Finite Range of Large Perturbations in Hamiltonian Dynamics

Abstract

The dynamics defined by the Hamiltonian $H = p^2 /2 + A\sum\nolimits_{m = - M}^M {\cos (q - mt + \varphi m)}$, where the φ _m are fixed random phases, is investigated for large values of A, and for $M \gg A^{2/3}$. For a given P ^* and for $\Delta \upsilon \geqslant A^{2/3}$, 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 + φ _m) with $|n/k - P^ * | > \Delta \upsilon$, is a random variable whose r.m.s. with respect to the φ _m is exponentially small in the parameter $\varepsilon = A/\Delta \upsilon ^{3/2}$. 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 $|m - p(t)| \leqslant \alpha A^{2/3}$, 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.