Errors, Correlations and Fidelity for noisy Hamilton flows. Theory and numerical examples
Résumé
We compare the decay of correlations and fidelity for some prototype noisy Hamiltonian flows on a compact phase space. The results obtained for maps on the torus T2 or on the cylinder T×I are recovered, in a simpler way, in the limit of vanishing time step Δt→0, if these maps are the symplectic integrators of the proposed flows. The mean square deviation σ(t) of the noisy flow asymptotically diverges, following a power law if the unperturbed flow is integrable, exponentially if it is chaotic. Correspondingly the fidelity, which measures the correlation at time t of the noisy flow with respect to unperturbed flow, decays as exp(−2π2σ2(t)). For chaotic flows the fidelity exhibits a plateau, followed by a super-exponential decay starting at t∗∝−logϵ where ϵ is the noise amplitude. We analyze numerically two simple models: the anharmonic oscillator the H\'enon-Heiles Hamiltonian and the N vortex system for N=3,4 The round-off error on the symplectic integrator acts as a (single realization of a) random perturbation provided that the map has a sufficiently high computational complexity. This can be checked by considering the reversibility error. Finally we consider the effect of the observational noise showing that the decrease of correlations or fidelity can only be observed, after a sequence of measurements. The multiplicative noise is more effective at least for long enough delay between two measurements.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...