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Ddet(a, a+δa)  ≈  0.394 ∣δa∣aD_{\rm det}(a,\, a+\delta a) \;\approx\; 0.394\, \frac{|\delta a|}{a}

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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\,u2u^2/(e2πue^{2\pi u}-1) means solving e2πu(1−πu)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∗u^\ast ≈\approx 0.25363 , at which u∗u^\ast g(u∗u^\ast) ≈\approx 0.3937 . The leading-order sensitivity is…

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DdetD_{\rm det}

Symbol D_rm det

well defined, it can be evaluated, and the cleanest case where it is well defined and non-trivial is a comparison between two accelerated trajectories, both with finite normalizable spectra, differing only slightly in acceleration.

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aa

Symbol a

a is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

Ddet(a, a+δa)  ≈  0.394 ∣δa∣a.D_{\rm det}(a,\, a+\delta a) \;\approx\; 0.394\, \frac{|\delta a|}{a}.

Equation 89 · Evolutionary Physics

No Particle Without a Cosigner

This equation gives an approximation: it relates the quantities while allowing an approximation.

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\,u2u^2/(e2πue^{2\pi u}-1) means solving e2πu(1−πu)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∗u^\ast ≈\approx 0.25363 , at which u∗u^\ast g(u∗u^\ast) ≈\approx 0.3937 . The leading-order sensitivity is…

Meanings in this article

  • DdetD_{\rm det}: well defined, it can be evaluated, and the cleanest case where it is well defined and non-trivial is a comparison between two accelerated trajectories, both with finite normalizable spectra, differing only slightly in acceleration.
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