Exact solution to ideal chain with fixed angular momentum

The statistical mechanics of a noninteracting polymer chain in the limit of a large number of monomers is considered when the total angular momentum $\mathbf{L}$ is fixed. The radius of gyration for a ring polymer in this situation is derived exactly in closed form by functional integration techniques. Even when $L=0$ the radius of gyration differs from that of a random walk by a prefactor of order unity. The dependence on $L$ is discussed qualitatively and the large-$L$ limit can be understood by physical arguments, which can also be extended to self-avoiding chains.

Figure 1

Radius of gyration versus rescaled angular momentum for an ideal ring chain.

Figure 2

Probability density function for finding a chain in thermal equilibrium with rescaled angular momentum $L^{\prime }$ for an ideal ring chain.