= Entropic spring
{title2=$k_{\rm eff}=3k_BT/(Na^2)$}
For an <ideal chain> of $N$ independent statistical segments of length $a$, the end-to-end <probability density> is proportional to $\exp[-3R^2/(2Na^2)]$. Its constrained <Helmholtz free energy> has extension-dependent part $3k_BTR^2/(2Na^2)$, so the restoring force is $-3k_BT\mathbf R/(Na^2)$. 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.
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