Hamiltonian dynamics with a weak noise and the echo effect for the rotator model

Abstract

We analyze the effect of a weak noise on the Hamiltonian transport from the analytical and numerical viewpoint. A solvable model, the noisy rotator, is proposed to illustrate the basic phenomena. In the absence of noise, the phase space evolution is a shear flow, whose angular correlations decay following a power law, which depends on the smoothness of the initial action distribution. If the action has a fluctuating component, given by a Wiener process, then the angular correlations decay exponentially according to e − 2 t 3 /6 or faster, where is the noise amplitude. The echo effect is well suited to investigate the competition between the decorrelation due to filamentation and noise. The noisy rotator model allows an exhaustive analytical investigation of the process for a wide class of initial conditions and a generic disturbance. The echo time is proportional to the delay τ of the disturbance and its amplitude is proportional to λτ where λ is the amplitude of the disturbance. The noise reduces the echo amplitude by e −c 2 τ 3 , where c depends on the Fourier components of the initial angular distribution, and of the disturbance applied at time τ . The analytical results, derived in the limit λ → 0, τ → ∞, with λτ finite and sufficiently small to justify a first order expansion, are checked numerically. For more realistic models the analytical procedure would provide qualitative results and scaling laws. Quantitative results are obtained by solving the Fokker-Planck equation with a numerical scheme based on splitting: back propagation and biquadratic interpolation for the integrable part, implicit finite difference scheme for the noise component. The application to a noisy pendulum describing the longitudinal dynamics in a particle accelerator is considered, and we determine the value of the noise amplitude , below which the echo cannot be detected.

Analysis of Round Off Errors With Reversibility Test as a Dynamical Indicator

“…where ξ n and χ n are random uncorrelated variables the growth of the variances σ 2 y and σ 2 x follows a linear and cubic law respectively (Turchetti et al, 2006). The random perturbation of a standard map with a small value of λ (shown in Figure 7-right) shows exactly the same growth.…”