Thermal Fluctuations of Singular Bar-Joint Mechanisms*

A bar-joint mechanism is an assembly of joints connected by bars that can deform and alter its shape without changing the bar lengths. DNA origami has enabled the fabrication of such mechanisms at the nanoscale, where thermal fluctuations cannot be neglected. Here we develop a general formalism to study the consequences of equilibrating such mechanisms with a thermal bath. A meaningful description of the free-energy landscape of a mechanism is in terms of its shape coordinates that parameterize its shape space, i.e., the set of all its lowest-energy configurations with distinct shapes. Shape spaces of bar-joint mechanisms are often characterized by singularities due to states of self stress -- equilibrium configurations capable of supporting nonzero tensions in the bars. We show that in the presence of such singularities, the landscape is dominated by their neighborhoods. We successfully apply our formalism to understand the free-energy landscape of the four-bar linkage, which is the canonical example of a mechanism with shape-space singularities, as well as larger mechanisms like a triangulated origami. For these two mechanisms, the free-energy landscape suggests that configurations closer to the singularity are more likely to be found than others.