$$u\,g(u) = 24\,u^2/(e^{2\pi u}-1)$$

Equation 85 · Evolutionary Physics

No Particle Without a Cosigner

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

This is the same trick used to derive Wien’s displacement law: because u\,g(u) rises from zero, passes through a single interior maximum, and falls back to zero, the integral of the absolute value of its derivative over all u is exactly twice that maximum, with no need to evaluate the integral directly. Finding the maximum of u\,g(u) = 24\,$u^2$/($e^{2\pi u}$-1) means solving $e^{2\pi u}(1-\pi u)$=1 , the same class of transcendental equation that fixes the peak of a Planck spectrum weighted by an extra power of frequency. Solved by Newton’s method to five-figure precision, the nontrivial root is $u^\ast$ $\approx$ 0.25363 , at which $u^\ast$ g($u^\ast$) $\approx$ 0.3937 . The leading-order sensitivity is…

Meanings in this article

• $u$: exactly twice that maximum.
• $u^2$: the square of u; exactly twice that maximum.

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