The twist map, the extended Frenkel-Kontorova model and the devil’s staircase.

*(English)*Zbl 0559.58013
Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica 7D, 240-258 (1983).

[For the entire collection see Zbl 0536.00007.]

The paper reviews exact results on the extended discrete Frenkel- Kontorova (FK) model and the associated area preserving twist maps of the cylinder onto itself. The structure of the ground states in the FK model is studied. For a fixed irrational rotation number the union of all the ground states (called incommensurate in this case) is either a KAM circle or, beyond a transition parameter, becomes a minimal Cantor set with zero length measure. A series of rigorous upper bounds for these breaking of the KAM curve parameters is obtained.

For a rational rotation number the periodic ground states (called commensurate) are studied. The existence of heteroclinic orbits (discommensurate states) is asserted. The Peierls-Nabarro barrier is studied in this situation. Necessary conditions for the trajectories to represent metastable configurations, which can be chaotic, are given. For the FK model it is asserted that the atomic mean distance versus a chemical potential is a Cantor function (the devil’s staircase).

Independently a similar theory was recently developed by J. Mather, see the survey paper by A. Chenciner [Astérisque 121/122, 147-170 (1985)].

The paper reviews exact results on the extended discrete Frenkel- Kontorova (FK) model and the associated area preserving twist maps of the cylinder onto itself. The structure of the ground states in the FK model is studied. For a fixed irrational rotation number the union of all the ground states (called incommensurate in this case) is either a KAM circle or, beyond a transition parameter, becomes a minimal Cantor set with zero length measure. A series of rigorous upper bounds for these breaking of the KAM curve parameters is obtained.

For a rational rotation number the periodic ground states (called commensurate) are studied. The existence of heteroclinic orbits (discommensurate states) is asserted. The Peierls-Nabarro barrier is studied in this situation. Necessary conditions for the trajectories to represent metastable configurations, which can be chaotic, are given. For the FK model it is asserted that the atomic mean distance versus a chemical potential is a Cantor function (the devil’s staircase).

Independently a similar theory was recently developed by J. Mather, see the survey paper by A. Chenciner [Astérisque 121/122, 147-170 (1985)].

Reviewer: F.Przytycki

##### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

##### References:

[1] | Ying, S. C.: Phys. rev.. 3, 4160 (1971) |

[2] | Aubry, S.: Solitons and condensed matter physics. Solid state sciences 8, 264 (1978) |

[3] | Aubry, S.: For a detailed version of this paper seelect. Notes in math.. Lect. notes in math. 925, 221 (1980) |

[4] | Aubry, S.; André, G.: Colloquium on group theoretical methods in physics. Annals of the Israel phys. Soc. 3, 133 (1980) |

[5] | Moser, J.: Stable and random motions in dynamical systems. (1973) · Zbl 0271.70009 |

[6] | Aubry, S.: On modulated crystallographic structures, exact results on the classical ground-states of a one-dimensional model. (1978) |

[7] | S. Aubry, Exact models with a complete devil’s staircase, preprint, to be published in J. of Phys. C. |

[8] | S. Aubry, P.Y. Le Daeron and G. André, in preparation. |

[9] | Greene, J.: J. math. Phys.. 20, 1183 (1979) |

[10] | Mcmillan, W. L.: Phys. rev.. 14, 1496 (1976) |

[11] | Marsden; Mckracken: The Hopf bifurcation and its application. Applied math. Sci. 19 (1976) |

[12] | Smale, S.: Bull. of AMS. 73, 747 (1967) · Zbl 0202.55202 |

[13] | Akhiezer, N. I.: The classical moment problem and some related questions in analysis. (1965) · Zbl 0135.33803 |

[14] | Peyrard, M.; Aubry, S.: To be published inj. Phys. C. J. phys. C (1983) |

[15] | R. Shilling, private communication (1982). |

[16] | Aubry, S.: Journal de physique (Paris). 44, 147 (1983) |

[17] | Aubry, S.: Physics of defects. LES houches 35, 431 (1981) |

[18] | Bak, P.: Phys. rev. Lett.. 46, 791 (1981) |

[19] | Bak, P.; Pokrovsky, V. L.: Phys. rev. Lett.. 47, 958 (1981) |

[20] | P.Y. Le Daeron and S. Aubry, Metal insulator transition in the Peierls chain, submitted to J. of Phys. C. · Zbl 1237.37059 |

[21] | Aubry, S.: On the dynamic of structural phase transition. Lattice locking and ergodic theory (1977) |

[22] | Nabarro, F.: Theory of crystal dislocations. (1967) |

[23] | Aubry, S.: G.lavald.gresillonintrinsic stochasticity in plasmas. Intrinsic stochasticity in plasmas (1979) |

[24] | J. MacKay, this conference. |

[25] | Mandelbrot, B.: Fractals. (1977) · Zbl 0359.76050 |

[26] | Riesz, F.; Nagy, B.: Functional analysis. (1965) · Zbl 0070.10902 |

[27] | D. Ruelle, this conference. |

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