Scaling Laws and Universal Features of Tethered Polymer Distributions in Confined Geometries

Abstract

We develop a unified scaling framework for the end-position distributions of tethered polymers confined in finite cylindrical geometries. Two observables are analysed: the longitudinal distribution P(x), along the confinement axis, and the transverse distribution P(y), perpendicular to the confinement axis. Using exact Fourier-sine and image-method representations with adaptive numerical schemes, we construct and test six scaling strategies for P(x) and five for P(y), encompassing geometric similarity, tether-position sweeps, confinement-strength crossovers, persistence-length effects, boundary-layer scaling near absorbing walls, and tether-centered coil scaling. Quantitative collapse diagnostics such as RMS residuals on common support, modal-energy fractions, and survival probabilities are combined with limiting-regime analysis and direct numerical evaluation to distinguish genuine universality from visually misleading overlap. From these tests we obtain a kappa-based confinement diagram and a two-parameter (kappa, a/L) regime map that link classical theories such as Flory/de Gennes blobs, Odijk deflection segments, and wormlike-chain behaviour within a single spectral picture. Gaussian, multimode, and eigenmode-dominated regimes are identified by explicit thresholds in modal composition and collapse error, providing operational criteria for when Gaussian or single-mode descriptions are valid and when full multimode structure is required. The resulting framework provides a compact, reproducible toolkit for analysing confined-polymer statistics, with applications to simulations and experiments on DNA, chromatin, and other biopolymers where confinement, stiffness, and tethering jointly control spatial organization.