  A bar-joint mechanism is a deformable assembly of freely rotating joints
connected by stiff bars. Here we develop a formalism to study the equilibration
of common bar-joint mechanisms with a thermal bath. When the constraints in a
mechanism cease to be linearly independent, singularities can appear in its
shape space, which is the part of its configuration space after discarding
rigid motions. We show that the free-energy landscape of a mechanism at low
temperatures is dominated by the neighborhoods of points that correspond to
these singularities. We consider two example mechanisms with shape-space
singularities and find that they are more likely to be found in configurations
near the singularities than others. These findings are expected to help improve
the design of nanomechanisms for various applications.
