Partition sum of thermal, under-constrained systems

Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain. Following our recently developed analytical theory for the athermal limit, here and in the companion paper, we extend this theory to under-constrained systems at finite temperatures. Close to the athermal transition point, we derive from first principles the partition sum for a broad class of under-constrained systems, from which we obtain analytic expressions for elastic material properties such as isotropic tension $t$ and shear modulus $G$ in terms of isotropic strain $\varepsilon$, shear strain $\gamma$, and temperature $T$. These expressions contain only three parameters, entropic rigidity $\kappa_S$, energetic rigidity $\kappa_E$, and a parameter $b_\varepsilon$ describing the interaction between isotropic and shear strain. We provide analytical expressions for these parameters based on the microscopic structure of the system. Our work unifies the physics of systems as diverse as polymer fibers & networks, membranes, and vertex models for biological tissues.