Thinking in Waves Episode 56 / Part XIX — Back to Physics, with the Tools
To the longest stretch of decades, with new instruments
Back to \(1/f\) ──
“many time constants” came with a price tag
Episode 2 said “\(1/f\) is a superposition of many time constants.” ★ How many is “many”?
★ Measured: decades of \(1/f\) = decades of \(\tau\) − 2.2 (slope 0.9916, intercept \(-2.17\)). You always lose 2.2 decades at the ends.
★★ And “\(1/f\) has no low-frequency cutoff” turns out not to be a measurable claim ── seeing the knee requires an observation longer than \(\tau_{\max}\) (Episode 5).
Part XVIII assembled the tools for “how much can we actually say.”
★ Carrying them, we go back to the longest stretch of decades in this series ── the \(1/f\) of Episode 2.
01Use Episode 2’s construction as it stands
| \(\omega\) | local \(\beta\) | \(|\beta-1|\) |
|---|---|---|
| \(10^{-5}\) | 0.931017 | 6.9e−2 |
| \(10^{-3}\) | 0.999349 | 6.5e−4 |
| 1 | 1.000000 | ★ 2.7e−15 |
| \(10^{3}\) | 1.000651 | 6.5e−4 |
| \(10^{5}\) | 1.068983 | 6.9e−2 |
★ With \(\tau\) spanning 12 decades (\(10^{-6}\) to \(10^{6}\)), the middle sits cleanly on \(\beta=1\) and the ends fall off.
02★★★ How much the ends cost ── “decades of \(\tau\) minus \(c\)”
If \(\tau\) spans \(T\) decades, over how many decades does \(\beta=1\) hold to within \(\pm0.05\)?
| decades of \(\tau\), \(T\) | ★ decades of \(1/f\) | ★ lost |
|---|---|---|
| 3 | 0.862 | 2.138 |
| 4 | 1.761 | 2.239 |
| 6 | 3.748 | 2.252 |
| 8 | 5.745 | 2.255 |
| 10 | 7.746 | 2.254 |
| 12 | 9.751 | 2.249 |
The core of §02 ── ★★★ the core of this episode
decades of \(1/f\) \(=0.9916\times T-2.17\)
── slope essentially 1, intercept \(-2.2\).
★ So what you get is “the decades of \(\tau\), minus about 2.2.” Those 2.2 decades at the ends are lost no matter how large \(T\) becomes.
★★ Episode 2 wrote “a superposition of many time constants,” and that “many” came with a price ── you need the decades you want to see, plus 2.2.
★ To report “we saw \(1/f\) over 6 decades,” \(\tau\) must span at least 8.2 decades.
03Re-measuring Episode 27’s “ratio that fixes \(\beta\)”
| \(\gamma\) | predicted \(\beta=2-\gamma\) | ★ measured \(\beta\) (at centre) | difference |
|---|---|---|---|
| 0.6 | 1.400000 | 1.399870 | 1.3e−4 |
| 0.8 | 1.200000 | 1.199990 | 9.7e−6 |
| 1.0 | 1.000000 | 1.000000 | 2.7e−15 |
| 1.2 | 0.800000 | 0.800010 | 9.7e−6 |
| 1.4 | 0.600000 | 0.600130 | 1.3e−4 |
The core of §03
★★ \(\beta=2-\gamma\) confirmed numerically.
★ In Episode 27’s language ── how much \(D(E)\) changes per decade of \(\tau\) fixes \(\gamma\), and that fixes \(\beta\).
★★★ And this is falsifiable ── measure \(D(E)\) (the distribution of activation energies) independently, and \(\beta\) is predicted.
04★ Is “\(1/f\) goes on forever” a measurable claim?
The low-frequency knee sits at \(\tau_{\max}\). To see it, the observation must be longer than \(\tau_{\max}\).
| observation time | lowest visible \(f\) [Hz] | ★ visible decades (upper edge 1 kHz) |
|---|---|---|
| 1 second | 1.0e+0 | 3.00 |
| 1 hour | 2.8e−4 | 6.56 |
| 1 day | 1.2e−5 | 7.94 |
| 1 year | 3.2e−8 | 10.50 |
| 10 years | 3.2e−9 | ★ 11.50 |
The core of §04
★ Ten years of measuring buys 11.5 decades. Experiments typically report around 6.
