Condition for $1/f$ noise to occur along with an example for a diffusion equation
While $1/f$ noise is ubiquitous and has been found in various systems, its physics remains uncertain. From an analytical study of an ordinary diffusion equation, we find an additional example of the $1/f$ noise. The formula for this example, together with existing knowledge about scaling in fluid turbulence, implies a necessary and sufficient condition for the occurrence of any stationary $1/f$ noise. That is, the noise needs to be characterized by two constant frequencies of $f_{\rm low} \ll f_{\rm high}$. For a frequency range from $f = f_{\rm low}$ to $f_{\rm high}$, it is further needed that, except for the mean amplitude of the noise, there is no other constant parameter. Then, at $f_{\rm low} \ll f \ll f_{\rm high}$, the noise scales asymptotically as $1/f$. Being statistical and simple, our condition applies to any system and hence explains the ubiquity of the $1/f$ noise. It is also applicable to some systems with noise of $\alpha \ne 1.0$ for $1/f^{\alpha}$, via intermittency analogous to that of the turbulence.