Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer-Edwards-Prager-Frisch model revisited

摘要

Using a general Green function formulation, we re-derive, both, (i) Spitzerand his followers results for the winding angle distribution of the planarBrownian motion, and (ii) Edwards-Prager-Frisch results on the statisticalmechanics of a ring polymer entangled with a straight bar. In the statisticalmechanics part, we consider both cases of quenched and annealed topology. Amongnew results, we compute exactly the (expectation value of) the surface area ofthe locus of points such that each of them has linking number $n$ with a givenclosed random walk trajectory ($=$ ring polymer). We also consider thegeneralizations of the problem for the finite diameter (disc-like) obstacle andwinding within a cavity.