. B 105, 6592 (2000).
47. H. Sakai, F.H. Stillinger, and S. Torquato, J. Chem. Phys. 117, 297 (2002).
48. S. Torquato, T.M. Truskett, and P.G. Debenedetti, Phys. Rev. Lett. 84, 2064 (2000).
49. A.R. Kansal, S. Torquato, and F.H. Stillinger, Phys. Rev. E 66, 041109 (2002).
50. S. Torquato and F.H. Stillinger, J. Phys. Chem. B 105, 11849 (2001); S. Torquato, A. Donev, and F. H. Stillinger, Int. J. Solids Struct. 40, 7143 (2003).
51. S. Torquato and F.H. Stillinger, J. Phys. Chem. B 105, 11849 (2001); S. Torquato, A. Donev, and F. H. Stillinger, Int. J. Solids Struct. 40, 7143 (2003).
52. J. Crawford, S. Torquato, and F. H. Stillinger, J. Chem. Phys. 119, 7065 (2003).
53. (d-1)/d among all infinite hyperuniform point patterns in two and three dimensions, respectively, for spherical windows.
54. A.R. Kansal, S. Torquato, and F.H. Stillinger, J. Chem. Phys. 117, 8212 (2002).
55. D. Levine and P.J. Steinhardt, Phys. Rev. B 34, 596 (1986).
56. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973); B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41, 121 (1978); A.P. Young, Phys. Rev. B 19, 1855 (1979).
57. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973); B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41, 121 (1978); A.P. Young, Phys. Rev. B 19, 1855 (1979).
58. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973); B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41, 121 (1978); A.P. Young, Phys. Rev. B 19, 1855 (1979).
59. F.H. Stillinger and R. Lovett, J. Chem. Phys. 48, 3858 (1968).
60. Y. Fan, J.K. Percus, D.K. Stillinger, and F.H. Stillinger, Phys. Rev. A 44, 2394 (1991).
61. F. Zernike and J.A. Prins, Z. Phys. 41, 184 (1927).