Finite range of large perturbations in Hamiltonian dynamics

D. Bénisti, D.F. Escande

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Abstract

The dynamics defined by the Hamiltonian H = p2/2 + A ∑Mm = -M cos(q - mt + φm), where the φm are fixed random phases, is investigated for large values of A, and for M ≫ A2/3. For a given P* and for Δv ≥ A2/3, this Hamiltonian is transformed through a rigorous perturbative treatment into a Hamiltonian where the sum of all the nonresonant terms, having a Q dependence of the kind cos(kQ - nt + φm) with |n/k - P*| > Δv, is a random variable whose r.m.s. with respect to the φm is exponentially small in the parameter ε = A/Δv3/2. Using this result, a rationale is provided showing that the statistical properties of the dynamics defined by H. and of the reduced dynamics including at each time t only the terms in H such that |m -p(t)| ≤ αA2/3, can be made arbitrarily close by increasing α. For practical purposes α close to 5 is enough, as confirmed numerically. The reduced dynamics being nondeterministic, it is thus analytically shown, without using the random-phase approximation, that the statistical properties of a chaotic Hamiltonian dynamics can be made arbitrarily close to that of a stochastic dynamics. An appropriate rescaling of momentum and time shows that the statistical properties of the dynamics defined by H can be considered as independent of A, on a finite time interval, for A large. The way these results could generalize to a wider class of Hamiltonians is indicated.
Original languageEnglish
Pages (from-to)909 - 972
Number of pages64
JournalJournal of Statistical Physics
Volume92
Issue number5-6
Publication statusPublished - Sep 1998
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Bénisti, D., & Escande, D. F. (1998). Finite range of large perturbations in Hamiltonian dynamics. Journal of Statistical Physics, 92(5-6), 909 - 972.