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In this study, we propose a universal scenario explaining the $1/f$fluctuation, including pink noises, in Hamiltonian dynamical systems with manydegrees of freedom under long-range interaction. In the thermodynamic limit,the dynamics of such systems can be described by the Vlasov equation, which hasan infinite number of Casimir invariants. In a finite system, they becomepseudoinvariants, which yield quasistationary states. The dynamics then exhibitslow motion over them, up to the timescale where the pseudo-Casimir-invariantsare effective. Such long-time correlation leads to $1/f$ fluctuations ofcollective variables, as is confirmed by direct numerical simulations. Theuniversality of this collective $1/f$ fluctuation is demonstrated by taking avariety of Hamiltonians and changing the range of interaction and number ofparticles.

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