Writhing Geometry at Finite Temperature: Random Walks and Geometric phases for Stiff Polymers

These lecture notes are a guided tour through the fascinating world of polymer chains interacting with themselves and/or with their environment. The focus is on the mathematical description of a number of physical and chemical phenomena, with particular emphasis on phase transitions and space-time scaling. The topics covered, though only a selection, are typical for the area. Sections 1-3 describe models of polymers without disorder, Sections 4-6 models of polymers with disor...

The Fourier-Bessel expansion of a function on a circular disc yields a simple series representation for the end-to-end probability distribution function w(R,phi) encountered in a planar persistent random walk, where the direction taken in a step depends on the relative orientation towards the preceding step. For all but the shortest walks, the proposed method provides a rapidly converging, numerically stable algorithm that is particularly useful for the precise study of inter...

We introduce a simple geometric model for a double-stranded and double-helical polymer. We study the statistical mechanics of such polymers using both analytical techniques and simulation. Our model has a single energy-scale which determines both the bending and twisting rigidity of the polymer. The helix melts at a particular temperature T_c below which the chain has a helical structure and above which this structure is disordered. Under extension we find that for small forc...

We describe a simple meanfield variational approach to study a number of properties of intrinsically stiff chains which are appropriate models for a large class of biopolymers. We present the calculation of the distribution of end-to-end distance and the elastic response of stiff chains under tension using this approach. In the former example we find that the simple expression almost quantitatively fits the results of computer simulation. For the case of the stiff chain under...

Employing Monte Carlo simulations of semiflexible polymer rings in weak spherical confinement a conformational transition to figure eight shaped, writhed configurations is discovered and quantified.

We propose a stochastic method to generate exactly the overdamped Langevin dynamics of semi-flexible Gaussian chains, conditioned to evolve between given initial and final conformations in a preassigned time. The initial and final conformations have no restrictions, and hence can be in any knotted state. Our method allows the generation of statistically independent paths in a computationally efficient manner. We show that these conditioned paths can be exactly generated by a ...

We have studied the compact phase conformations of semi-flexible polymer chains confined in two dimensional nonhomogeneous media, modelled by fractals that belong to the family of modified rectangular (MR) lattices. Members of the MR family are enumerated by an integer $p$ $(2\leq p<\infty)$, and fractal dimension of each member of the family is equal to 2. The polymer flexibility is described by the stiffness parameter $s$, while the polymer conformations are modelled by wei...

This paper generalizes the Gaussian random walk and Gaussian random polygon models for linear and ring polymers to polymer topologies specified by an arbitrary multigraph $G$. Probability distributions of monomer positions and edge displacements are given explicitly and the spectrum of the graph Laplacian of $G$ is shown to predict the geometry of the configurations. This provides a new perspective on the James-Guth-Flory theory of phantom elastic networks. The model is based...

The constraints imposed by nano- and microscale confinement on the conformational degrees of freedom of thermally fluctuating biopolymers are utilized in contemporary nano-devices to specifically elongate and manipulate single chains. A thorough theoretical understanding and quantification of the statistical conformations of confined polymer chains is thus a central concern in polymer physics. We present an analytical calculation of the radial distribution function of harmoni...

We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves. Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any singl...