A density-functional closure for the two-dimensional discrete wormlike chain

We develop a density-functional description of the two-dimensional (2D) discrete wormlike chain model for a single semiflexible polymer under tension. The central result is an exact closure relation that connects the pair angular distribution Ci,i+1rs to the single-site angular density ρir; it takes the compact form Ci,i+1rsCi,i+100/(Ci,i+1r0Ci,i+10s)=eβ̄rs, with the bond coupling β̄=βκε2/a following transparently from the quadratic bending energy. The closure permits an exact integration of the entropy functional, and minimization of the free energy yields a coupled set of self-consistent equations for ρir and Ci,i+1rs that we solve by fixed-point iteration. Because the single 2D chain is a nearest-neighbor one-dimensional system, its partition function is also accessible by an elementary transfer matrix; we use this as an exact benchmark and show that the self-consistent density-functional solution reproduces the transfer-matrix marginals and force-extension curve to machine precision. We then compare with the continuum 2D wormlike chain, recover the rigid-rod and random-coil limits through the mean-square size ⟨R2⟩, and benchmark bend-angle statistics against the short-DNA molecular-dynamics data of Mazur. The value of the closure is not that it outperforms the transfer matrix for one chain but that it is a density functional, and therefore a natural starting point for the interacting multi-chain regime where transfer-matrix methods become intractable; we identify this as the principal direction for future development.