★★★ So “\(1/f\) has no low-frequency cutoff” may be saying no more than “\(\tau\) longer than the observation is invisible.”
★★ In Episode 55’s words ── this is “cannot measure,” not “does not exist.” It has exactly the shape of Episode 5’s “a period longer than the age of the universe is indistinguishable from a constant.”
★ Combined with §02 ── if you saw 6 decades of \(1/f\), then \(\tau\) spans at least 8.2 decades, which means there are \(\tau\) more than 2 decades longer than the lowest frequency you can see.
05So what would settle it?
| what to do | verdict | why |
|---|---|---|
| ★ measure \(D(E)\) independently | ◎ settles it | ★ \(\beta\) becomes predicted (§03) |
| observe for longer | △ grows only as a log | Episode 53. 10× buys 1 decade |
| estimate \(\tau_{\max}\) separately | ◎ promising | from the upper limit on activation energy |
| ★ change the temperature | ◎ promising | ★ \(\tau=\tau_0e^{E/kT}\), so the whole window shifts |
| shrink the sample | ◎ promising | ★ fewer defects; it should go discrete |
| \(E\) [eV] | \(T=100\)K | \(T=200\)K | \(T=300\)K | \(T=400\)K |
|---|---|---|---|---|
| 0.3 | 1.3e+15 | 3.6e+7 | 1.1e+5 | 6.0e+3 |
| 0.5 | 1.6e+25 | 4.0e+12 | 2.5e+8 | 2.0e+6 |
| 1.0 | 2.5e+50 | 1.6e+25 | ★ 6.3e+16 | 4.0e+12 |
The core of §05 (the table is \(\tau/\tau_0\))
★★★ At 300 K with \(E=1.0\) eV, \(\tau\) is \(6\times10^{16}\) times \(\tau_0\) ── with \(\tau_0=10^{-13}\) s that is \(\tau\approx6\times10^3\) s. Observable.
★ Lower the temperature and that \(\tau\) lengthens sharply, so the \(1/f\) knee should move down.
★★ Whether it moves is the most direct test of Episode 2’s “superposition of activated processes” ── changing temperature beats observing longer. Episode 53’s “range cannot be bought with time” is doing the work here too.
06What we can and cannot say as waves
| item | verdict | basis |
|---|---|---|
| \(\beta=2-\gamma\) | ◎ numerical | agreement to \(10^{-6}\) |
| ★ decades of \(1/f\) \(=T-2.2\) | ◎ numerical | ★ intercept constant over \(T=3\)–12 |
| Episode 27’s ratio re-confirmed | ◎ numerical | \(D(E)\) fixes \(\beta\) |
| ★ “no cutoff” is not measurable | ◎ logical | ★ invisible unless observation exceeds \(\tau_{\max}\) |
| 10 years buys 11.5 decades | ◎ numerical | reports are typically around 6 |
| ★ temperature is the strongest handle | ○ proposal | ★ \(\tau=\tau_0e^{E/kT}\) |
| ★ test against real data | × not attempted | ★ this episode is synthetic only |
Exercises
- Where do the “2.2 decades” of §02 come from?
Show answer
From the width of a single Lorentzian’s bend. One Lorentzian \(\tau/(1+\omega^2\tau^2)\) takes about a decade on each side of \(\omega\tau=1\) to turn its slope from 0 to \(-2\) (the same structure as the “5.36 decades each” measured in Episode 48 §03).
★ At the ends of the superposition, those not-yet-fully-bent Lorentzians are what you see.
★★ So the 2.2 is “1.1 decades at each end” ── about the width of one Lorentzian’s bend.
★ Loosen the criterion (to \(\pm0.1\)) and the 2.2 shrinks ── the 2.2 here is the value under a \(\pm0.05\) criterion. ★ As Episode 55 insisted, the criterion is part of the number and should be written with it. - §04 says “no cutoff is not measurable.” Was Episode 2’s claim therefore empty?
Show answer
No ── Episode 2 did not claim “no cutoff.” What it said was “\(1/f\) cannot be made from a single time constant; it takes many superposed,” and that holds over the decades actually observed (§02: if you saw 6 decades, \(\tau\) spans 8.2).
