  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.
