Worm-like polymer loops and Fourier knots

Every smooth closed curve can be represented by a suitable Fourier sum as a function of an arbitrary parameter τ. We show that the ensemble of curves generated by randomly chosen Fourier coefficients with amplitudes inversely proportional to spatial frequency (with a smooth exponential cutoff) can be accurately mapped on the physical ensemble of inextensible worm-like polymer loops. The τ → s mapping of the curve parameter τ on the arc length s of the inextensible polymer is achieved at the expense of coupling all Fourier harmonics in a non-trivial fashion. We characterize the obtained ensemble of conformations by looking at tangent-tangent and position-position correlations. Measures of correlation on the scale of the entire loop yield a larger persistence length than that calculated from the tangent-tangent correlation function at small length scales. The topological properties of the ensemble, randomly generated worm-like loops, are shown to be similar to those of other polymer models.