The Analyticity Breakdown for Frenkel-kontorova Models in Quasi-periodic Media: Numerical Explorations
نویسندگان
چکیده
We study numerically the “analyticity breakdown” transition in 1-dimensional quasi-periodic media. This transition corresponds physically to the transition between pinned down and sliding ground states. Mathematically, it corresponds to the solutions of a functional equation losing their analyticity properties. We implemented some recent numerical algorithms that are efficient and backed up by rigorous results so that we can compute with confidence even close to the breakdown. We have uncovered several phenomena that we believe deserve a theoretical explanation: A) The transition happens in a smooth surface. B) There are scaling relations near breakdown. C) The scaling near breakdown is very anisotropic. Derivatives in different directions blow up at different rates. Similar phenomena seem to happen in other KAM problems. Quasi-periodic solutions, quasi-crystals, hull functions, KAM theory [2000] 70K43, 52C23, 37A60, 37J40, 82B20
منابع مشابه
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...
متن کاملEquilibrium Quasi-periodic Configurations with Resonant Frequencies in Quasi-periodic Media I: Perturbative Expansions
We consider 1-D quasi-periodic Frenkel-Kontorova models (describing, for example, deposition of materials in a quasi-periodic substratum). We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions ...
متن کاملEquilibrium Quasi-periodic Configurations with Resonant Frequencies in Quasi-periodic Media Ii: Kam Theory
We develop an a-posteriori KAM theory for the equilibrium equations for quasi-periodic solutions in a quasi-periodic Frenkel-Kontorova model when the frequency of the solutions resonates with the frequencies of the substratum. The KAM theory we develop is very different both in the methods and in the conclusions from the more customary KAM theory for Hamiltonian systems or from the KAM theory i...
متن کاملThe static properties of multi-chain Frenkel–Kontorova model: ground state and static friction
The static properties of the multi-chain Frenkel–Kontorova model are studied numerically. In the case of incommensurate structure, the transition by breaking of analyticity appears in each chain. A new hull function h(x), which also shows the transition by breaking of analyticity, is defined. The static friction (depinning force) depends strongly on the strength of the substrate potential, the ...
متن کاملComputation of the breakdown of analyticity in statistical mechanics models: numerical results and a renormalization group explanation
We consider one dimensional systems of particles interacting with each other through long range interactions that are translation invariance. We seek quasi-periodic equilibrium states. Standard arguments show that if there are continuous families of quasi-periodic ground states, the system can have large scale motion, if the family of ground states is discontinuous, the system is pinned down. T...
متن کامل