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

Volumn 82, Issue 5, 2010, Pages

Reformulation of the covering and quantizer problems as ground states of interacting particles

(1) Torquato, S a,b

a PRINCETON UNIVERSITY (United States)

b Princeton University (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIGITAL COMMUNICATIONS; ENERGY MINIMIZATION; ENERGY MINIMIZATION PROBLEM; EUCLIDEAN SPACES; GRAVITATIONAL WAVES; HIGH DIMENSIONAL DATA; INFINITE NUMBERS; INTERACTING PARTICLES; LARGE DIMENSIONS; NEAREST-NEIGHBORS; NUMERICAL OPTIMIZATION TECHNIQUES; OPTIMIZATION PROBLEMS; PAIR POTENTIAL; PARAMETER SPACES; POINT PARTICLE; QUANTIZERS; RADIATION THERAPY; RANDOM MEDIA; SPHERE PACKINGS; SPHERE-PACKING PROBLEM; THREE SPACE DIMENSIONS; UPPER BOUND; WIRELESS COMMUNICATION NETWORK;

DATA COMPRESSION; DIGITAL COMMUNICATION SYSTEMS; GROUND STATE; NUMBER THEORY; NUMERICAL ANALYSIS; OPTIMIZATION; PACKING; STATISTICAL MECHANICS; WIRELESS TELECOMMUNICATION SYSTEMS;

SPHERES;

EID: 78651365744 PISSN: 15393755 EISSN: 15502376 Source Type: Journal
DOI: 10.1103/PhysRevE.82.056109 Document Type: Article

Times cited : (65)

References (63)

1
- 0003644127
- Academic Press, New York
- J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, New York, 1986).
- (1986) Theory of Simple Liquids
- Hansen, J.P.¹ McDonald, I.R.²

2
- 0003581254
- Springer-Verlag, New York
- S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York, 2002).
- (2002) Random Heterogeneous Materials: Microstructure and Macroscopic Properties
- Torquato, S.¹

3
- 0036640590
- 10.1063/1.1480863
- F. H. Stillinger, H. Sakai, and S. Torquato, J. Chem. Phys. 117, 288 (2002). 10.1063/1.1480863
- (2002) J. Chem. Phys. , vol.117 , pp. 288
- Stillinger, F.H.¹ Sakai, H.² Torquato, S.³

4
- 0003707206
- World Scientific, Riveredge, NJ
- D. Ruelle, Statistical Mechanics: Rigorous Results (World Scientific, Riveredge, NJ, 1999).
- (1999) Statistical Mechanics: Rigorous Results
- Ruelle, D.¹

5
- 38349171557
- 10.1103/PhysRevLett.100.020602
- S. Torquato and F. H. Stillinger, Phys. Rev. Lett. 100, 020602 (2008). 10.1103/PhysRevLett.100.020602
- (2008) Phys. Rev. Lett. , vol.100 , pp. 020602
- Torquato, S.¹ Stillinger, F.H.²

6
- 0003517281
- Cambridge University Press, New York
- P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, New York, 1995).
- (1995) Principles of Condensed Matter Physics
- Chaikin, P.M.¹ Lubensky, T.C.²

7
- 0033717788
- 10.1088/0953-8984/12/24/302
- A. Lang, C. N. Likos, M. Watzlawek, and H. Lowën, J. Phys.: Condens. Matter 12, 5087 (2000). 10.1088/0953-8984/12/24/302
- (2000) J. Phys.: Condens. Matter , vol.12 , pp. 5087
- Lang, A.¹ Likos, C.N.² Watzlawek, M.³ Lowën, H.⁴

8
- 0035883479
- 10.1063/1.1394922
- F. H. Stillinger, J. Chem. Phys. 115, 5208 (2001). 10.1063/1.1394922
- (2001) J. Chem. Phys. , vol.115 , pp. 5208
- Stillinger, F.H.¹

9
- 33144486158
- 10.1103/PhysRevLett.96.045701
- B. M. Mladek, D. Gottwald, G. Kahl, M. Neumann, and C. N. Likos, Phys. Rev. Lett. 96, 045701 (2006). 10.1103/PhysRevLett.96.045701
- (2006) Phys. Rev. Lett. , vol.96 , pp. 045701
- Mladek, B.M.¹ Gottwald, D.² Kahl, G.³ Neumann, M.⁴ Likos, C.N.⁵

