Fractional correlation functions in simple viscoelastic liquids

We develop a hydrodynamic formulation of fractional fluctuations in a viscoelastic liquid whose longitudinal modulus is a scalar and only depends on time. The method is based on the introduction of fractional time derivatives in the hydrodynamic equations due to the viscoelastic memory. Coupled generalized Langevin equations for the fluctuations of the state variables are formulated and solved analytically for a power-law memory kernel with long correlation noise. The associated fractional Fokker–Planck equation (FFPE) is also derived and it is shown that it exhibits a fluctuation–dissipation theorem (FDT). The explicit analytic diffusion coefficient is calculated for power law viscoelasticity and is shown to be subdiffusive. The fractional correlations for longitudinal velocity and density fluctuations are analytically calculated and used to obtain the light scattering spectrum and the intermediate scattering function of the viscoelastic fluid. Our model calculation predicts that the fractional effects on these properties are not small effects and might be measurable.