Marginal Distributions at the Tip of a Grafted Stiff Polymer: Analytical and Monte Carlo Investigations

This study investigates the marginal distributions of grafted stiff polymer tips in a 2D embedding space using both analytical methods and Monte Carlo simulations. By mapping active Brownian particle (ABP) trajectories in the short-time regime, analytical expressions for the elongation of the free end of the polymer under horizontal and vertical forces are derived and these expressions are validated using Monte Carlo simulations. These results indicate that the theoretical predictions match well with the simulation results when the chain length is short or the force is large. However, a slight discrepancy is observed between the theoretical and simulation results when the chain is extremely long, although the qualitative asymptotic results remain valid. Additionally, expressions are provided for the horizontal and vertical force versus displacement for the wormlike chain under the weakly bending approximation. This research provides insights into how the trajectories of an ABP correspond to the equilibrium configuration of a semiflexible polymer. These findings have potential applications in various fields, including biophysics and materials science, where understanding the behavior of grafted polymers is crucial.