10
- 34250831282
- 10.1090/S0894-0347-06-00546-7
- H. Cohn and A. Kumar, J. Am. Math. Soc. 20, 99 (2007). 10.1090/S0894-0347-06-00546-7
- (2007) J. Am. Math. Soc. , vol.20 , pp. 99
- Cohn, H.¹ Kumar, A.²

11
- 54749141939
- 10.1103/PhysRevE.78.036706
- S. Gravel and V. Elser, Phys. Rev. E 78, 036706 (2008). 10.1103/PhysRevE.78.036706
- (2008) Phys. Rev. e , vol.78 , pp. 036706
- Gravel, S.¹ Elser, V.²

12
- 79051470505
- 10.1209/0295-5075/78/46004
- M. A. Glaser, G. M. Grason, R. D. Kamien, A. A. Kosmrlj, C. D. Santangelo, and P. Ziherl, EPL 78, 46004 (2007). 10.1209/0295-5075/78/46004
- (2007) EPL , vol.78 , pp. 46004
- Glaser, M.A.¹ Grason, G.M.² Kamien, R.D.³ Kosmrlj, A.A.⁴ Santangelo, C.D.⁵ Ziherl, P.⁶

13
- 41349086026
- 10.1103/PhysRevE.70.046122
- O. U. Uche, F. H. Stillinger, and S. Torquato, Phys. Rev. E 70, 046122 (2004). 10.1103/PhysRevE.70.046122
- (2004) Phys. Rev. e , vol.70 , pp. 046122
- Uche, O.U.¹ Stillinger, F.H.² Torquato, S.³

14
- 49749132375
- 10.1063/1.2961314
- R. D. Batten, F. H. Stillinger, and S. Torquato, J. Appl. Phys. 104, 033504 (2008). 10.1063/1.2961314
- (2008) J. Appl. Phys. , vol.104 , pp. 033504
- Batten, R.D.¹ Stillinger, F.H.² Torquato, S.³

15
- 33644883967
- 10.1103/PhysRevE.73.031106
- S. Torquato and F. H. Stillinger, Phys. Rev. E 73, 031106 (2006). 10.1103/PhysRevE.73.031106
- (2006) Phys. Rev. e , vol.73 , pp. 031106
- Torquato, S.¹ Stillinger, F.H.²

16
- 33750536978
- S. Torquato and F. H. Stillinger, Exp. Math. 15, 307 (2006).
- (2006) Exp. Math. , vol.15 , pp. 307
- Torquato, S.¹ Stillinger, F.H.²

17
- 43049116152
- 10.1063/1.2897027
- A. Scardicchio, F. H. Stillinger, and S. Torquato, J. Math. Phys. 49, 043301 (2008). 10.1063/1.2897027
- (2008) J. Math. Phys. , vol.49 , pp. 043301
- Scardicchio, A.¹ Stillinger, F.H.² Torquato, S.³

18
- 0003643241
- Springer-Verlag, New York
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer-Verlag, New York, 1998).
- (1998) Sphere Packings, Lattices and Groups
- Conway, J.H.¹ Sloane, N.J.A.²

19
- 33744537759
- 10.1007/s00222-005-0488-2
- P. Sarnak and A. Strömbergsson, Invent. Math. 165, 115 (2006). 10.1007/s00222-005-0488-2
- (2006) Invent. Math. , vol.165 , pp. 115
- Sarnak, P.¹ Strömbergsson, A.²

20
- 0037769863
- 10.4007/annals.2003.157.689
- H. Cohn and N. Elkies, Ann. Math. 157, 689 (2003). 10.4007/annals.2003. 157.689
- (2003) Ann. Math. , vol.157 , pp. 689
- Cohn, H.¹ Elkies, N.²

21
- 22044441063
- 10.4007/annals.2005.162.1065
- T. C. Hales, Ann. Math. 162, 1065 (2005). 10.4007/annals.2005.162.1065
- (2005) Ann. Math. , vol.162 , pp. 1065
- Hales, T.C.¹

