 08189 Renato Calleja, Rafael de la Llave
 Fast numerical computation of quasiperiodic equilibrium states in 1D
statistical mechanics, including twist maps
(1781K, postscript)
Oct 17, 08

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Abstract. We develop fast algorithms to compute quasiperiodic
equilibrium states of one dimensional models in statistical
mechanics.
The models considered include as particular cases,
FrenkelKontorova models, possibly with long range
interactions, Heisenberg XY models, possibly with
longrage interactions as well as problems
from dynamical systems such as twist mappings and
monotone recurrences. In the dynamical cases, the
quasiperiodic solutions are KAM tori.
The algorithms developed are highly efficient. If we discretize
a quasiperiodic 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 FrenkelKontorova 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.
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