 Entropic spring

For an ideal chain of $N$ independent statistical segments of length $a$, the end-to-end probability density is proportional to $\exp [-3R^2/(2N{a}^2)]$. Its constrained Helmholtz free energy has extension-dependent part $3k_{B}T{R}^2/(2N{a}^2)$, so the restoring force is $-3k_{B}T\mathbf{R}/(N{a}^2)$. This is an entropic spring because extension removes available chain configurations, and its stiffness grows with temperature. The Gaussian law applies at extensions small compared with the contour length $Na$; near full extension, finite extensibility matters.