22
- 71649112588
- 10.4007/annals.2009.170.1003
- H. Cohn and A. Kumar, Ann. Math. 170, 1003 (2009). 10.4007/annals.2009. 170.1003
- (2009) Ann. Math. , vol.170 , pp. 1003
- Cohn, H.¹ Kumar, A.²

23
- 0346304806
- 10.1103/PhysRevE.68.041113
- S. Torquato and F. H. Stillinger, Phys. Rev. E 68, 041113 (2003). 10.1103/PhysRevE.68.041113
- (2003) Phys. Rev. e , vol.68 , pp. 041113
- Torquato, S.¹ Stillinger, F.H.²

24
- 78651357382
- The support of a function is the set of points where the function is not zero. A function has compact support if it is zero outside a compact set.
- The support of a function is the set of points where the function is not zero. A function has compact support if it is zero outside a compact set.

25
- 0036722531
- 10.1080/07408170208928915
- D. Adickes, IIE Trans. 34, 823 (2002). 10.1080/07408170208928915
- (2002) IIE Trans. , vol.34 , pp. 823
- Adickes, D.¹

26
- 34748877313
- 10.1088/0264-9381/24/19/S11
- R. Prix, Class. Quantum Grav. 24, S481 (2007). 10.1088/0264-9381/24/19/ S11
- (2007) Class. Quantum Grav. , vol.24 , pp. 481
- Prix, R.¹

27
- 67650489153
- 10.1007/s11590-008-0095-4
- L. Liberti, N. Maculan, and Y. Zhang, Optim. Lett. 3, 109 (2009). 10.1007/s11590-008-0095-4
- (2009) Optim. Lett. , vol.3 , pp. 109
- Liberti, L.¹ MacUlan, N.² Zhang, Y.³

28
- 4143094282
- McGraw Hill, New Delhi
- R. Bose, Information Theory, Coding and Cryptography (McGraw Hill, New Delhi, 2002).
- (2002) Information Theory, Coding and Cryptography
- Bose, R.¹

29
- 0033317391
- 10.1137/S0036144599352836
- Q. Du, V. Faber, and M. Gunzburger, SIAM Rev. 41, 637 (1999). 10.1137/S0036144599352836
- (1999) SIAM Rev. , vol.41 , pp. 637
- Du, Q.¹ Faber, V.² Gunzburger, M.³

30
- 0003628866
- 10.1063/1.1730361; this paper considered void nearest-neighbor functions for the special case of identical hard spheres in equilibrium Gibbs ensemble.
- H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959) 10.1063/1.1730361
- (1959) J. Chem. Phys. , vol.31 , pp. 369
- Reiss, H.¹ Frisch, H.L.² Lebowitz, J.L.³

31
- 0000983452
- 10.1103/PhysRevA.41.2059. This paper considered two types of nearest-neighbor functions void and particle quantities for a general "nonequilibrium" case of identical of spheres with arbitrary interactions, e.g., spheres with variable interpenetrability that interact with repulsive and attractive forces.
- S. Torquato, B. Lu, and J. Rubinstein, Phys. Rev. A 41, 2059 (1990) 10.1103/PhysRevA.41.2059
- (1990) Phys. Rev. A , vol.41 , pp. 2059
- Torquato, S.¹ Lu, B.² Rubinstein, J.³

32
- 78651360785
- F [i.e., without the factor of (2π) d], in which case self-duality is defined with respect to unit density; see Ref
- F [i.e., without the factor of (2 π) d], in which case self-duality is defined with respect to unit density; see Ref..

33
- 78651377848
- d. This is the common notation for both objects, and therefore we adhere to this convention.
- d. This is the common notation for both objects, and therefore we adhere to this convention.

