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For a planar elastic filament under positive constant extension force $F$, the quadratic force-controlled energy isThe higher-order Euler-Lagrange equation is $B{\zeta }^{\prime \prime \prime \prime }-F{\zeta }^{\prime \prime }=B{\kappa }_0^{\prime \prime }$. On a periodic interval or in a bulk Fourier transform treatment, every nonzero mode obeys the displayed transfer relation. The length deficit is $\frac{1}{2}\int ({\zeta }^{\prime }{)}^{2}dx$, so its ensemble value is determined by the power spectrum of the quenched intrinsic curvature. Boundary conditions must be specified for a finite filament.

For a stationary quenched intrinsic curvature with power spectrum $S_{\kappa }(q)$ and finite second spectral moment, the bulk deficit satisfiesThe moment is $⟨({\kappa }_0^{\prime }{)}^2⟩$ for mean-square differentiable curvature. Infinite-moment disorder and finite endpoint boundary layers can change the exponent, so this is a smooth bulk law rather than a universal law for all random polymers.