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The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior.

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  • 1Università degli Studi di Padova, Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121 Padova, Italy. ponno@math.unipd.it

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Area of Science:

  • Nonlinear dynamics
  • Statistical mechanics
  • Computational physics

Background:

  • The Fermi-Pasta-Ulam (FPU) paradox highlights the complex energy redistribution in nonlinear systems.
  • Integrable systems like the Toda model offer a baseline for understanding non-integrable dynamics.
  • Understanding relaxation to equilibrium is crucial in statistical mechanics.

Purpose of the Study:

  • To compare the relaxation dynamics of the FPU α-model and the integrable Toda model.
  • To elucidate the mechanisms of energy transfer and equipartition in these systems.
  • To identify universal behaviors and model-specific deviations during the approach to equilibrium.

Main Methods:

  • Numerical simulations of both FPU α-model and Toda model.
  • Analytical investigation of mode energy evolution.
  • Comparison of time-averaged modal energy spectra.

Main Results:

  • Both models show similar short-term dynamics characterized by a secular avalanche process.
  • The Toda model's equilibrium spectrum is well-approximated by a q-breather, serving as a quasi-equilibrium for the FPU system.
  • FPU system's long-term dynamics diverge due to diffusive growth of tail modes, eventually reaching a flat equilibrium spectrum.

Conclusions:

  • The Toda model provides a valuable approximation for the quasi-equilibrium state of the FPU system.
  • A simple law governs the growth of tail modes in the FPU system, enabling timescale estimation.
  • This study offers insights into energy equipartition and relaxation in nonlinear systems.