Recent results on the Fermi-Pasta-Ulam problem. (English. Russian original) Zbl 1120.37049

Galgani, L.; Giorgilli, A.

J. Math. Sci., New York 128, No. 2, 2761-2766 (2005); translation from Zap. Nauchn. Semin. POMI 300, 145-154 (2003).

Summary: We revisit the celebrated model of Fermi, Pasta, and Ulam with the purpose of investigating the thresholds to equipartition in the thermodynamic limit. Starting with a particular class of initial conditions, i.e., with all the energy on the first mode, we observe that in a short time the system splits into two separate subsystems. We conjecture the existence of a function \(\varepsilon_c(\omega)\), independent of the number \(N\) of particles in the chain, such that if the initial energy \(E\) satisfies \(E/N< \varepsilon_c(\omega)\), then only the packet of modes with frequency not exceeding \(\omega\) shares most of the energy.

MSC:

37N20	Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
82C05	Classical dynamic and nonequilibrium statistical mechanics (general)

Keywords:

Hamilton system; stable states; equipartition; thermodynamic limit; subsystems

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