We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to exploit the complete resonance, while keeping the extensive nature of the system. Next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear Schrödinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of the system size, for a large set of initial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory, with fluctuation much smaller than expected according to standard statistical estimates.

Extensive Adiabatic Invariants for Nonlinear Chains / A. Giorgilli, S. Paleari, T. Penati. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 148:6(2012 Sep), pp. 1106-1134. [10.1007/s10955-012-0568-9]

Extensive Adiabatic Invariants for Nonlinear Chains

A. Giorgilli
Primo
;
S. Paleari
Secondo
;
T. Penati
Ultimo
2012

Abstract

We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to exploit the complete resonance, while keeping the extensive nature of the system. Next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear Schrödinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of the system size, for a large set of initial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory, with fluctuation much smaller than expected according to standard statistical estimates.
Adiabatic invariant; Ergodicity; Extensive Hamiltonian lattice; Resonant normal form; Thermodynamic limit
Settore MAT/07 - Fisica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/214596
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