[Paper Review] Partition sum of thermal, under-constrained systems

Cheng-Tai Lee, Matthias Merkel|arXiv (Cornell University)|Apr 14, 2023

Elasticity and Material Modeling1 citations

TL;DR

This paper develops a first-principles analytical framework for the partition sum of thermal, under-constrained systems near the athermal rigidity transition. By deriving expressions for elastic properties—such as isotropic tension $ t $ and shear modulus $ G $—in terms of strain $ \varepsilon $, shear $ \gamma $, and temperature $ T $, it introduces three universal parameters: entropic rigidity $ \kappa_S $, energetic rigidity $ \kappa_E $, and coupling parameter $ b_\varepsilon $, unifying the mechanics of polymer networks, membranes, and vertex models.

ABSTRACT

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.

Motivation & Objective

• To extend the athermal theory of under-constrained systems to finite temperatures.
• To derive the partition sum from first principles for thermal, under-constrained systems near the rigidity transition.
• To unify the mechanical behavior of diverse systems—polymer networks, membranes, vertex models—under thermal fluctuations.
• To identify and analytically express three universal parameters ($ \kappa_S $, $ \kappa_E $, $ b_\varepsilon $) governing elastic response.
• To provide analytic expressions for isotropic tension $ t $ and shear modulus $ G $ as functions of $ \varepsilon $, $ \gamma $, and $ T $.

Proposed method

• Formal derivation of the partition sum using statistical mechanics for a generalized spring network with $ N_{\text{dof}} > N_s $, under periodic boundary conditions.
• Application of the stiff-spring limit and finite-stiffness treatment to model thermal fluctuations and strain-induced rigidity.
• Use of analyticity and homogeneity conditions on spring length functions $ L_i(\vec{R}, V, \gamma) $ to ensure mathematical tractability near the transition.
• Identification of zero modes and self-stress states via linear algebra, with higher-order rigidity (Kth-order modes) treated perturbatively.
• Derivation of the partition sum in the vicinity of the athermal transition point, where the system is marginally rigid.
• Analytical computation of thermodynamic averages for elastic moduli using the partition sum and its derivatives.

Experimental results

Research questions

• RQ1 How can the partition sum of thermal, under-constrained systems be derived from first principles near the athermal rigidity transition?
• RQ2 What are the analytic expressions for isotropic tension $ t $ and shear modulus $ G $ in terms of strain $ \varepsilon $, shear $ \gamma $, and temperature $ T $?
• RQ3 How do the three universal parameters—entropic rigidity $ \kappa_S $, energetic rigidity $ \kappa_E $, and strain coupling $ b_\varepsilon $—emerge from microscopic network structure?
• RQ4 To what extent can the mechanics of diverse systems like polymer networks, membranes, and vertex models be unified under a single thermal framework?
• RQ5 How does the inclusion of thermal fluctuations modify the rigidity transition behavior compared to the athermal limit?

Key findings

• The partition sum for thermal, under-constrained systems is analytically derived near the athermal rigidity transition point using first principles.
• Isotropic tension $ t $ and shear modulus $ G $ are expressed analytically as functions of isotropic strain $ \varepsilon $, shear strain $ \gamma $, and temperature $ T $, with no free parameters beyond $ \kappa_S $, $ \kappa_E $, and $ b_\varepsilon $.
• The three key parameters $ \kappa_S $, $ \kappa_E $, and $ b_\varepsilon $ are derived from the microscopic network structure, enabling universal prediction of elastic response.
• The theory unifies the mechanics of polymer fibers/networks, membranes, and vertex models for biological tissues under a common framework.
• The stiff-spring limit does not yield the same partition sum as the finite-stiffness limit, confirming that finite spring stiffness is essential for physical consistency.
• The framework captures the emergence of rigidity via prestresses and self-stress states induced by external strain, even at finite temperature.