Dynamics of AN Incommensurate Harmonic Chain.

The discrete Frenkel-Kontorova model, which describes a one-dimensional solid in a periodic potential, is examined in the incommensurate region. The ground state undergoes a transition from an unpinned to a pinned phase, as Aubry first discovered. The first half of the thesis describes analytic and numerical investigations of the system near this transition. A disorder parameter and a correlation length are defined and shown numerically to obey scaling relations on both sides of the transition. Some renormalization group descriptions of the transition are described. The system studied is equivalent to the "standard map" of dynamical systems theory: this relationship is discussed. In particular, the results described here extend the scaling behavior found by Shenker and Kadanoff into the "chaotic" regime. The second half of the thesis concerns numerical studies of the incommensurate pinned phase as a uniform destabilizing force is applied. A transition from a stationary to a moving phase occurs as a function of the applied force which is qualitatively different from the pinning transition but which also appears to satisfy scaling relations. Possible applicability of this work to experimental observations of charge density wave systems is discussed. Studies of the incommensurate harmonic chain help to reveal the essential differences between systems with finitely many and infinitely many interacting degrees of freedom. The complexity of the dynamic response serves to indicate the richness of models with competing interactions.