Slowing down of relaxation in a complex system by constraint dynamics

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Abstract
In this paper we view a relaxing complex system such as entangled polymer melt to consist of three parts: (1) an individual primary species PS of interest; (2) a heat bath (HB) whose interaction with the PS provides the primary mechanism of relaxation; (3) other relaxing species whose interactions with the PS, the PS-C coupling, are for us the principal characteristic of complexity. The PS-C coupling is represented by time dependent constraints whose effect begins only after the primary relaxation process due to the PS-HB interaction is already underway. The overall process is described both physically and theoretically. The latter is described classically by means of time dependent Dirac constraint theory applied to a Liouville operator formalism. The physical and theoretical discussion leads to a time dependent relaxation rate W(t). The specific form of W(t) is adduced based on the requirement of time-temperature equivalence or thermorheological simplicity. The result is a time independent relaxation rate W0 for times short compared to the onset of the effect of the time dependent constraints at tc=ω−1c, and a time dependent rate W0(ωct)−n for times long compared to tc. The case W0tctc.