1/f fluctuations in diffusion processes of effective space dimension less than one

Abstract

It is shown that the stationary fluctuations described by the diffusion equation of an effective space dimension less than one have a 1/f spectrum. A physical diffusion model equivalent to such a low dimensional diffusion process is proposed. Here, a singular disturbance of temperature fluctuation is assumed to diffuse with the space-dependent diffusion coefficient D(r)=D0(1+κr−m)θ. Here, D0 is the constant part of the diffusion coefficient and κr−m represents the effect of local temperature enhancement of the background medium caused by the singular disturbance itself, where r is the distance from the singular disturbance at r=0, and m and θ positive constants.