Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions

Journal of Statistical Physics

Volumn 109, Issue 5-6, 2002, Pages 945-965

  Widom, M ^a   Mosseri, R ^b   Destainville, N ^c   Bailly, F ^d  

^a CARNEGIE MELLON UNIVERSITY   (United States)

^b SORBONNE UNIVERSITÉ   (France)

^c CNRS   (France)

^d Lab de Physique des Solides de   (France)

Author keywords

Boundary effects; Configurational entropy; Integer partitions; Random tilings; Transition matrix Monte Carlo algorithms

Indexed keywords

EID: 0036388161     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1020464224385     Document Type: Article

References (46)

1. D. Shechtman et al., Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53:1951 (1984).
2. V. Elser, Comment on "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 54:1730 (1985).
3. C. L. Henley, Random tiling models, in Quasicrystals, the State of the Art, D. P. Di Vincenzo and P. J. Steinhart, eds. (World Scientific, 1991), p. 429.
4. M. Mihalkovic et al., Total-energy-based prediction of a quasicrystal structure, Phys. Rev. B (to appear, 2002).
5. M. Widom, K. J. Strandburg, and R. H. Swendsen, Quasicrystal equilibrium state, Phys. Rev. Lett., 58:706 (1987).
6. H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. of Math. 4:137 (1998); H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14:297 (2001).
7. H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. of Math. 4:137 (1998); H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14:297 (2001).
8. M. Luby, D. Randall, and A. Sinclair, Markov chain algorithms for planar lattice structures, Proc. 36th Ann. Symp. on the Foundations of Comp. Sci. (IEEE, 1995), p. 150
9. V. Elser, Solution of the dimer problem on an hexagonal lattice with boundary, J. Phys. A: Math. Gen. 17:1509 (1984).
10. R. Mosseri, F. Bailly, and C. Sire, Configurational entropy in random tiling models, J. Non-Cryst. Solids 153-154:201 (1993).
11. R. Mosseri and F. Bailly, Configurational entropy in octagonal tiling models, Int. J. Mod. Phys. B 6&7:1427 (1993).
12. N. Destainville, R. Mosseri, and F. Bailly, Configurational entropy of codimension-one tilings and directed membranes, J. Stat. Phys. 87:697 (1997).
13. N. Destainville, R. Mosseri, and F. Bailly, Fixed-boundary octagonal random tilings: A combinatorial approach, J. Stat. Phys. 102:147 (2001).
14. D. Grensing and G. Grensing, Boundary effects in the dimer problem on a non-bravais lattice, J. Math. Phys. 24:620 (1983); D. Grensing, I. Carlsen, and H. Chr. Zapp, Some exact results for the dimer problem on plane lattices with nonstandard boundaries, Phil. Mag. A 41:777 (1980)
15. D. Grensing and G. Grensing, Boundary effects in the dimer problem on a non-bravais lattice, J. Math Phys. 24:620 (1983); D. Grensing, I. Carlsen, and H. Chr. Zapp, Some exact results for the dimer problem on plane lattices with nonstandard boundaries, Phil. Mag. A 41:777 (1980)
16. N. Destainville, Entropy and boundary conditions in random rhombus tilings, J. Phys. A: Math. Gen 31:6123 (1998).
17. J. Linde, C. Moore, and M.G. Nordahl, An n-dimensional generalization of the rhombus tiling, in Proceedings of the 1st International conference on Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG'01), M. Morvan, R. Cori, J. Mazoyer and R. Mosseri, eds., Discrete Math. and Theo. Comp. Sc. AA:23 (2001).
18. J. Linde, C. Moore, and M.G. Nordahl, An n-dimensional generalization of the rhombus tiling, in Proceedings of the 1st International conference on Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG'01), M. Morvan, R. Cori, J. Mazoyer and R. Mosseri, eds., Discrete Math. and Theo. Comp. Sc. AA:23 (2001).
