Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps

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-range 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 and accuracy with other algorithms. More substantially, we use the algorithms to study the analyticity breakdown transition, which for twist mappings becomes the breakdown of KAM tori. 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