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A polymer chain pinned in space exerts a fluctuating force on the pin pointin thermal equilibrium. The average of such fluctuating force is wellunderstood from statistical mechanics as an entropic force, but little is knownabout the underlying force distribution. Here, we introduce two phase spacesampling methods that can produce the equilibrium distribution of instantaneousforces exerted by a terminally pinned polymer. In these methods, both thepositions and momenta of mass points representing a freely jointed chain areperturbed in accordance with the spatial constraints and the Boltzmanndistribution of total energy. The constraint force for each conformation andmomentum is calculated using Lagrangian dynamics. Using terminally pinnedchains in space and on a surface, we show that the force distribution is highlyasymmetric with both tensile and compressive forces. Most importantly, the meanof the distribution, which is equal to the entropic force, is not the mostprobable force even for long chains. Our work provides insights into themechanistic origin of entropic forces, and an efficient computational tool forunbiased sampling of the phase space of a constrained system.

Calculation of a fluctuating entropic force by phase space sampling