Errors, Correlations and Fidelity for noisy Hamilton flows. Theory and numerical examples

TL;DR

This paper investigates how small random perturbations affect Hamiltonian flows, analyzing error growth, correlations, and fidelity through theoretical analysis and numerical examples, distinguishing behaviors in integrable and chaotic systems.

Contribution

It provides a comprehensive comparison of forward and reversibility errors, characterizes their growth laws, and examines the impact of observational noise on correlations in Hamiltonian systems.

Findings

Error grows as a power law in integrable systems.

Error grows exponentially in chaotic systems.

Correlations decay exponentially with the square of the error.

Abstract

We analyse the asymptotic growth of the error for Hamiltonian flows due to small random perturbations. We compare the forward error with the reversibility error, showing their equivalence for linear flows on a compact phase space. The forward error, given by the root mean square deviation $\sigma(t)$ of the noisy flow, grows according to a power law if the system is integrable and according to an exponential law if it is chaotic. The autocorrelation and the fidelity, defined as the correlation of the perturbed flow with respect to the unperturbed one, exhibit an exponential decay as $\exp\left(-\sigma^2(t)\right)$. Some numerical examples such as the anharmonic oscillator and the H\'enon Heiles model confirm these results. We finally consider the effect of the observational noise on an integrable system, and show that the decay of correlations can only be observed after a sequence of measurements and that the multiplicative noise is more effective if the delay between two measurements is large.