19. N. Destainville and R. Mosseri (Unpublished, 2001)
20. J. Propp (Unpublished, 2001)
21. K. J. Strandburg, Entropy of a three-dimensional random-tiling quasicrystal, Phys. Rev. B 44:4644 (1991).
22. 0.14, J. Phys. Lett. France 46:L601 (1985).
23. 0.14, J. Phys. Lett. France 46:L601 (1985).
24. 0.14, J. Phys. Lett. France 46:L601 (1985).
25. Recall that the configurational entropy per tile is the logarithm of the number of configurations, divided by the number of tiles.
26. N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane, Ned. Akad. Wetensch. Proc. A84:39 (1981); Dualization of multigrids, J. Phys. France 47:C3-9 (1986).
27. N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane, Ned. Akad. Wetensch. Proc. A84:39 (1981); Dualization of multigrids, J. Phys. France 47:C3-9 (1986).
28. To make the "free-boundary" entropy well defined, we need to add some weak constraint to ensure the tiling remains connected and reasonably compact. We presume that the free-boundary entropy equals that for strain-free periodic boundary conditions.
29. M. Widom, Bethe ansatz solution of the square-triangle random tiling model, Phys. Rev. Lett. 70:2094 (1993); P. Kalugin, Square-triangle random-tiling model in the thermodynamic limit, J. Phys. A 27:3599 (1994); J. de Gier and B. Nienhuis, Exact solution of an octagonal random tiling model, Phys. Rev. Lett. 76:2918 (1996); J. de Gier and B. Nienhuis, Bethe ansatz solution of a decagonal rectangle-triangle random tiling, J. Phys. A 31:2141 (1998);
30. M. Widom, Bethe ansatz solution of the square-triangle random tiling model, Phys. Rev. Lett. 70:2094 (1993); P. Kalugin, Square-triangle random-tiling model in the thermodynamic limit, J. Phys. A 27:3599 (1994); J. de Gier and B. Nienhuis, Exact solution of an octagonal random tiling model, Phys. Rev. Lett. 76:2918 (1996); J. de Gier and B. Nienhuis, Bethe ansatz solution of a decagonal rectangle-triangle random tiling, J. Phys. A 31:2141 (1998);
31. M. Widom, Bethe ansatz solution of the square-triangle random tiling model, Phys. Rev. Lett. 70:2094 (1993); P. Kalugin, Square-triangle random-tiling model in the thermodynamic limit, J. Phys. A 27:3599 (1994); J. de Gier and B. Nienhuis, Exact solution of an octagonal random tiling model, Phys. Rev. Lett. 76:2918 (1996); J. de Gier and B. Nienhuis, Bethe ansatz solution of a decagonal rectangle-triangle random tiling, J. Phys. A 31:2141 (1998);
32. M. Widom, Bethe ansatz solution of the square-triangle random tiling model, Phys. Rev. Lett. 70:2094 (1993); P. Kalugin, Square-triangle random-tiling model in the thermodynamic limit, J. Phys. A 27:3599 (1994); J. de Gier and B. Nienhuis, Exact solution of an octagonal random tiling model, Phys. Rev. Lett. 76:2918 (1996); J. de Gier and B. Nienhuis, Bethe ansatz solution of a decagonal rectangle-triangle random tiling, J. Phys. A 31:2141 (1998);
33. K. J. Strandburg, L.-H. Tang, and M. V. Jaric, Phason elasticity in entropic quasicrystals, Phys. Rev. Lett. 63:314 (1989); L.-H. Tang, Random-tiling quasicrystal in three dimensions, Phys. Rev. Lett. 64:2390 (1990) ; D. Joseph and M. Baake, Boundary conditions, entropy and the signature of random tilings, J. Phys. A 29:6709 (1996); M. Oxborrow and C. L. Henley, Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals, Phys. Rev. B 48:6966 (1993).
34. K. J. Strandburg, L.-H. Tang, and M. V. Jaric, Phason elasticity in entropic quasicrystals, Phys. Rev. Lett. 63:314 (1989); L.-H. Tang, Random-tiling quasicrystal in three dimensions, Phys. Rev. Lett. 64:2390 (1990) ; D. Joseph and M. Baake, Boundary conditions, entropy and the signature of random tilings, J. Phys. A 29:6709 (1996); M. Oxborrow and C. L. Henley, Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals, Phys. Rev. B 48:6966 (1993).
