Thinking in Waves Episode 48 / Part XVII — Looking for the Edge of the Axis
Where your own tool stops working
The Edge of the Order Axis —
There is exactly one condition for using it
Episode 43's exercises said a third test would come from somewhere with no power law at all. ★ Here we go looking for it deliberately.
★ Measuring the local order \(\alpha(\omega)=d\ln|H|/d\ln\omega\) — an exponential gives \(\alpha=-\omega\), with no upper bound (it runs off the scale).
★★ And a marginal coupling approaches 0 only as \(1/\ln\mu\) — still \(-0.0130\) after 32 decades. That is not "a power law of order 0."
Episode 41 produced a quantity that does not sit on the axis (an integer); Episode 42 produced a place where the axis is not single (a distribution).
★ The third is somewhere with no power law at all. This episode is not a defence of the axis but a determination of its conditions of use.
01First, confirm the case where the axis works
| \(\omega\) | \(H=\omega^{0.5}\) | \(H=\omega^{-5/6}\) | constant? |
|---|---|---|---|
| \(10^{-3}\) | 0.50000000 | −0.83333333 | ★ yes |
| 1 | 0.50000000 | −0.83333333 | ★ yes |
| \(10^{6}\) | 0.50000000 | −0.83333333 | ★ yes |
★ Here "order" is a good coordinate — the same value at every frequency. \(-5/6\) was Episode 42's Kolmogorov order.
02★ Edge 1 — the local order becomes a function of frequency
| \(\omega\) | Lorentzian \(\frac1{1+\omega^2}\) | log \(\ln(1+1/\omega)\) | exponential \(e^{-\omega}\) | stretched \(e^{-\sqrt\omega}\) |
|---|---|---|---|---|
| \(10^{-2}\) | −0.00020 | −0.21453 | −0.01000 | −0.05000 |
| 1 | −1.00000 | −0.72135 | −1.00000 | −0.50000 |
| \(10^{1}\) | −1.98020 | −0.95382 | −10.00000 | −1.58114 |
| \(10^{3}\) | −2.00000 | −0.99950 | ★ −1000.00000 | −15.81139 |
The point of §02
★ The Lorentzian moves continuously from 0 to \(-2\). At either end it returns to a power law — which is why "zeroth order below, second order above" is a legitimate description.
★★ The exponential: \(\alpha=-\omega\), with no upper bound. ★ It does not fail to sit on the axis; it runs off the scale — a different kind of ending.
★ Episode 28 sorted "two sources of exponentials"; in either case the result leaves the order axis.
03How wide a range earns the word "power law"?
The measure from Episode 27 — how many decades keep \(\alpha(\omega)\) within \(\pm0.1\) of a target (out of 12).
| \(H\) | target order | ★ decades within \(\pm0.1\) |
|---|---|---|
| \(\omega^{0.5}\) | 0.50 | ★ 12.000 (all of them) |
| Lorentzian (low side) | 0.00 | 5.361 |
| Lorentzian (high side) | −2.00 | 5.360 |
| log \(\ln(1+1/\omega)\) | 0.00 | 1.658 |
| exponential \(e^{-\omega}\) | −1.00 | ★ 0.087 |
The point of §03
★ An exponential earns almost no decades at any target order (0.087) — "it is not a power law" has become a number.
★★ A Lorentzian is two power laws of 5.36 decades each — 10.7 together, with the remaining 1.3 being the bend.
★ What is good about this measure is that it answers "is it a power law" in decades rather than yes/no — the same idea as Episode 27's ratio that sets \(\beta\).
04★★ Edge 2 — the marginal case in the renormalization group
| \(\mu\) | \(g(\mu)\) | ★ local order | \(\alpha\times\ln\mu\) |
|---|---|---|---|
| \(10^{1}\) | 0.17743337 | −0.17743337 | −0.408555 |
| \(10^{4}\) | 0.07972146 | −0.07972146 | −0.734262 |
| \(10^{16}\) | 0.02489129 | −0.02489129 | −0.917029 |
| \(10^{32}\) | 0.01298431 | ★ −0.01298431 | −0.956719 |
The point of §04 — ★★★
"Marginal" does not mean the order is 0.
It means the order approaches 0 logarithmically.
★ After 32 decades it is still \(-0.0130\) — not "a power law of order 0."
★★ \(\alpha\times\ln\mu\) approaching \(-1/b=-1\) is the signature of a \(1/\ln\mu\) form.
