Stochastic fractal behavior in concentration fluctuation and fluorescence correlation spectroscopy

Area of Science:

• Physics
• Physical Chemistry
• Complex Systems

Background:

• Brownian motion is fundamental to understanding particle dynamics.
• Fluctuations in particle concentration are observed in various scientific measurements.
• Fractal behavior and long-memory effects are observed in complex systems.

Purpose of the Study:

• To investigate the nature of concentration fluctuations of Brownian particles.
• To determine the power spectrum of these fluctuations across different dimensions.
• To establish a physical mechanism for fractal behavior and long-memory effects.

Main Methods:

• Analyzing fluctuations in particle concentration in one and two dimensions.
• Calculating the power spectrum S(omega) of these fluctuations.
• Comparing results for one, two, and three-dimensional diffusion.

Main Results:

• Concentration fluctuations demonstrate stochastic fractal properties with a long tail.
• The power spectrum S(omega) shows a singularity at omega = 0.
• Specific power-law dependencies were identified: S(omega) ~ omega-1/2 in 1D and S(omega) ~ log omega in 2D, with no singularity in 3D.

Conclusions:

• A simple physical mechanism for long-memory fractal behavior has been identified.
• The findings have significant implications for understanding biological processes involving particle diffusion.
• The dimensional dependence of fluctuation spectra provides a new perspective on diffusion dynamics.