On the stupendous beauty of closure

Abstract

Closure seems to be something rheologists would prefer to avoid. Here, the story of closure is told in such a way that one should enduringly forget any improper undertone of "uncontrolled approximation" or "necessary evil" which might arise, for example, in reducing a diffusion equation in configuration space to moment equations. In its widest sense, closure is associated with the search for self-contained levels of description on which time-evolution equations can be formulated in a closed, or autonomous, form. Proper closure requires the identification of the relevant structural variables participating in the dominant processes in a system of interest, and closure hence is synonymous with focusing on the essence of a problem and consequently with deep understanding. The derivation of closed equations may or may not be accompanied by the elimination of fast processes in favor of dissipation. As a general requirement, any closed set of evolution equations should be thermodynamically admissible. Thermodynamic admissibility comprises much more than the second law of thermodynamics, most notably, a clear separation of reversible and irreversible effects and a profound geometric structure of the reversible terms as a hallmark of reversibility. We discuss some implications of the intimate relationship between nonequilibrium thermodynamics and the principles of closure for rheology, and we illustrate the abstract ideas for the rod model of liquid crystal polymers, bead-spring models of dilute polymer solutions, and the reptation model of melts of entangled linear polymers.