Apparent violation of equipartition of energy in constrained dynamical systems

We propose a planar chain system, which is a simple mechanical system with a constraint. It is composed of $N$ masses connected by $N-1$ light links. It can be considered as a model of a chain system, e.g., a polymer, in which each bond is replaced by a rigid link. The long time average of the kinetic energies of the masses in this model is numerically computed. It is found that the average kinetic energies of the masses are different and masses near the ends of the chain have large energies. We explain that this result is not in contradiction with the principle of equipartition. The apparent violation of equipartition is observed not only in the planar chain systems but also in other constrained systems. We derive an approximate expression for the average kinetic energy, which is in qualitative agreement with the numerical results.