34
- 68949093616
- 10.1038/nature08239;
- S. Torquato and Y. Jiao, Nature (London) 460, 876 (2009) 10.1038/nature08239
- (2009) Nature (London) , vol.460 , pp. 876
- Torquato, S.¹ Jiao, Y.²

35
- 70350075655
- 10.1103/PhysRevE.80.041104. In these papers, the asphericity of a nonspherical solid body is defined to be ratio of the circumradius to the inradius of the circumsphere and insphere of the nonspherical particle, respectively, which provides a measure of the spherical asymmetry of a solid body. An asphericity equal to unity corresponds to a perfect sphere.
- S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009) 10.1103/PhysRevE.80.041104
- (2009) Phys. Rev. e , vol.80 , pp. 041104
- Torquato, S.¹ Jiao, Y.²

36
- 0004279601
- Cambridge University Press, Cambridge, England
- C. A. Rogers, Packing and Covering (Cambridge University Press, Cambridge, England, 1964).
- (1964) Packing and Covering
- Rogers, C.A.¹

37
- 84940644968
- C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).
- (1948) Bell Syst. Tech. J. , vol.27 , pp. 379
- Shannon, C.E.¹

38
- 84891683949
- H. Minkowski, J. Reine Angew. Math. 129, 220 (1905).
- (1905) J. Reine Angew. Math. , vol.129 , pp. 220
- Minkowski, H.¹

39
- 33845776611
- 10.1103/PhysRevE.74.061308
- S. Torquato, O. U. Uche, and F. H. Stillinger, Phys. Rev. E 74, 061308 (2006). 10.1103/PhysRevE.74.061308
- (2006) Phys. Rev. e , vol.74 , pp. 061308
- Torquato, S.¹ Uche, O.U.² Stillinger, F.H.³

40
- 0002468745
- G. A. Kabatiansky and V. I. Levenshtein, Probl. Inf. Transm. 14, 1 (1978).
- (1978) Probl. Inf. Transm. , vol.14 , pp. 1
- Kabatiansky, G.A.¹ Levenshtein, V.I.²

41
- 0001926808
- 10.1007/BF02392553
- J. Beck, Acta Math. 159, 1 (1987). 10.1007/BF02392553
- (1987) Acta Math. , vol.159 , pp. 1
- Beck, J.¹

42
- 78651352660
- 10.1088/1742-5468/2009/12/P12015
- C. E. Zachary and S. Torquato, J. Stat. Mech.: Theory Exp. (2009), P12015. 10.1088/1742-5468/2009/12/P12015
- J. Stat. Mech.: Theory Exp. , vol.2009 , pp. 12015
- Zachary, C.E.¹ Torquato, S.²

43
- 84973954613
- 10.1017/S2040618500035668
- R. A. Rankin, Proc. Glasg. Math. Assoc. 1, 149 (1953). 10.1017/S2040618500035668
- (1953) Proc. Glasg. Math. Assoc. , vol.1 , pp. 149
- Rankin, R.A.¹

44
- 84975969601
- 10.1017/S0305004100038330
- V. Ennola, Proc. Cambridge Philos. Soc. 60, 855 (1964). 10.1017/S0305004100038330
- (1964) Proc. Cambridge Philos. Soc. , vol.60 , pp. 855
- Ennola, V.¹

45
- 21744437271
- 10.1090/S0002-9939-97-03872-0
- P. Chiu, Proc. Am. Math. Soc. 125, 723 (1997). 10.1090/S0002-9939-97- 03872-0
- (1997) Proc. Am. Math. Soc. , vol.125 , pp. 723
- Chiu, P.¹

46
- 30744446481
- 10.1007/s00454-005-1202-2
- A. Schürmann and F. Vallentin, Discrete Comput. Geom. 35, 73 (2006). 10.1007/s00454-005-1202-2
- (2006) Discrete Comput. Geom. , vol.35 , pp. 73
- Schürmann, A.¹ Vallentin, F.²

47
- 0020100081
- 10.1109/TIT.1982.1056490
- P. L. Zador, IEEE Trans. Inf. Theory 28, 139 (1982). 10.1109/TIT.1982. 1056490
- (1982) IEEE Trans. Inf. Theory , vol.28 , pp. 139
- Zador, P.L.¹

48
- 0032166070
- 10.1109/18.705561
- E. Agrell and T. Eriksson, IEEE Trans. Inf. Theory 44, 1814 (1998). 10.1109/18.705561
- (1998) IEEE Trans. Inf. Theory , vol.44 , pp. 1814
- Agrell, E.¹ Eriksson, T.²

