Thinking in Waves Episode 57 / Part XIX — Back to Physics, with the Tools
Tracking down where the 2.2 decades came from
What happens at an edge ──
every edge was a leading correction term
Episode 56’s “2.2 decades lost at the ends” is not measured here but derived. For \(\gamma=1\) there is an exact solution, and the answer is \(2\log_{10}\!\big(2/(\pi\varepsilon)\big)=2.2098\).
★ From there, a general law ── decades lost \(=(1/n)\log_{10}(1/\varepsilon)+\)const, with \(n\) the power of the leading correction.
★★★ And a prediction failed ── the slope is not \(2\) but \(2/[\gamma(2-\gamma)]\), which is minimised at \(\gamma=1\) ── exactly \(1/f\).
Episode 56 measured “2.2 decades lost at the ends.” ★ This time we find out where that 2.2 comes from, and fold every edge in this series into one formula.
01Start with one Lorentzian’s bend
| \(\varepsilon\) | flat-side edge \(x\) | \(-2\)-side edge \(x\) | width [dec] | \(\log_{10}(2/\varepsilon)\) |
|---|---|---|---|---|
| 0.2 | 0.3333 | 3.000 | 0.9542 | 1.0000 |
| 0.05 | 0.1601 | 6.245 | 1.5911 | 1.6021 |
| 0.01 | 0.0709 | 14.107 | 2.2989 | 2.3010 |
| 0.005 | 0.0501 | 19.975 | 2.6010 | 2.6021 |
★ The slope of width against \(\log_{10}(1/\varepsilon)\) is 1.000299 (predicted 1). Both sides have \(x^2\) as their leading correction, so each contributes \(1/2\) decade and the total is 1.
02★★★ Deriving Episode 56’s “2.2 decades”
For \(\gamma=1\) the superposition integral can be done exactly.
| \(\varepsilon\) | \(1/f\) [dec] | ★ lost [dec] | \(2\log_{10}\!\frac{2}{\pi\varepsilon}\) |
|---|---|---|---|
| 0.2 | 10.894 | 1.106 | 1.006 |
| 0.1 | 10.327 | 1.673 | 1.608 |
| 0.05 | 9.753 | ★ 2.2475 | ★ 2.2098 |
| 0.01 | 8.384 | 3.616 | 3.608 |
| 0.005 | 7.786 | 4.214 | 4.210 |
The core of §02 ── what the 2.2 was
lost decades at \(\varepsilon=0.05\) \(=2\log_{10}\dfrac{2}{\pi\varepsilon}=2.2098\)
★ This matches Episode 56’s measured 2.249 / 2.255 ── that 2.2 came from the \(2/(\pi x)\) tails of the two \(\arctan\)s.
★★ A measured number became a derived one. It is Episode 49’s handbook rule — “when a number appears, look for the closed form” — applied to this series itself.
★ The slope, fitted for \(\varepsilon\le0.002\), is 1.9988 (predicted 2): each end carries a \(1/x\) correction (\(n=1\)) and costs one decade.
03★★ A prediction failed ── the slope depends on \(\gamma\)
I expected the slope to be 2 for every \(\gamma\). It is not.
| \(\gamma\) | lost at \(\varepsilon=0.05\) | ★ slope (measured) | ★ predicted \(\frac{2}{\gamma(2-\gamma)}\) |
|---|---|---|---|
| 0.6 | 2.4695 | 2.3695 | 2.3810 |
| 0.8 | 2.2980 | 2.0751 | 2.0833 |
| 1.0 | 2.2475 | ★ 1.9926 | ★ 2.0000 |
| 1.2 | 2.2980 | 2.0751 | 2.0833 |
| 1.4 | 2.4695 | 2.3695 | 2.3810 |
Because it failed I went back to the asymptotics, and the edges showed themselves ── the effect of cutting at \(\tau_{\max}\) is \((\omega\tau_{\max})^{-\gamma}\) on the low side and \((\omega\tau_{\min})^{2-\gamma}\) on the high side (inverting \(\tau\to1/\tau\) sends \(\gamma\to2-\gamma\)). So the slope is \(1/\gamma+1/(2-\gamma)=2/[\gamma(2-\gamma)]\).
The core of §03 ── ★★★ the best thing this episode found
\(2/[\gamma(2-\gamma)]\) takes its minimum value 2 at \(\gamma=1\).
★★★ And \(\gamma=1\) is exactly \(\beta=1\) ── \(1/f\).
★★ So \(1/f\) is also the spectrum whose edges are cheapest ── move \(\beta\) either way and the same span of \(\tau\) yields fewer decades.
★ This came out only because the prediction (slope 2) failed. A prediction that misses can teach more than one that lands.
