Partition sum of thermal, underconstrained systems

Area of Science:

• Physics
• Materials Science
• Soft Matter Physics

Background:

• Athermal (zero-temperature) underconstrained systems exhibit floppy behavior.
• External strain can induce rigidity in these systems.
• A prior analytical theory addressed the athermal limit.

Purpose of the Study:

• Extend the analytical theory to underconstrained systems at finite temperatures.
• Derive first-principles expressions for elastic material properties.
• Unify the physics across diverse systems like polymers, membranes, and biological tissues.

Main Methods:

• Derivation of the partition sum for underconstrained systems near the athermal transition.
• Analytic expression development for isotropic tension (t) and shear modulus (G).
• Formulation of parameters (entropic rigidity κS, energetic rigidity κE, strain interaction parameter bɛ) based on microscopic structure.

Main Results:

• Analytic expressions for elastic properties (t, G) derived in terms of strain (ɛ, γ) and temperature (T).
• Elastic properties are described by three universal parameters: κS, κE, and bɛ.
• Expressions are valid for systems near the athermal transition point.

Conclusions:

• The developed theory provides a unified framework for understanding the mechanical behavior of various underconstrained systems.
• The finite-temperature theory accurately describes rigidity transitions driven by strain.
• Microscopic structure dictates the key parameters governing macroscopic elastic response.