Event-Driven Power-Law Relaxation in Weak Turbulence

We characterize the spectral properties of weak turbulence in a liquid crystal sample driven by an external electric field, as a function of the applied voltage, and we find a $1/f$ noise spectrum S(f)∝1/f^η within the whole range 0<η<2. We theoretically explore the hypothesis that the system complexity is driven by non-Poisson events resetting the system through creation and annihilation of coherent structures, retaining no memory of previous history (crucial events). We study the time asymptotic regime by means of the density ψ(τ) of the time distances between two crucial events, yielding η=3-μ, where μ is defined through the long-time form ψ(τ)∝1/τ^μ, with 1<μ<3. The system regression to equilibrium after an abrupt voltage change experimentally confirms the theory, proving violations of the ordinary linear response theory for both η>1 and η<1.