Legendre Spectrum

Mode Amplitudes Bl

The Legendre spectrum bl shows how thermal fluctuation energy is distributed across angular modes. Higher modes are suppressed as bending energy cost grows as ~l⁴. Compare multiple experimental conditions.

Primary Parameters
40.0
10.0
10.0
Compare Experimental Conditions
Scale
Derived Quantities
Legendre Coefficients Bl vs Mode l
\[ b_l = \frac{2l+1}{4\pi}\,\frac{k_B T}{\kappa(l-1)(l+2)\left[l(l+1)+\Sigma\right]} \]
Mode Stiffness αl/κ = (l−1)(l+2)[l(l+1)+Σ]
\(\alpha_l/\kappa=(l-1)(l+2)\left[l(l+1)+\Sigma\right]\): higher modes carry less thermal energy.
Numerical Values
l pl ql αl/κ bl Σ bl