Multifractal analysis of light scattering-intensity fluctuations

We provide a simple interpretation of non-Gaussian nature of the light scattering-intensity fluctuations from an aging colloidal suspension of Laponite using the multiplicative cascade model, Markovian method, and volatility correlations. The cascade model and Markovian method enable us to reproduce most of recent empirical findings: long-range volatility correlations and non-Gaussian statistics of intensity fluctuations. We provide evidence that the intensity increments $\Delta x\left(\tau \right)=I\left(t+\tau \right)-I\left(t\right)$, upon different delay time scales $\tau$, can be described as a Markovian process evolving in $\tau$. Thus, the $\tau$ dependence of the probability density function $p\left(\Delta x,\tau \right)$ on the delay time scale $\tau$ can be described by a Fokker-Planck equation. We also demonstrate how drift and diffusion coefficients in the Fokker-Planck equation can be estimated directly from the data.

Figure 1

(Color online) Typical light scattering-intensity fluctuations measured from an interacting colloidal suspension of Laponite with concentration 3.2 wt % as a function of time. In this paper we express intensities in terms of the detector count rate in kHz.

Figure 2

(Color online) Upper panel shows the scaling behavior of structure function $M\left(q,\tau \right)$ for various values of $q$. Lower left panel corresponds to the scaling exponent ${\xi }_q$ of light scattering-intensity data (filled circle symbol) as a function of moment order $q$. In this panel, solid lines are given for a monofractal process with Hurst exponent equal to $H=0.92$. Long-dashed line is given by Eq. (19). Lower right panel indicates ${\zeta }_q$ versus $q$ (the exponents derived using the extended self-similarity method). Solid line shows the exponents for a Gaussian process, ${\zeta }_q=q/3$.

Figure 3

(Color online) Continuous deformation of the intensity increments’ PDFs across scales (from bottom to top) $\tau =2000,1000,500,250,150,25\left(406\mu s\right)$. Solid lines for each $\tau$’s are given by Eq. (7) and symbols are directly computed by data set. Long-dashed curve corresponds to the Gaussian probability density function. Curves are shifted in vertical direction for clarity of presentation.

Figure 4

(Color online) Upper panel corresponds to the correlation function of light intensity increments, $C_q\left(l,\tau \right)$ for $l=1$ versus $\tau$ for $q=1$ (filled circle), $q=2$ (filled square), and $q=3$ (filled diamond). Lower panel shows the estimation of the power-law exponent $C_q\left(l,\tau \right)\sim {\tau }^{{\nu }_q}$ derived by direct calculation for small value of $\tau$ (filled circle symbol) and theoretical prediction using ${\lambda }_0=0.077\pm 0.054$ and given by Eq. (52).