Doctor of Philosophy, The Ohio State University, 2011, Mathematics

In late 1950's M.V. Volkenstein and other chemists suggested a discretized model of polymers called Rotational Isomeric State approximation model (RIS model), in which torsional angles at each step of polymer configuration take values from a fixed finite set of angles. This model was further studied by P. Flory and others. One of the natural questions is what happens with polymer when number of its bonds tends to infinity. Investigation related to this question was not completely done at the time, while some results found then were not quite rigorously proved and remained justified by intuitive or heuristic arguments. The reason for this is because some mathematical techniques and results were either not known at the time RIS model was developed, or were discovered not long before. The work presented in this thesis is continuation of study on the RIS model done by Volkenstein, Flory and others. We consider what happens with the RIS polymer when the number of its bonds tends to infinity, and show that, under suitable scaling, it converges to the Kratky-Porod model. We rigorously prove (already known) convergence of the sequence of torsional angles of the polymer, which forms an inhomogeneous Markov chain, to some homogeneous Markov chain. We also show that the rate of this convergence is geometric. To prove that the RIS model converges to the Kratky-Porod model, we use sequence of stochastic rotations whose limit satisfies linear Stratonovich stochastic equation. Driving process of this equation is antisymmetric Gaussian stochastic matrix, which rises from the sequence of torsional angles.

Committee: Peter March PhD (Advisor); Saleh Tanveer PhD (Committee Member); Yuan Lou PhD (Committee Member); Dorinda Gallant PhD (Committee Member) Subjects: Mathematics; Molecular Chemistry