★ Everything physics calls a marginal coupling — \(\phi^4\) in four dimensions, QCD's \(\alpha_s\), weak localization in 2D (Episode 37) — lives here. That is why logarithms appear.
05★ Edge 3 — in a finite system, not even an exponent is definable
The 1D Ising chain with periodic boundary and size \(L\) is exactly solvable — \(\langle s_0s_r\rangle=\dfrac{t^r+t^{L-r}}{1+t^L}\), \(t=\tanh K\). An infinite chain gives a pure exponential; a finite one folds back.
| \(r\) | \(L=16\) | \(L=32\) | \(L=64\) | infinite |
|---|---|---|---|---|
| 2 | 0.251472 | 0.272063 | 0.272341 | 0.272341 |
| 4 | 0.215003 | 0.271513 | 0.272341 | 0.272341 |
| 8 | ★ 0.000000 | 0.265120 | 0.272340 | 0.272341 |
| 12 | ★ −0.215003 | 0.215003 | 0.272331 | 0.272341 |
The point of §05 (the table is the local decay rate \(-d\ln\langle ss\rangle/dr\))
★★ At \(L=16\) the decay rate hits 0 at \(r=8\) and goes negative beyond. The correlation grows with distance — "decay constant" has stopped meaning anything.
★ In a finite system neither a single exponent nor a single order is definable.
★★ It is the spatial version of Episode 5's "a period longer than the age of the universe is indistinguishable from a constant" — a correlation longer than the system cannot be seen inside it.
06Critical exponents as orders — and how good the ε-expansion is
| \(\epsilon=4-d\) | \(\nu\) (to first order) | \(\eta\) (to second order) | 3D literature values |
|---|---|---|---|
| 0.00 | 0.500000 | 0.000000 | |
| 0.50 | 0.541667 | 0.004630 | |
| 1.00 | 0.583333 | 0.018519 | \(\nu=0.6300,\ \eta=0.0363\) |
The point of §06
★ At \(\epsilon=1\) (three dimensions) the discrepancies are 7.4 % for \(\nu\) and 49.0 % for \(\eta\).
★★ "The order varies continuously with dimension" is a good approximation near \(d=4\), but first order is not enough by the time you reach three.
★ Episode 30's \(\alpha=(d-3)/2\) was about free fields. With interactions, the order stops being a linear function of dimension. ★ This too is not "the axis fails" but "the axis's scale becomes non-linear."
07All the ways the axis ends, side by side
| How it ends | Where it appeared | What is happening |
|---|---|---|
| ① a quantity that does not fit | Ep. 41 | integers (winding, Chern) |
| ② the axis is not single | Ep. 42 | the order becomes a distribution |
| ★ ③a it runs off the scale | §02 here | ★ exponential: \(\alpha\) diverges |
| ★ ③b it approaches 0 logarithmically | §04 here | ★ marginal coupling: \(1/\ln\mu\) |
| ★ ③c the system is finite | §05 here | ★ folds back; no exponent at all |
The point of §07 — ★★★ the conclusion
None of these means the axis is wrong.
They make its condition of use explicit —
that scale invariance holds over some number of decades.
★ And the reason power laws are so common in physics is that this condition is so often met — near a critical point, between the correlation length and the system size, in the inertial range, outside a horizon. All of them regions with no characteristic length.
★★ Turned around — let a single characteristic length in, and the axis is at best a local coordinate.
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Power law ⟹ constant \(\alpha\) | ◎ numerics | where the axis works |
| ★ Exponential ⟹ \(\alpha\) diverges | ◎ numerics | ★ \(\alpha=-\omega\), unbounded |
| Lorentzian: 0 → −2 continuously | ◎ numerics | power law only at the ends |
| ★ Marginal coupling is \(1/\ln\mu\) | ◎ numerics | ★ still −0.0130 after 32 decades |
| ★ Finite size kills the exponent | ◎ numerics | ★ shown on the 1D Ising chain |
| Accuracy of the ε-expansion | ◎ numerics | 3D: \(\nu\) 7 %, \(\eta\) 49 % |
| ★ A replacement for the axis | × none found | ★ this article did not find one |
Exercises
- §02 had the exponential's \(\alpha\) diverge. Does that mean an exponential "has no order"?
Show answer
Not "none" but "no single value." At each point \(\alpha(\omega)=-\omega\) is perfectly well defined; definability is not the problem.
