Total curvature and total torsion of a freely jointed circular polymer with n ≫ 1 segments

A study was conducted to find that both total curvature and total torsion of the closed random polygon are not equal to π/2 per bond, but have corrections proportional to 1/n with universal coefficients close to 1.2 by using different methods. Many geometrical characteristics of closed polymer loops modeled as polygons of n > 1 edges. The joint probability distribution of all vectors was considered in the closed polygon to ensure the correct normalization. The study suggested that nπ/2n is an average angle of n turn for the polygon, because of the connection of smooth curves. It is concluded that both total curvature an total torsion deviates from the expected linear in n behavior and correction to both curvature and torsion do not depend on n because of opposite signs.