49
- 6144273465
- 10.1103/PhysRevLett.74.2156
- S. Torquato, Phys. Rev. Lett. 74, 2156 (1995). 10.1103/PhysRevLett.74. 2156
- (1995) Phys. Rev. Lett. , vol.74 , pp. 2156
- Torquato, S.¹

50
- 65549094121
- 10.1088/1742-5468/2008/11/P11019
- S. Torquato, A. Scardicchio, and C. E. Zachary, J. Stat. Mech.: Theory Exp. (2008), P11019. 10.1088/1742-5468/2008/11/P11019
- J. Stat. Mech.: Theory Exp. , vol.2008 , pp. 11019
- Torquato, S.¹ Scardicchio, A.² Zachary, C.E.³

51
- 78651369478
- V (0) =2ρ.
- V (0) = 2 ρ.

52
- 0001059552
- n.
- S. Torquato, J. Stat. Phys. 45, 843 (1986) 10.1007/BF01020577
- (1986) J. Stat. Phys. , vol.45 , pp. 843
- Torquato, S.¹

53
- 78651368462
- V (R); see the text for further explanation.
- V (R); see the text for further explanation.

54
- 0039823454
- 10.1063/1.1726548
- B. Widom, J. Chem. Phys. 44, 3888 (1966). 10.1063/1.1726548
- (1966) J. Chem. Phys. , vol.44 , pp. 3888
- Widom, B.¹

55
- 0001753855
- A. Reńyi, Sel. Trans. Math. Stat. Prob. 4, 203 (1963).
- (1963) Sel. Trans. Math. Stat. Prob. , vol.4 , pp. 203
- Reńyi, A.¹

56
- 0019186347
- 10.1016/0022-5193(80)90358-6
- J. Feder, J. Theor. Biol. 87, 237 (1980). 10.1016/0022-5193(80)90358-6
- (1980) J. Theor. Biol. , vol.87 , pp. 237
- Feder, J.¹

57
- 0001525706
- 10.1103/PhysRevA.38.522
- D. W. Cooper, Phys. Rev. A 38, 522 (1988). 10.1103/PhysRevA.38.522
- (1988) Phys. Rev. A , vol.38 , pp. 522
- Cooper, D.W.¹

58
- 78651368986
- note
- Note that leading-order term in either Eq. or Eq. involving 1 / 2 d is a dominant dimensional contribution for RSA saturation densities in relatively low dimensions for good theoretical reasons. The fact that this term does not appear in the upper bound is another reason why it could only be realizable by RSA saturated packings in high dimensions. If true, then the d ln (d) and d ln [ln (d)] terms in Eq. are high-dimensional asymptotic corrections.

59
- 78651377014
- note
- Observe that the extrapolation of the conservative fit function to d = 24 gives a density that is about 3.7% larger than that predicted by Eq.. This percentage difference between the two fit functions decreases as d decreases to d = 7.

60
- 78651379077
- note
- This includes the operational ground-based Laser Interferometer Gravitational-Wave Observatory and the space-based Laser Interferometer Space Antenna, which is expected to be operational in the next six years. See the link "Astro2010: The Astronomy and Astrophysics Decadal Survey."

61
- 56349133234
- 10.1088/0264-9381/25/18/184027
- I. W. Harry, S. Fairhurst, and B. S. Sathyaprakash, Class. Quantum Grav. 25, 184027 (2008). 10.1088/0264-9381/25/18/184027
- (2008) Class. Quantum Grav. , vol.25 , pp. 184027
- Harry, I.W.¹ Fairhurst, S.² Sathyaprakash, B.S.³

62
- 66749135292
- 10.1103/PhysRevD.79.104017
- C. Messenger, R. Prix, and M. A. Papa, Phys. Rev. D 79, 104017 (2009). 10.1103/PhysRevD.79.104017
- (2009) Phys. Rev. D , vol.79 , pp. 104017
- Messenger, C.¹ Prix, R.² Papa, M.A.³

63
- 84945618459
- 10.1080/00268976300101411
- M. J. D. Powell, Mol. Phys. 7, 591 (1964). 10.1080/00268976300101411
- (1964) Mol. Phys. , vol.7 , pp. 591
- Powell, M.J.D.¹

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