Shape of Random Walk and Size of SelfAvoiding Manifold.
Abstract
This thesis studies the anisotropy of random walks and the size of an ndimensional manifold with interactions. In particular, we look at a free selfavoiding manifold and one that is immersed in a medium consisting of randomly distributed impurities, and a polymer (n = 1) confined to a region between two infinite, parallel, repelling plates. The anisotropy is characterized by the quantity Delta = < tr{ bf Q}^2/(tr{bf Q})^2 >, averaged over an ensemble of random walks, where Q is the 'shape tensor', and {bf Q} = {bf Q}  {1 over d} {rm tr}{bf Q}. The approximation Delta _{o}equiv< tr{ bf Q}^2>/<(tr{ bf Q})^2> is frequently used in the literature. This differs significantly from Delta. In this thesis we use the integral representation 1/({rm tr} {bf Q})^2 = int_sp{0 }{infty} dx x exp( {x} {rm tr}{bf Q}) together with Gegenbauer polynomials to calculate Delta. We find that the asphericity parameter has a minimum near d = 4, implying that the random walk is more spherical in intermediate dimensions than in low or high dimensions. Treating y trQ (informally) as a free energy provides a bridge connecting polymers with surfaces (n = 2), examples of which include the inner layer of red blood cells, plaquette ensembles in lattice gauge theories, and interfaces. A particular (continuous) model of a surface, and its generalization, the (n>2)dimensional manifold, is based on Edwards' polymer model. Renormalization group (RG) calculations, in the literature, of the size exponent, nu, for such a surface are patently incorrect, and are ambiguous when specialized to twodimensional polymers (n = 1). We analyze this model using the "effective step length" method. We find that manifolds are more extended than RG indicates. A tethered membrane in an annealed random environment undergoes a transition from an extended state, with size exponent nu = 4/5, to a 'crumpled' state, when the density of the impurities approaches a certain critical value. This is consistent with the behavior of polymers reported previously in the literature. For a polymer in a slit of width D, we find that the size scales as R~ N^{3/4 }D^{1/4} as D/R_ {F}to0, R_{F} the Flory radius of the chain. For D/R_ {F}toinfty, we recover the threedimensional result, R~ N^ {3/5}..
 Publication:

Ph.D. Thesis
 Pub Date:
 1993
 Bibcode:
 1993PhDT.......172M
 Keywords:

 Physics: General