Stochastic fractal behavior in concentration fluctuation and fluorescence correlation spectroscopy

Fluctuations in the concentration of Brownian particles in one and two dimensions, or any reasonable measurement of the concentration such as in fluorescence correlation spectroscopy, is shown to be a stochastic fractal with a long tail. Being singular at omega = 0, the power spectrum of the fluctuation S(omega) approximately omega-1/2 for diffusion in one dimension, approximately log omega in two dimensions, but non-singular in three dimensions. This discovery provides one simple physical mechanism for possible long-memory fractal behavior, and its implications to various biological processes are discussed.