Collective $1/f$ fluctuation by pseudo-Casimir-invariants

TL;DR

This paper explains the universal emergence of $1/f$ noise in Hamiltonian systems with many degrees of freedom and long-range interactions, attributing it to pseudo-Casimir invariants causing slow dynamics and long-time correlations.

Contribution

It introduces a universal scenario linking pseudo-Casimir invariants in finite systems to $1/f$ fluctuations, supported by numerical simulations across various Hamiltonians.

Findings

$1/f$ fluctuations observed in collective variables.

Pseudo-Casimir invariants cause slow dynamics and correlations.

Universality confirmed across different Hamiltonians and interaction ranges.

Abstract

In this study, we propose a universal scenario explaining the $1/f$ fluctuation, including pink noises, in Hamiltonian dynamical systems with many degrees of freedom under long-range interaction. In the thermodynamic limit, the dynamics of such systems can be described by the Vlasov equation, which has an infinite number of Casimir invariants. In a finite system, they become pseudoinvariants, which yield quasistationary states. The dynamics then exhibit slow motion over them, up to the timescale where the pseudo-Casimir-invariants are effective. Such long-time correlation leads to $1/f$ fluctuations of collective variables, as is confirmed by direct numerical simulations. The universality of this collective $1/f$ fluctuation is demonstrated by taking a variety of Hamiltonians and changing the range of interaction and number of particles.