Generalized Frenkel-Kontorova models

Abstract

We have studied two generalized Frenkel-Kontorova (FK) models. In one of them the interatomic potential is changed from simple harmonic to Toda; in the other the external potential is changed from sinusoidal to nonanalytic. Their critical behaviors are found to be markedly different. In the Toda case, the local critical exponents are the same as those in the standard FK model. However, the global singularity spectrum and the generalized dimension depend on the anharmonicity parameter. In the nonanalytic FK model, the critical exponents are found to vary with the degree of inflection z for 2≤z<3, but remain the same for z≥3. The degree of inflection thus plays a role quite similar to that of dimensionality in phase transitions with z = 2 and 3 corresponding respectively to the lower and upper critical dimensions. For z>3, a sequence of pinning and depinning transitions, reflecting the recurrence of Kolmogorov-Arnol'd-Moser tori, has also been observed. © 1994.