Abstract
We compare two proposals for the study of positional long-range order: one in terms of the spectrum of the translation operator, the other in terms of the Fourier spectrum. We point out that only the first one allows for the consideration of molecular, as opposed to atomic, (weakly) periodic structures. We illustrate this point on the Thue-Morse system.
This is a preview of subscription content, log in via an institution to check access.
References
C. Radin, Low temperature and the origin of crystalline symmetry,Int. J. Mod. Phys. B 1:1157 (1987).
C. Radin, Global order from local sources,Bull. Am. Math. Soc., to appear.
D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translation symmetry,Phys. Rev. Lett. 53:1951 (1984).
D. Levine and P. J. Steinhardt, Quasicrystals: A new class of ordered structures,Phys. Rev. Lett. 53:2477 (1984).
D. Ruelle, Do turbulent crystals exist?,Physica 113A:619 (1982).
S. Aubry, Devil's staircase and order without periodicity in classical condensed matter,J. Phys. (Paris)44:147 (1983).
D. Ruelle,Statistical Mechanics; Rigorous Results (Benjamin, Reading, Massachusetts, 1969), esp. Chapter 6.
D. Ruelle, States of physical systems,Commun. Math. Phys. 3:133 (1966).
D. Ruelle, Integral representation of states on aC * algebra,J. Funct. Anal. 6:116 (1970).
D. Kastler and D. W. Robinson, Invariant states in statistical mechanics,Commun. Math. Phys. 3:151 (1966).
G. G. Emch, TheC *-algebra to phase transitions, inPhase Transitions and Critical Phenomena, Vol. 1, C. Domb and M. L. Green, eds. (Academic Press, New York, 1972).
G. G. Emch, H. J. F. Knops, and E. J. Verboven, Breaking of Euclidean symmetry with an application to the theory of crystallization,J. Math. Phys. 11:1165 (1970).
O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics, Vols. 1 and 2 (Springer, Berlin, 1979/1981).
C. Radin, Correlations in classical ground states,J. Stat. Phys. 43:707 (1986).
E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory,Contemp. Math. 64 (1987).
C. Radin, Disordered ground states of classical lattice models,Rev. Math. Phys. 3:125 (1991).
R. I. Jewett, The prevalence of uniquely ergodic systems,J. Math. Mech. 19:717 (1970).
W. Krieger, On unique ergodicity, inProceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (1970), pp. 327–346.
M. Denker, C. Grillenberger, and K. Sigmund,Ergodic Theory on Compact Spaces (Springer, Berlin, 1976).
M. Queffélec,Substitution Dynamical Systems—Spectral Analysis (Springer, Berlin, 1987).
S. Aubry, Weakly periodic structures and example,J. Phys. (Paris)Coll. C3-50:97 (1984).
S. Aubry, Weakly periodic structures with a singular continuous spectrum, inProceedings of the NATO Advanced Research Workshop on Common Problems of Quasi-Crystals,Liquid-Crystals and Incommensurate Insulators, Preveza 1989, J. I. Toledano, ed.
S. Aubry, C. Godréche, and J. M. Luck, Scaling properties of a structure intermediate between quasiperiodic and random,J. Stat. Phys. 51:1033 (1988).
J. W. Cahn and J. E. Taylor, An introduction to quasicrystals,Contemp. Math. 64 (1987).
Z. Cheng, R. Savit, and R. Merlin, Structure and electronic properties of Thue-Morse lattices,Phys. Rev. B 37:4375 (1988).
Z. Cheng and R. Savit, Structure factor of substitutional sequences,J. Stat. Phys. 60:383 (1990).
M. Kolar, M. K. Ali, and F. Nori, Generalized Thue-Morse chains and their physical properties,Phys. Rev. B 43:1034 (1991).
M. Keane, Generalized Morse sequences,Z. Wahr. 10:335 (1968).
C. Gardner, J. Micekisz, C. Radin, and A. C. D. van Enter, Fractal symmetry in an Ising model,J. Phys. A 22:L1019 (1989).
A. C. D. van Enter and J. Micekisz, Breaking of periodicity at positive temperatures,Commun. Math. Phys. 134:647 (1990).
R. Merlin, K. Bajemu, J. Nagle, and K. Ploog, Raman scattering by acoustic phonons and structural properties of Fibonacci, Thue-Morse, and random superlattices,J. Phys. Coll. (Paris)C5:503 (1987).
F. Axel and H. Terauchi, High resolutionX-ray diffraction spectra of Thue-Morse GaAs-AlAs heterostructures: Towards a novel description of disorder,Phys. Rev. Lett. 66:2223 (1991).
C. Radin, Crystals and quasicrystals: A lattice gas model,Phys. Lett. 114A:385 (1986).
K. Mahler, On the translation properties of a simple class of arithmetical functions,J. Math. Phys. 6:150 (1927).
S. Kakutani, Ergodic properties of shift transformations, inProceedinigs of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (1967), pp. 404–414.
S. Mozes, Tilings, substitutions, and dynamical systems generated by them,J. Anal. Math. 53:139 (1989).
J. Slawny, Ergodic properties of equilibrium states,Commun. Math. Phys. 80:477 (1981).
Author information
Authors and Affiliations
Additional information
Communicated by D. Stauffer
Rights and permissions
About this article
Cite this article
van Enter, A.C.D., Miekisz, J. How should one define a (weak) crystal?. J Stat Phys 66, 1147–1153 (1992). https://doi.org/10.1007/BF01055722
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01055722