Abstract
We construct the first model of particles in the plane with completely symmetric, short range, two body interactions which has quasiperiodic, but no periodic, ground states.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson, P. W.: Basic notions of condensed matter physics. pp. 11–15 Menlo Park: Benjamin/Cummings 1984
Brush, S. G.: Statistical physics and the atomic theory of matter. p. 277 Princeton: Princeton University Press 1983
Simon, B.: Fifteen problems in mathematical physics. In: Perspectives in mathematics: Anniversary of Oberwolfach 1984. pp. 442–443 Basel: Birkhäuser Verlag 1984
Uhlenbeck, G. E.: An outline of statistical mechanics. In: Fundamental problems in statistical mechanics II. p. 16 Cohen, E.G.D. (ed.) New York: Wiley 1968
Uhlenbeck, G. E.: Summarizing remarks. In: Statistical mechanics; Foundations and applications. p. 581 Bak, T. A. (ed). New York: Benjamin 1967
Schechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.53, 1951–1953 (1984)
Kunz, H.: The one-dimensional classical electron gas. Ann. Phys. (N.Y.)85, 303–335 (1974)
Ventevogel, W. J.: On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Physica (Utrecht)92A, 343–361 (1978)
Ventevogel, W. J., Nijboer, B. R. A.: On the configuration of systems of interacting particles with minimum potential energy per particle. Physica (Utrecht)98A, 274–288 (1979)
Ventevogel, W. J., Nijboer, B. R. A.: On the configuration of systems of interacting particles with minimum potential energy per particle. Physica (Utrecht)99A, 569–580 (1979)
Gardner, C. S., Radin, C.: The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys.20, 719–724 (1979)
Hamrick, G. C., Radin, C.: The symmetry of ground states under perturbation. J. Stat. Phys.21, 601–607 (1979)
Heitmann, R. C., Radin, C.: The ground state for sticky disks. J. Stat. Phys.22, 281–287 (1980)
Borden, B., Radin, C.: The crystal structure of the noble gases. J. Chem. Phys.75, 2012–2013 (1981)
Radin, C.: The ground state for soft disks. J. Stat. Phys.26, 365–373 (1981)
Wagner, H. -J.: Crystallinity in two dimensions: A note on a paper of C. Radin. J. Stat. Phys.33, 523–526 (1983)
Radin, C.: Crystalline symmetry and surface tension. Physica (Utrecht)113A, 338–342 (1982)
Nicolò, F., Radin, C.: A first-order phase transition between crystal phases in the shift model. J. Stat. Phys.28, 473–478 (1982)
Radin, C., Schulman, L.: Periodicity of classical ground states. Phys. Rev. Lett.51, 621–622 (1983)
Radin, C.: Classical ground states in one dimension. J. Stat. Phys.35, 109–117 (1984)
Radin, C.: Tiling, periodicity and crystals. J. Math. Phys.26, 1342–1344 (1985)
Lieb, E., Kennedy, T.: To be published
Radin, C.: Crystals and quasicrystals: a lattice gas model. Phys. Lett.114A, 381–383 (1986)
Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc.66 (1966)
Gardner, M.: Mathematical games. Sci. Am.236, (1), 110–121 (1977)
Robinson, R. M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math.12, 177–209 (1971)
Robinson, R. M.: personal communication
Grünbaum, B., Shephard, G. C.: Tilings and patterns. To be published
Fejes-Tóth, L.: Über einen geometrischen satz. Math. Z.46, 79–83 (1940)
Author information
Authors and Affiliations
Additional information
Communicated by J. Fröhlich
Supported in part by NSF Grant No. DMS-8501911
Rights and permissions
About this article
Cite this article
Radin, C. Crystals and quasicrystals: A continuum model. Commun.Math. Phys. 105, 385–390 (1986). https://doi.org/10.1007/BF01205933
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01205933