End-point probability distributions of tethered polymers: Theory and Monte Carlo simulations in confined geometries

A combined theoretical and numerical study of end-point probability distributions [P(x)] and [P(y)] for a polymer tethered inside a cylindrical confinement is presented here. Analytical solutions of the diffusion equation (Fourier series and image methods) are benchmarked against large-scale Monte Carlo simulations of three canonical polymer architectures: the Freely Jointed Chain (FJC), Self-Avoiding Walk (SAW), and Worm-Like Chain (WLC). To overcome the severe sampling attrition for off-lattice SAW in confinement, a chain-length-adaptive hybrid sampling strategy has been developed by employing kinetic growth for short chains, an enhanced pruned-enriched Rosenbluth method in the crossover regime, and an off-lattice pivot algorithm for long chains, all implemented with Numba JIT (Just-In-Time) acceleration to reach statistically converged ensembles. The agreement is excellent for the FJC vs the Fourier propagator, while the SAW and the WLC show systematic, model-specific departures driven by excluded volume and bending rigidity. Using information-theoretic (Kullback-Leibler), distributional (Kolmogorov-Smirnov, moments), and several other statistical metrics, three distinct regimes (short-chain, crossover, and long-chain saturation) have been identified. In addition, the longitudinal probability distribution has been analyzed for different parameter regimes, such as the location of the tethering point and persistence lengths. The work delivers a validated, multi-scale computational toolkit along with the development of a Streamlit-based interactive application for interpreting confined-polymer experiments and for selecting appropriate polymer models in biological and nanoconfined settings.