35. K. J. Strandburg, L.-H. Tang, and M. V. Jaric, Phason elasticity in entropic quasicrystals, Phys. Rev. Lett. 63:314 (1989); L.-H. Tang, Random-tiling quasicrystal in three dimensions, Phys. Rev. Lett. 64:2390 (1990) ; D. Joseph and M. Baake, Boundary conditions, entropy and the signature of random tilings, J. Phys. A 29:6709 (1996); M. Oxborrow and C. L. Henley, Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals, Phys. Rev. B 48:6966 (1993).
36. K. J. Strandburg, L.-H. Tang, and M. V. Jaric, Phason elasticity in entropic quasicrystals, Phys. Rev. Lett. 63:314 (1989); L.-H. Tang, Random-tiling quasicrystal in three dimensions, Phys. Rev. Lett. 64:2390 (1990) ; D. Joseph and M. Baake, Boundary conditions, entropy and the signature of random tilings, J. Phys. A 29:6709 (1996); M. Oxborrow and C. L. Henley, Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals, Phys. Rev. B 48:6966 (1993).
37. H.-C. Jeong and P. J. Steinhardt, Finite-temperature elasticity phase transition in decagonal quasicrystals, Phys. Rev. B 48:9394 (1993)
38. F. Gähler, Thermodynamics of random tiling quasicrystals, in Proceedings of the 5th International Conference on Quasicrystals, C. Janot and R. Mosseri, eds. (World Scientific, 1995), p. 236.
39. L. J. Shaw and C. L. Henley, A transfer-matrix Monte Carlo study of random Penrose tilings, J. Phys. A 24:4129 (1991).
40. M. Widom, D. P. Deng, and C. L. Henley, Transfer-matrix analysis of a two-dimensional quasicrystal, Phys. Rev. Lett. 63:310 (1989); W. Li, H. Park, and M. Widom, Phase diagram of a random tiling quasicrystal, J. Stat. Phys. 66: 1 (1992); M. E. J. Newman and C. L. Henley, Phason elasticity of a three-dimensional quasicrystal: A transfer matrix study, Phys. Rev. B 52:6386 (1995)
41. M. Widom, D. P. Deng, and C. L. Henley, Transfer-matrix analysis of a two-dimensional quasicrystal, Phys. Rev. Lett. 63:310 (1989); W. Li, H. Park, and M. Widom, Phase diagram of a random tiling quasicrystal, J. Stat. Phys. 66: 1 (1992); M. E. J. Newman and C. L. Henley, Phason elasticity of a three-dimensional quasicrystal: A transfer matrix study, Phys. Rev. B 52:6386 (1995)
42. M. Widom, D. P. Deng, and C. L. Henley, Transfer-matrix analysis of a two-dimensional quasicrystal, Phys. Rev. Lett. 63:310 (1989); W. Li, H. Park, and M. Widom, Phase diagram of a random tiling quasicrystal, J. Stat. Phys. 66: 1 (1992); M. E. J. Newman and C. L. Henley, Phason elasticity of a three-dimensional quasicrystal: A transfer matrix study, Phys. Rev. B 52:6386 (1995)
43. J. S. Wang, T. K. Tay, and R. H. Swendsen, Transition matrix Monte Carlo reweighting and dynamics, Phys. Rev. Lett. 82:476 (1999); R. H. Swendsen, B. Diggs, J.-S. Wang, S.-T. Li, and J. B. Kadane, Transition matrix Monte Carlo, Int. J. Mod. Phys. C 10:1563 (1999);
44. J. S. Wang, T. K. Tay, and R. H. Swendsen, Transition matrix Monte Carlo reweighting and dynamics, Phys. Rev. Lett. 82:476 (1999); R. H. Swendsen, B. Diggs, J.-S. Wang, S.-T. Li, and J. B. Kadane, Transition matrix Monte Carlo, Int. J. Mod. Phys. C 10:1563 (1999);
45. The transition matrix Monte Carlo method(29) differs from the transfer matrix Monte Carlo method.(27) The former constructs a time series of different tilings, each of identical compact finite size, while the latter constructs a single highly elongated structure whose length grows proportionally to the duration of the simulation.
46. max is unique provided the free-boundary entropy is a concave function of the phason gradient.