# FCS and RICS Spectra of Probes in Complex Fluids

## TL;DR

This paper derives the relationships between particle displacement distributions and FCS/RICS spectra, highlighting how non-Gaussian behaviors in complex fluids can lead to significant errors in diffusion measurements if Gaussian assumptions are incorrectly applied.

## Contribution

It provides the first general theoretical framework linking P(x,t) to FCS and RICS spectra in complex fluids, including non-Gaussian displacement distributions.

## Key findings

- Gaussian and exponential P(x,t) lead to different spectral forms
- Misinterpreting non-Gaussian P(x,t) causes errors in diffusion coefficients
- Theoretical comparison shows importance of correct P(x,t) assumptions

## Abstract

The Fluorescence Correlation Spectroscopy (FCS) spectrum G(t) and Raster Image Correlation Spectroscopy (RICS) spectrum R(t) of dilute diffusing particles are determined by the displacement distribution function P(x,t) of the particles and by the experimental parameters of the associated optical trains. This letter obtains the general relationships between P(x,t) and these spectra. For dilute diffusing molecules in simple liquids, P(x,t) is a Gaussian in the displacement x; the corresponding G(t) is a Lorentzian in (<(x(t))^2>)^(1/2). In complex fluids such as polymer solutions, colloid and protein solutions, and the interior of living cells, P(x,t) may have a non-Gaussian dependence on x, for example an exponential in |x|. We compare theoretical forms for FCS and for RICS spectra of two systems in which P(x,t) is a Gaussian or an exponential in x, but in which the mean-square displacements are precisely equal at all times. If the G(t) and R(t) arising from an exponential P(x,t) are interpreted by using the forms for G(t) and R(t) that are appropriate for a Gaussian P(x,t), the inferred diffusion coefficient may be substantially in error.