ISSN:
1572-9613
Keywords:
Perturbation theory
;
Hamiltonian dynamics
;
wave–particle interaction: transport properties
;
chaos
;
plasma turbulence
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1023092526620
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