 05310 Rafael de la Llave
 KAM theory for equilibrium states in 1D statistical mechanics models
(473K, pdf)
Sep 8, 05

Abstract ,
Paper (src),
View paper
(auto. generated pdf),
Index
of related papers

Abstract. We extend the Lagrangian proof of KAM for twist mappings
[D Salamon, E. Zehnder 89, M. Levi, J. Moser 99]
to show persistence of
quasiperiodic equilibrium solutions in
statistical mechanics models. The interactions in the
models considered here do not need to
be of finite range but they
have to decrease sufficiently with the distance.
When the interactions are range $R$, the models admit the dynamical
interpretation of recurrences in $(\real)^{2R}$.
Note that the small perturbations
in the Lagrangian are singular
from the dynamical systems point of view since
they may increase the dimension of phase space.
We show that in these models, given an approximate
solution of the equilibrium equation with one
Diophantine frequency, which is not too degenerate, there is a true solution nearby.
As a consequence, we deduce that quasiperiodic solutions
of the equilibrium equation with
one Diophantine frequency persist under small modifications of
the model.
The main result can also be used to
validate numerical calculations or perturbative
expansions.
We also show that Lindstedt series can be computed to
all orders in these models.
 Files:
05310.src(
05310.keywords ,
kamtheory.pdf.mm )