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

– 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 > of G(r, N) can be calculated exactly by calculating the (1,1) matrix element of 2n-th power of a truncated matrix of rank n + 1. The theory of flexible polymers is now understood [1][4]. Many polymer molecules have internal stiffness and cannot be modeled by the model of flexible molecules developed by Edwards. This is especially true for several important biopolymers such as actin, DNA, and microtubules [5]. Models of semiflexible polymers have also applications in different topics besides polymer physics [6]. If the chain length decreases, the chain stiffness becomes an important factor. A quantitative measure for the stiffness of the polymer is the persistence length lp, which is the correlation length for the tangent-tangent correlation function along the polymer. Polymers with contour length L much larger than lp are flexible and are described by using the tools of quantum mechanics and quantum field theory [1][4]. The standard coarse-graining model of a wormlike or a semiflexible polymer was proposed by Kratky and Porod [7]. A few first moments of G(r,N) were computed in [8][10]. The literature on the computation of G(r,N) and its moments can be found in the book by Yamakawa [11]. For recent work see [12][13]. In this Letter we will study the problem of computation of the distribution functionG(r,N) of the end-to-end distance of a semiflexible polymer, which is described by Kratky-Porod model, by using the analogy of the worm like chain with the quantum rigid rotator in an external homogeneous field within the quantum mechanical propagator method [14]. Relating the combinatorics of counting the paths contributing to G(r,N) to random walks with