Scaling theory for the $1/f$ noise

Abstract: We show that in a broad class of processes that show a $1/f^{\alpha}$ spectrum, the power also explicitly depends on the characteristic time scale. Despite an enormous amount of work, this generic behavior remains so far overlooked and poorly understood. An intriguing example is how the power spectrum of a simple random walk on a ring with $L$ sites shows $1/f^{3/2}$ not $1/f^2$ behavior in the frequency range $1/L^2 \ll f \ll 1/2$. We address the fundamental issue by a scaling method and discuss a class of solvable processes covering physically relevant applications.