Finite extensibility of a polymer chain ($R\le Na$)

The end-to-end distance of a polymer chain with $N$ fixed-length links of length $a$ cannot exceed its polymer contour length $Na$. A Gaussian chain spring has no such bound, so it eventually becomes an unphysical extrapolation. For a three-dimensional freely jointed chain under force $f$, the mean extension along the force is $Na[\coth x-1/x]$, where $x=fa/(k_{B}T)$. Its small-$x$ limit is $Nf{a}^2/(3k_{B}T)$, and its large-$x$ limit approaches $Na$ from below.