04A different edge ── Episode 16’s dispersion, Episode 44’s band edge
The discretised string has \(\omega=(2c/a)\sin(ka/2)\). Over what range does it stay within \(\varepsilon\) of the straight line \(\omega=ck\)?
| \(\varepsilon\) | \(x_{\max}=ka/2\) | \(\sqrt{6\varepsilon}\) | difference |
|---|---|---|---|
| 0.2 | 1.13110 | 1.09545 | 3.6e−2 |
| 0.05 | 0.55191 | 0.54772 | 4.2e−3 |
| 0.01 | 0.24532 | 0.24495 | 3.7e−4 |
| 0.005 | 0.17334 | 0.17321 | 1.3e−4 |
★ The slope of lost decades is 0.5030 (predicted \(1/2\)). Since \(\sin x/x=1-x^2/6+\cdots\), the leading correction is \(x^2\), so \(n=2\). The band edge (Episode 44), going as \((k-k_c)^2\), is in the same family.
05Episode 51’s “window edge” is in that family too
Episode 51 §04 found that a symmetric window with symmetric curvature has its bias cancel. What if a cubic term survives?
| half-width \(h\) | measured bias | \(\frac{3}{5}dh^2\) | ratio |
|---|---|---|---|
| 0.1 | 0.0060030 | 0.0060000 | 1.0005 |
| 0.5 | 0.1500750 | 0.1500000 | 1.0005 |
| 2.0 | 2.4011998 | 2.4000000 | 1.0005 |
★ The bias goes as \(h^2\), so the window that keeps it within \(\varepsilon\) has \(h\propto\sqrt{\varepsilon}\) ── slope 0.500000, exactly the same family as Episode 44’s edge.
06★★★ The general law
| \(n\) | \(A\) | measured slope | predicted \(1/n\) | difference |
|---|---|---|---|---|
| 1 | 1 / 0.1 | 1.000000 | 1.000000 | 2.4e−15 |
| 2 | 1 / 0.1 | 0.500000 | 0.500000 | 0.0 |
| 3 | 1 / 0.1 | 0.333333 | 0.333333 | 6.7e−16 |
| 4 | 1 / 0.1 | 0.250000 | 0.250000 | 6.1e−16 |
| 6 | 1 / 0.1 | 0.166667 | 0.166667 | 3.3e−16 |
★ \(A\) — the size of the correction — moves only the intercept; the slope is set by \(n\) alone. An edge’s character lives in the power, not in the magnitude.
07Laying out every edge in the series
| edge | leading correction | \(n\) | dec/dec | from |
|---|---|---|---|---|
| superposition, low end | \((\omega\tau_{\max})^{-\gamma}\) | \(\gamma\) | 1.0 | Ep. 2, 56 |
| superposition, high end | \((\omega\tau_{\min})^{2-\gamma}\) | \(2-\gamma\) | 1.0 | Ep. 2, 56 |
| a single Lorentzian | \(x^2\) | 2 | 0.5 | Ep. 5, 48 |
| discrete-string dispersion | \(x^2/6\) | 2 | 0.5 | Ep. 16 |
| band edge | \((k-k_c)^2\) | 2 | 0.5 | Ep. 44 |
| window fitting | \(\frac{3}{5}h^2\) | 2 | 0.5 | Ep. 51 |
| frequency resolution | \(1/N\) | 1 | 1.0 | Ep. 51, 53 |
The core of §07
★ Edges are not types ── they lie along a continuum in \(n\). (The dec/dec column is for \(\gamma=1\).)
★★ An \(n=1\) edge (superposition, resolution) costs one decade for every 10× tightening of \(\varepsilon\); an \(n=2\) edge (Taylor expansion, window) costs only 0.5.
★★★ But a small \(n\) also means the information outside the edge reaches far inside it ── an expensive edge and an edge that lets you see past it are the same thing from two sides.
08Relation to Episode 53
| what you spend | how decades grow | per 10× | from |
|---|---|---|---|
| observation time \(T\) | \(\log_{10}T\) | +1.0 dec | Ep. 53 |
| sample count \(N\) (precision) | \(-\frac12\log_{10}N\) | 3.2× better | Ep. 53 |
| tolerance \(\varepsilon\) (\(n=1\)) | \(\log_{10}(1/\varepsilon)\) | −1.0 dec | here |
| tolerance \(\varepsilon\) (\(n=2\)) | \(\frac12\log_{10}(1/\varepsilon)\) | −0.5 dec | here |
| both ends of \(1/f\) (\(\gamma=1\)) | \(2\log_{10}(1/\varepsilon)\) | −2.0 dec | here |
| temperature (activation) | \(E/kT\) directly | orders of magnitude | Ep. 56 |
★ Episode 53’s “range can only be bought logarithmically” holds at edges too. But the coefficient is \(1/n\) ── not knowing an edge’s \(n\) means misjudging the precision you need by a factor of two.
Exercises
- §03’s prediction (slope 2 for every \(\gamma\)) failed. What did the way it failed reveal?
Show answer
That the slope is minimised at \(\gamma=1\) ── another way in which \(1/f\) is special.
★ Had the slope been 2 as predicted, the story would have ended at “edges do not depend on \(\gamma\).” ★★ Because it failed, I re-derived the asymptotics and found the asymmetric pair \((\omega\tau_{\max})^{-\gamma}\) and \((\omega\tau_{\min})^{2-\gamma}\).
