The distribution function of a semiflexible polymer and random walks with constraints

TL;DR

This paper derives an exact mathematical framework for the end-to-end distribution function of a semiflexible polymer, linking it to constrained random walks and algebraic structures, enabling precise calculations of moments.

Contribution

It introduces a novel mapping of the polymer distribution problem onto constrained random walks related to Temperley-Lieb algebra, providing exact formulas and recursion relations for the distribution function.

Findings

Derived an exact expression for the Fourier-Laplace transform of the distribution function.

Established a recursion relation for computing the distribution function directly.

Calculated moments of the distribution function exactly using matrix methods.

Abstract

In studying the end-to-end distribution function $G(r,N)$ of a worm like chain by using the propagator method we have established that the combinatorial problem of counting the paths contributing to $G(r,N)$ can be mapped onto the problem of random walks with constraints, which is closely related to the representation theory of the Temperley-Lieb algebra. By using this mapping we derive an exact expression of the Fourier-Laplace transform of the distribution function, $G(k,p)$, as a matrix element of an inverse of an infinite rank matrix. Using this result we also derived a recursion relation permitting to compute $G(k,p)$ directly. We present the results of the computation of $G(k,N)$ and its moments. The moments $<r^{2n}>$ of $% G(r,N)$ can be calculated \emph{exactly} by calculating the (1,1) matrix element of $2n$-th power of a truncated matrix of rank $n+1$.