05-310 Rafael de la Llave
KAM theory for equilibrium states in 1-D statistical mechanics models (473K, pdf) Sep 8, 05
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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 quasi-periodic 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 quasi-periodic 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.

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