This paper establishes the precise relationship between the macroscopic class of factorized Rivlin–Sawyers equations and a class of microscopic-based stochastic models. The former is a well-established and popular class of rheological models for polymeric fluids, while the latter is a more recently introduced class of rheological models which combines aspects of network and reptation theory with aspects of continuum mechanic models. It is shown that the two models are equivalent in a defined sense under certain unrestrictive assumptions. The first part of the proof gives the functional relationship between the linear viscoelastic memory function of the Rivlin–Sawyers model and the probability density for creation times of random variables in the stochastic model. The main part of the proof establishes the relationship between the strain descriptions in each model by showing that the difference in corresponding strain expressions can be made arbitrarily small using the appropriate weighted norm from spectral approximation theory.
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