Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli

We calculate the transverse velocity fluctuations correlation function of a linear and homogeneous viscoelastic liquid by using a generalized Langevin equation (GLE) approach. We consider a long-ranged (power-law) viscoelastic memory and a noise with a long-range (power-law) auto-correlation. We first evaluate the transverse velocity fluctuations correlation function for conventional time derivatives C ^ N F ( k → , t ) and then introduce time fractional derivatives in their equations of motion and calculate the corresponding fractional correlation function. We find that the magnitude of the fractional correlation C ^ F ( k → , t ) is always lower than the non-fractional one and decays more rapidly. The relationship between the fractional loss modulus G F ″ ( ω ) and C ^ F ( k → , t ) is also calculated analytically. The difference between the values of G ″ ( ω ) for two specific viscoelastic fluids is quantified. Our model calculation shows that the fractional effects on this measurable quantity may be three times as large as compared with its non-fractional value. The fact that the dynamic shear modulus is related to the light scattering spectrum suggests that the measurement of this property might be used as a suitable test to assess the effects of temporal fractional derivatives on a measurable property. Finally, we summarize the main results of our approach and emphasize that the eventual validity of our model calculations can only come from experimentation.