We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β^(1/a) for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.

An extensive adiabatic invariant for the Klein–Gordon model in the thermodynamic limit / A. Giorgilli, S. Paleari, T. Penati. - In: ANNALES HENRI POINCARE'. - ISSN 1424-0637. - 16:4(2015 Apr), pp. 897-959. [10.1007/s00023-014-0335-3]

An extensive adiabatic invariant for the Klein–Gordon model in the thermodynamic limit

A. Giorgilli
Primo
;
S. Paleari
Secondo
;
T. Penati
Ultimo
2015

Abstract

We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β^(1/a) for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.
thermodynamic limit; extensivity; long time scales; adiabatic invariant; statistical estimates
Settore MAT/07 - Fisica Matematica
Teoria delle perturbazioni ed applicazioni alla Meccanica Statistica ed all'Elettrodinamica
Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite
29-apr-2014
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/234712
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