★★★ Inverting \(\tau\to1/\tau\) sends \(\gamma\to2-\gamma\), and the fixed point of that symmetry is \(\gamma=1\). \(1/f\) has the cheapest edges because it is the one self-dual point.
★ Episode 30 called \(1/f\) “the scale-invariant fixed point”; here we arrive at the same place from the measurement side. - §07 says a smaller \(n\) means outside information reaches further in. Is that good or bad?
Show answer
Bad for measuring, good for learning about the system.
★ The bad side: an \(n=1\) edge has a \(1/x\) tail that falls off slowly, so the \(\tau\) outside the window keep affecting the measurement and the clean power-law range shrinks.
★★ The good side: for the same reason, information about the \(\tau_{\max}\) outside the window is leaking inside it. Episode 56 said “\(\tau\) longer than the observation are invisible” ── strictly, the knee is invisible but the tail is doing work.
★★★ In principle, measuring the edge shape precisely would let you estimate \(\tau_{\max}\) from outside the window ── but only by assuming a model for that shape, which puts the result in Episode 55’s “conditional on a model” category. - Does §06’s law still work when \(n\) is not an integer?
Show answer
Yes ── §03 is exactly that case. At \(\gamma=0.6\) the low-end \(n\) is 0.6 and the high-end \(n\) is 1.4, neither an integer. The measured slope 2.3695 agrees with \(1/0.6+1/1.4=2.3810\) to 0.5 %.
★ \(n\) is an integer only where a Taylor expansion applies; at edges that come from power-law tails, \(n\) is a continuous quantity.
★★ The structure is the same as Episodes 41–42: the order leaves the integers for the same reason the edge’s \(n\) does. ★ Both are the one fact that a power law does not require integers.
Summary — every edge was a leading correction term
Episode 56’s “2.2 decades lost at the ends” is derived here rather than measured.
For \(\gamma=1\), \(S(\omega)=[\arctan(\omega\tau_{\max})-\arctan(\omega\tau_{\min})]/\omega\) is exact, and the decades lost are \(2\log_{10}[2/(\pi\varepsilon)]\) ── 2.2098 at \(\varepsilon=0.05\), matching Episode 56’s measured 2.249 / 2.255. That 2.2 came from the tails of the two \(\arctan\)s.
★★★ The general law ──
decades lost \(=\dfrac{1}{n}\log_{10}\dfrac{1}{\varepsilon}+\)const,
with \(n\) the power of the leading correction.
★ Every edge in this series fits it ── one Lorentzian (\(n=2\)), the discrete-string dispersion (\(n=2\), Episode 16), the band edge (\(n=2\), Episode 44), window fitting (\(n=2\), Episode 51), frequency resolution (\(n=1\)), and the superposition (\(n=\gamma\) and \(2-\gamma\)). Edges are not types; they lie along a continuum in \(n\).
★★ And a prediction failed. I expected the slope to be 2 for every \(\gamma\); it is \(2/[\gamma(2-\gamma)]\). That takes its minimum value 2 at \(\gamma=1\) ── exactly \(1/f\). ★★★ Inverting \(\tau\to1/\tau\) sends \(\gamma\to2-\gamma\), and \(\gamma=1\) is the fixed point ── so \(1/f\) is also the spectrum with the cheapest edges.
★ Episode 53’s “range can only be bought logarithmically” holds at edges too ── but with a coefficient of \(1/n\), so not knowing an edge’s \(n\) means misjudging the precision you need by a factor of two.
★ Next we turn that around ── how what lies outside a window shows up inside it.
★★ Episode 40’s window functions, Episode 51’s leakage, and Episode 45’s Bogoliubov transformation ── in the sense that “what is outside appears inside,” these may all have the same shape. ★ We put it in measurable form and check.
This document is Episode 57 of the “Thinking in Waves” series. The computations are in kenshou/hashi.js. On the numbers in this episode: the Lorentzian bend width, the decades lost from the exact \(\gamma=1\) solution, the slopes over a range of \(\gamma\), the useful range of the discrete-string dispersion, the window-fitting bias, and the general \(Q=1+Ax^n\) law are all computed here. The closed form for \(\gamma=1\) via \(\arctan\), the expansion of \(\sin x/x\), and the \(h^2\) scaling of least-squares bias are standard results and are not discoveries of this episode ── what this episode did was collect them into the single form “decades lost \(=(1/n)\log_{10}(1/\varepsilon)\),” lay out the series’ edges along it, and show that the slope is \(2/[\gamma(2-\gamma)]\) with its minimum at \(\gamma=1\). The §03 slopes are fitted for \(\varepsilon\le0.01\) and those of §§01, 02 and 04 for \(\varepsilon\le0.002\) ── at larger \(\varepsilon\) higher-order corrections survive and the slope has not reached its asymptote (this is why the tabulated values sit about 4 % from the prediction). The “dec/dec” column in §07 is for \(\gamma=1\). This episode also uses synthetic data only, and has not been applied to real measurements. ── To print, use your browser’s Print → Save as PDF.