STATISTICAL MECHANICS OF SELF-AVOIDING CRUMPLED MANIFOLDS — Part II

Abstract

†We consider a model of a D-dimensional tethered manifold interacting by excluded volume in IR d with a single point. Use of intrinsic distance geometry provides a rigorous definition of the analytic continuation of the perturbative expansion for arbitrary D, 0 < D < 2. Its oneloop renormalizability is first established by direct resummation. A renormalization operation R is then described, which ensures renormalizability to all orders. The similar question of the renormalizability of the self-avoiding manifold (SAM) Edwards model is then considered, first at one-loop, then to all orders. We describe a short-distance multi-local operator product expansion, which extends methods of local field theories to a large class of models with non-local singular interactions. It vindicates the direct renormalization method used earlier in part I of these lectures, as well as the corresponding scaling laws. ⋆

Non-entropic theory of rubber elasticity: Flexible chains grafted on a rigid surface

“…Although the necessity to go beyond Gaussian chains has been realized for a long time (the seminal paper by Flory [2] appeared more than half a century ago), our knowledge of the mechanical behavior of networks of non-Gaussian chains remains rather limited, due to some difficulties in their mathematical treatment. Serious progress in the analysis of statistics of single polymer chains and membranes with excluded-volume interactions was reached by using the renormalization group technique [3,4,5,6]. These methods, however, have been employed for the analysis of the distribution functions only and have not yet been applied to determine the strain energy of a network, despite the importance of the latter problem for applications, see [7,8] and the references therein, as well as recent publications [9,10,11,12].…”