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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., Comment: 15 pages, 9 figures

A polymer chain pinned in space exerts a fluctuating force on the pin point in thermal equilibrium. The average of such fluctuating force is well understood from statistical mechanics as an entropic force, but little is known about the underlying force distribution. Here, we introduce two phase space sampling methods that can produce the equilibrium distribution of instantaneous forces exerted by a terminally pinned polymer. In these methods, both the positions and momenta of mass points representing a freely jointed chain are perturbed in accordance with the spatial constraints and the Boltzmann distribution of total energy. The constraint force for each conformation and momentum is calculated using Lagrangian dynamics. Using terminally pinned chains in space and on a surface, we show that the force distribution is highly asymmetric with both tensile and compressive forces. Most importantly, the mean of the distribution, which is equal to the entropic force, is not the most probable force even for long chains. Our work provides insights into the mechanistic origin of entropic forces, and an efficient computational tool for unbiased sampling of the phase space of a constrained system., Comment: 15 pages, 4 figures