We revisit the celebrated model of Fermi, Pasta and Ulam with the aim of investigating, by numerical computations, the trend towards equipartition in the thermodynamic limit. We concentrate our attention on a particular class of initial conditions, namely, with all the energy on the first mode or the first few modes. We observe that the approach to equipartition occurs on two different time scales: in a short time the energy spreads up by forming a packet involving all low--frequency modes up to a cutoff frequency $\omega_c$, while a much longer time is required in order to reach equipartition, if any. In this sense one has an energy localization with respect to frequency. The crucial point is that our numerical computations suggest that this phenomenon of a fast formation of a natural packet survives in the thermodynamic limit. More precisely we conjecture that the cutoff frequency $\omega_c$ is a function of the specific energy $\epsilon = E/N$, where $E$ and $N$ are the total energy and the number of particles, respectively. Equivalently, there should exist a function $\epsilon_c(\omega)$, representing the minimal specific energy at which the natural packet extends up to frequency $\omega$. The time required for the fast formation of the natural packet is also investigated

Berchialla, L., Galgani, L., Giorgilli, A. (2004). Localization of energy in FPU chains. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 11(4), 855-866 [10.3934/dcds.2004.11.855].

Localization of energy in FPU chains

BERCHIALLA, LUISA;
2004

Abstract

We revisit the celebrated model of Fermi, Pasta and Ulam with the aim of investigating, by numerical computations, the trend towards equipartition in the thermodynamic limit. We concentrate our attention on a particular class of initial conditions, namely, with all the energy on the first mode or the first few modes. We observe that the approach to equipartition occurs on two different time scales: in a short time the energy spreads up by forming a packet involving all low--frequency modes up to a cutoff frequency $\omega_c$, while a much longer time is required in order to reach equipartition, if any. In this sense one has an energy localization with respect to frequency. The crucial point is that our numerical computations suggest that this phenomenon of a fast formation of a natural packet survives in the thermodynamic limit. More precisely we conjecture that the cutoff frequency $\omega_c$ is a function of the specific energy $\epsilon = E/N$, where $E$ and $N$ are the total energy and the number of particles, respectively. Equivalently, there should exist a function $\epsilon_c(\omega)$, representing the minimal specific energy at which the natural packet extends up to frequency $\omega$. The time required for the fast formation of the natural packet is also investigated
Articolo in rivista - Articolo scientifico
FPU model, Non linear systems
English
2004
11
4
855
866
open
Berchialla, L., Galgani, L., Giorgilli, A. (2004). Localization of energy in FPU chains. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 11(4), 855-866 [10.3934/dcds.2004.11.855].
File in questo prodotto:
File Dimensione Formato  
localization_of_energy.pdf

accesso aperto

Tipologia di allegato: Author’s Accepted Manuscript, AAM (Post-print)
Dimensione 655.65 kB
Formato Adobe PDF
655.65 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/7489
Citazioni
  • Scopus 59
  • ???jsp.display-item.citation.isi??? 50
Social impact