★ The problem is that the range over which it stays the same is nearly zero (0.087 decades in §03) — the summary called "order" has stopped being useful.
★★ Episode 1 introduced dB/oct precisely because one number could summarize a spectrum. ★ Applying a summary to something unsummarizable only loses information. To summarize an exponential, use its time constant, not an order. - §04 found "marginal" to be \(1/\ln\mu\). How does that differ from Episode 42's "the order becomes a distribution"?
Show answer
Episode 42 was "at the same place, different moments give different answers." Different points carry different orders.
★ §04 here is "every moment gives the same answer, but that answer drifts slowly with \(\mu\)" — a drift, not a spread.
★★ They are distinguishable by measurement — a distribution bends the higher moments (Episode 42); a drift moves every moment alike, as \(\ln\mu\).
★ And both can of course happen at once — which this article does not treat. - §07 said power laws appear only where there is no characteristic length. So what have you learned when you see one?
Show answer
That no characteristic length or time lies within that range of decades. That is surprisingly strong information.
★ If \(1/f\) holds for six decades (Episode 2), then no special time constant lives anywhere in those six decades — a single relaxation process cannot explain it, so a superposition of many is required.
★★ Episode 27's ratio that sets \(\beta\) is the same reasoning — how many decades it holds is what narrows down the mechanism.
★ Conversely, calling something a power law when it holds for one decade says almost nothing — recall that the exponential in §03 managed 0.087.
Summary — the axis is not wrong; its conditions are now explicit
We went looking for the third test promised in Episode 43's exercises. Measuring the local order \(\alpha(\omega)=d\ln|H|/d\ln\omega\) —
★ The Lorentzian moves continuously from 0 to \(-2\), returning to a power law only at its ends (5.36 decades each).
★★ The exponential gives \(\alpha=-\omega\) with no upper bound — it does not fail to sit on the axis; it runs off the scale. At any target order it earns only 0.087 decades.
★★★ And what "marginal" really is —
not an order of 0, but an order approaching 0 logarithmically.
Still \(-0.0130\) after 32 decades.
In a finite system not even an exponent survives — in the 1D Ising chain at \(L=16\), the decay rate hits 0 at \(r=8\) and goes negative beyond. The spatial version of Episode 5.
Critical exponents too: at \(\epsilon=1\) (three dimensions) the first-order ε-expansion is off by 7.4 % in \(\nu\) and 49.0 % in \(\eta\) — Episode 30's \(\alpha=(d-3)/2\) was a free-field statement, and interactions make the scale non-linear.
★★★ The conclusion — none of these three endings means the axis is wrong.
They make its condition of use explicit:
scale invariance holding over some number of decades.
★ And power laws are common in physics because that condition is so often met — near critical points, in the inertial range, outside horizons: regions with no characteristic length.
★ Honestly: a replacement for the axis — one coordinate that works more widely — was not found here.
★ Next time we switch to using the tool — given a set of measurements, how do you read it?
★★ "Measure the slope. Count the decades. Check which moment you measured with. Look for a characteristic length." The procedures built across this series, collected into a single handbook. If Episodes 15 and 43 were scorecards of what we got right, next time is the manual for how to use it.
This document is Episode 48 of the "Thinking in Waves" series. Computations are in kenshou/hate.js. On the numbers here: the local-order measurements, the decade counts within \(\pm0.1\), the running of the marginal coupling, and the 1D Ising correlation function with its local decay rates were all computed here. The notion of a local power-law exponent, the logarithmic running of a marginal coupling (\(g\propto1/\ln\mu\)), the exact 1D Ising correlation function, finite-size scaling, and the ε-expansion \(\nu=\frac12+\frac{\epsilon}{12}\), \(\eta=\frac{\epsilon^2}{54}\) together with the 3D Ising literature values (\(\nu=0.6300,\ \eta=0.0363\)) are all well-known results or literature values and none are discoveries of this article — what this article did was gather them under the single question "where does this series' axis stop working" and compare them on one common measure, decades within \(\pm0.1\). The decade counts in §03 depend on the \(\pm0.1\) criterion chosen here; a different criterion gives different numbers. The 1D Ising example in §05 is at \(K=1\), away from criticality; finite-size scaling at a critical point itself is not treated. The ε-expansion in §06 is carried only to first order (second for \(\eta\)); higher orders are known to approach the 3D values. This article proposes no wider-reaching axis — that none was found is the result. — To print, use your browser's Print and "Save as PDF."