  Loops undergoing thermal fluctuations are prevalent in nature. Ring-like or
cross-linked polymers, cyclic macromolecules, and protein-mediated DNA loops
all belong to this category. Stability of these molecules are generally
described in terms of free energy, an average quantity, but it may also be
impacted by local fluctuating forces acting within these systems. The full
distribution of these forces can thus give us insights into mechanochemistry
beyond the predictive capability of thermodynamics. In this paper, we study the
force exerted by an inextensible semiflexible polymer constrained in a looped
state. By using a novel simulation method termed "phase-space sampling", we
generate the equilibrium distribution of chain conformations in both position
and momentum space. We compute the constraint forces between the two ends of
the loop in this chain ensemble using Lagrangian mechanics, and show that the
mean of these forces is equal to the thermodynamic force. By analyzing kinetic
and potential contributions to the forces, we find that the mean force acts in
the direction of increasing extension not because of bending stress, but in
spite of it. Furthermore, we obtain a distribution of constraint forces as a
function of chain length, extension, and stiffness. Notably, increasing contour
length decreases the average force, but the additional freedom allows
fluctuations in the constraint force to increase. The force distribution is
asymmetric and falls off less sharply than a Gaussian distribution. Our work
exemplifies a system where large-amplitude fluctuations occur in a way
unforeseen by a purely thermodynamic framework, and offers novel computational
tools useful for efficient, unbiased simulation of a constrained system.
