Generalized-Regression-of-Fluctuations Theory as Applied to the Second Viscosity

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Abstract
A generalized-regression-of-fluctuations (GRF) theory is developed for the study of the second viscosity, by making use of a generalized transport equation describing the evolution of a dynamical variable in a dense fluid. This method constitutes a generalization of the Mandelstam–Leontovitch model. We obtain a formal expression for the frequency-dependent second viscosity in terms of the kernel of the transport equation. We show that this result is equivalent to the expression obtained from the autocorrelation formalism approach to thermal transport coefficients. This equivalence is easily understood on the basis of the linear response theory; the above result is therefore not restricted to the case of the second viscosity, and constitutes a general exact result within the limits of validity of the theory of regression of fluctuations. We next derive a dispersion equation which is solved for temporal attenuation (light scattering) and for spatial attenuation (ultrasonics) to yield the first sound velocity and the absorption coefficient in both cases. As an example, a simple model is introduced to calculate explicitly the second viscosity, the sound velocity, and the attenuation. Comparison is also made with the single-relaxation-time theory.