New explicit solutions are obtained for the nonlinear equations of Kirchhoff’s theory of the dynamics of inextensible elastic rods without neglect of rotatory inertia. These exact solutions describe a class of motions possible in closed circular rings possessing a uniform distribution of intrinsic curvature ku and intrinsic torsion. When ku≠0, the motions in this class are such that the axial curve of the ring remains stationary while the cross sections rotate about their centers in such a way that the angle ψ of rotation is independent of axial location and is governed by the nonlinear pendulum equation. When ku=0, such uniform rotation of cross sections can occur at an arbitrary steady rate. The methods of classical equilibrium statistical mechanics yield the following conclusions for canonical ensembles of rings for which the motion is this type of pure homogeneous torsion. When 1/ku=11.85 nm (i.e., when the intrinsic curvature ku is among the highest observed in naturally occurring, approximately uniformly curved, stress-free DNA segments), if the flexural rigidity is assigned a value usually accepted for duplex DNA, at T=298 K the root-mean-square value, 〈ψ2〉1/2, of the angle ψ is 11.2°. For motions in this class, the heat capacity per ring, as a function of T/ku, shows a maximum which, when T=298 K, occurs where 1/ku=127 nm and corresponds to an ensemble of rings of which approximately 1% have sufficient energy for escape over the barrier associated with the separatrix between periodic and monotone solutions of the nonlinear pendulum equation; for that ensemble of rings, 〈ψ2〉1/2=43.3°.

Tobias, I., Coleman, B. D., & Lembo, M. (1996). A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids. The Journal of Chemical Physics, 105(6), 2517–2526.