The trimer paradox: the effect of stiff constraints on equilibrium distributions in overdamped dynamics

We reconsider the classical problem of a freely joined chain of Brownian particles connected by elastic springs and study its conformational probability distribution function in the overdamped regime in the limit of infinite stiffness of constraints. We show that the well-known solution by Fixman [Proc. Natl. Acad. Sci. USA 71, 3050 (1974)] is missing a shape-related term, later alluded to but not computed by Helfand [J. Chem. Phys 71, 5000 (1979)]. In our approach, the shape term, also termed zero-point energy, arises explicitly from a careful treatment of the distributional limit. We present a computationally feasible method for the calculation of the shape term and demonstrate its validity in a couple of examples.