Confinement deflection length ($\lambda \sim (R^2{\ell }_p{)}^{1/3}$)

A stiff chain's transverse wandering over contour length $s$ scales as $s^3/{\ell }_p$. Equating this to the squared confining radius gives the displayed deflection length when $R\ll {\ell }_p$. A free-energy cost of order $k_{B}T$ per deflection yields confinement free energy per inserted contour length of order $k_{B}T/\lambda$. Numerical prefactors depend on geometry and boundary constraints.