Thermal fluctuations of singular mechanical networks

Mechanical networks, like many constrained systems, are often characterized by singularities in their configuration manifolds due to states of self stress—equilibrium states with nonzero internal forces allowed by the geometry of the network. Examples of such systems include linkages and origami, which are typically modeled as ball-and-spring networks that often support self-stress states. In this study, we explore the geometric and topological origins of these singularities and their effects when such networks are equilibrated with a thermal bath. Self stresses can also give rise to additional infinitesimal zero modes, which are perturbations that preserve the constraints to linear order. Working with stiff networks at low temperatures, we use analytical calculations and simulations to find the existence of a nontrivial coupling between infinitesimal zero modes, harmonic modes, and coordinates on the configuration manifold. This coupling manifests itself as an enhancement of equilibrium probability density near these singularities. This, in turn, has a direct effect on measurable thermodynamic observables, which display atypical thermal scaling.