★ What cannot be measured is only “does it continue beyond that” ── the substance of the claim is intact.
★★ In the handbook’s phrasing from Episode 49 ── write “\(1/f\) holds over \(N\) decades,” never “it holds forever.”
★ This series never wrote “forever” in Episode 2 ── but it did not state the number of decades either, and that is what this episode fixes. - §05 concludes that changing the temperature is strongest. Which of Episode 54’s ways of buying range is that?
Show answer
③ change the system itself. Changing \(T\) shifts \(\tau=\tau_0e^{E/kT}\) bodily, so the window of visible \(\tau\) moves with the same sample and the same instrument.
★ And it moves exponentially ── at \(E=1.0\) eV, 300 K → 200 K takes \(\tau/\tau_0\) from \(6\times10^{16}\) to \(1.6\times10^{25}\), more than 8 decades.
★★ Episode 54 found that raising \(Re\) only helps as \(\log Re\); here the dependence is exponential, so the leverage is in a different league.
★★★ In other words, the very mechanism called “activation” hands us a way to buy range ── Episode 28’s “addition becomes multiplication,” now working on the measurement side.
Summary — “many” had a price tag
Carrying the tools assembled in Part XVIII, we went back to the \(1/f\) of Episode 2.
First, \(\beta=2-\gamma\) confirmed to \(10^{-6}\) (with \(g(\tau)\propto\tau^{-\gamma}\)). Episode 27’s “\(D(E)\) fixes \(\beta\)” survives, and measuring \(D(E)\) independently would predict \(\beta\) ── it remains falsifiable.
★★★ The core of this episode ──
decades of \(1/f\) \(=0.9916\times\)(decades of \(\tau\))\(-2.17\).
You always lose 2.2 decades at the ends.
★ Episode 2’s “superposition of many time constants” carried a price ── the decades you want to see, plus 2.2. To say “we saw 6 decades of \(1/f\)” requires \(\tau\) spanning at least 8.2 decades.
★★ And “\(1/f\) has no low-frequency cutoff” is not a measurable claim ── seeing the knee requires an observation longer than \(\tau_{\max}\), and ten years buys only 11.5 decades. It is the same shape as Episode 5’s “a period longer than the age of the universe is indistinguishable from a constant,” and in Episode 55’s words it is “cannot measure,” not “does not exist.”
★ The substance of Episode 2 is intact ── “a single time constant cannot do it” holds over the decades observed. ★ What needs fixing is only the phrasing: write “over \(N\) decades,” not “forever.”
★★ The strongest way to settle it is to change the temperature ── since \(\tau=\tau_0e^{E/kT}\), going from 300 K to 200 K moves \(\tau\) by more than 8 decades. That is Episode 54’s “change the system itself,” and it works exponentially ── far faster than observing longer.
★ Next we look at that “end” more generally ── what must always happen at the edge of a superposition.
★★ Episode 16’s dispersion, Episode 44’s band edge, Episode 51’s window edge ── “edges” have appeared in this series many times. ★ We ask whether they are the same thing.
This document is Episode 56 of the “Thinking in Waves” series. The computations are in kenshou/ichif.js. On the numbers in this episode: the numerical superposition integral, the measurement of local \(\beta\) (Episode 51’s window fit), the relation between decades of \(1/f\) and decades of \(\tau\), the observation time versus visible decades, and \(\tau/\tau_0\) from activation are all computed here. That a superposition of Debye relaxations produces \(1/f\), that \(\beta=2-\gamma\), and the activated form \(\tau=\tau_0e^{E/kT}\) are all well-known results and are not discoveries of this episode ── what this episode did was use the tools of Episodes 51–55 to measure “how many decades can we claim,” and put the relation between the decades of \(1/f\) and the decades of \(\tau\) into a number. The “2.2 decades” of §02 depends on the \(\pm0.05\) criterion and on a fitting window of 0.2 decades; with a \(\pm0.1\) criterion the value is smaller (as Episode 55 insisted, the criterion is part of the number). The “experiments typically report around 6 decades” in §04 describes the representative situation, not a value from a specific paper. The \(\tau_0=10^{-13}\) s in §05 is an estimate from a typical lattice-vibration period. 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.