 Solution

Write the end-to-end random vector as $\mathbf{r}={\sum }_{j=1}^{N}b{\mathbf{u}}_j$, with independent isotropically distributed unit vectors. At zero applied force, $⟨{\mathbf{u}}_j⟩=0$, ${\mathbf{u}}_j\cdot {\mathbf{u}}_j=1$, and for $i\ne j$ independence gives $⟨{\mathbf{u}}_i\cdot {\mathbf{u}}_j⟩=0$. ThereforeHere “size” is the root mean square end-to-end distance of a polymer chain. It is not the magnitude of the mean random vector, which is zero, nor the mean distance $⟨r⟩$. In the long-chain Gaussian limit of the end-to-end distribution of a freely jointed chain, $⟨r⟩\simeq b\sqrt{8N/(3\pi )}$, illustrating why these definitions cannot be interchanged.