Curvature Distribution of Worm-like Chains in Two and Three Dimensions

Bending of worm-like polymers carries an energy penalty which results in the appearance of a persistence length l such that the polymer is straight on length scales smaller than l and bends only on length scales larger than this length. Intuitively, this leads us to expect that the most probable value of the local curvature of a worm-like polymer undergoing thermal fluctuations in a solvent, is zero. We use simple geometric arguments and Monte Carlo simulations to show that while this expectation is indeed true for polymers on surfaces (in two dimensions), in three dimensions the probability of observing zero curvature anywhere along the worm-like chain, vanishes.