Renato Calleja, Rafael de la Llave
Fast numerical computation of quasi-periodic equilibrium states in 1-D
statistical mechanics, including twist maps
(1781K, postscript)
ABSTRACT. We develop fast algorithms to compute quasi-periodic
equilibrium states of one dimensional models in statistical
mechanics.
The models considered include as particular cases,
Frenkel-Kontorova models, possibly with long range
interactions, Heisenberg XY models, possibly with
long-rage interactions as well as problems
from dynamical systems such as twist mappings and
monotone recurrences. In the dynamical cases, the
quasi-periodic solutions are KAM tori.
The algorithms developed are highly efficient. If we discretize
a quasi-periodic function using $N$ Fourier coefficients,
the algorithms introduced here
require $O(N)$ storage and a Newton step for the
equilibrium equation requires only
$O(N \log(N))$ arithmetic operations.
These algorithms are also backed up by rigorous
``a posteriori estimates'' that give conditions
that ensure that approximate solutions correspond to
true ones.
We have implemented the algorithms and
present comparisons of timings, accuracy with other algorithms.
More substantially, we use the algorithms to study the
{\sl analyticity breakdown} transition, which for
twist mappings becomes the breakdown of KAM tori.
We argue that the method presented here gives
a method, independent of other previous methods
to compute the breakdowns.
We use this method to explore the analyticity breakdown
in some Frenkel-Kontorova models with extended interactions.
In some ranges of parameters, we find that the
breakdown presents scaling relations that, up to
the accuracy of our calculations are the same as those
for the standard map.
We also present results that indicate that, when
the interactions decrease very slowly, the
breakdown of analyticity is quantitatively very different.