The distribution of end-to-end distances of the weakly bending rod model

The weakly bending rod model is an approximation to a worm-like chain in the limit where the ratio L(0)/P of the contour length L(0) to the persistence length P is not too large. The range of validity of the weakly bending rod model is investigated by deriving analytical expressions for its distribution of end-to-end distances P(L) and its moments <L(m) > and numerically comparing the results with corresponding values for the worm-like chain model. No general, closed form analytical expression for either P(L) or the average length <L> of a worm-like chain exists, so those quantities are obtained by Monte Carlo simulations. Exact analytical expressions for <L(2)> and <L(4)> for the worm-like chain are employed in the comparison of the computed moments. Moments calculated for the approximate distributions of Daniels and Yamakawa and Stockmayer are also compared with the others. In addition, P(L) and its moments for the alternative model of Winkler et al. are compared with the others. The weakly bending rod model gives a reasonably good account of both P(L) and <L(m) >, m = 1,2,4, over the range L(0)/P </= 0.6, but deviates significantly for L(0)/P >/= 1.0. In contrast, the alternative model of Winkler et al. yields rather poor results in the rodlike domain.