An exact density-functional closure for the two-dimensional discrete wormlike chain

Abstract

We formulate an exact density-functional description of the angularly discretized two-dimensional (2D) discrete wormlike chain (DWLC) under tension. The central result is a bond-local closure connecting the nearest-neighbor pair distribution $C_{i,i+1}^{rs}$ to the single-site angular densities ${\rho }_i^r$: $C_{i,i+1}^{rs}{C}_{i,i+1}^{00}/(C_{i,i+1}^{r0}{C}_{i,i+1}^{0s})=e^{\bar{\beta }rs}$, with the bond coupling $\bar{\beta }=\beta \kappa {\epsilon }^2/a$. This relation follows directly from the connected Boltzmann weight of the quadratic bending interaction and permits exact integration of the entropy functional. Variational minimization then yields coupled self-consistent equations for the one- and two-site angular distributions, which we solve by fixed-point iteration. We establish the equivalence of this density-functional formulation to the exact transfer-matrix solution: the two approaches reproduce the angular marginals and force-extension curves to machine precision. The formulation also recovers the continuum 2D wormlike-chain behavior, including the rigid-rod and random-coil limits of the mean-square end-to-end distance. Using the segment length and persistence length taken directly from Mazur's short-DNA molecular-dynamics study, without additional fitting, the predicted bend-angle statistics agree with the simulation data within the estimated uncertainty. The closure structure extends naturally to nonharmonic local bending interactions and can also be interpreted inversely, allowing measured nearest-neighbor angular correlations to constrain effective coarse-grained bending potentials. These properties establish a direct connection between conformational statistics and local interactions and provide a density-level framework for treating interacting semiflexible-polymer systems.