Thinking in Waves Episode 2 ／ Part I · Building the tools
Making the gap between differentiation and integration continuous
Differentiate by one half ──
and it turns out to be real
If differentiation is \(\times(i\omega)^1\) and integration is \(\times(i\omega)^{-1}\), there is no reason to keep the exponent an integer.
Move \(\alpha\) continuously and you get dB/oct \(=6\alpha\), phase \(=90\alpha\) degrees ── a clean interpolation.
And \(\alpha=-1/2\) turns out to be a quantity measured daily in battery laboratories.
Last time we found that differentiation and integration are multiplications on the Fourier side. So is there anything in between?
There is. And it is not a mathematical game — it is a measured physical quantity.
01Make the exponent continuous
Let us actually sweep \(\alpha\) and measure. Same signal as last time, comparing two tones two octaves apart.
| \(\alpha\) | Amplitude ratio (2 oct) | dB/oct | Phase advance | |
|---|---|---|---|---|
| \(-1.00\) | 0.2500 | \(-6.02\) | \(-90.0°\) | integration |
| \(-0.50\) | 0.5000 | \(-3.01\) | \(-45.0°\) | ★ half-integration |
| \(0.00\) | 1.0000 | \(0.00\) | \(0.0°\) | identity |
| \(+0.25\) | 1.4142 | \(+1.51\) | \(+22.5°\) | |
| \(+0.50\) | 2.0000 | \(+3.01\) | \(+45.0°\) | ★ half-differentiation |
| \(+1.00\) | 4.0000 | \(+6.02\) | \(+90.0°\) | differentiation |
| \(+2.00\) | 16.0000 | \(+12.04\) | \(180.0°\) | second derivative |
Conclusion of §01
$$\text{dB/oct}=6\alpha,\qquad \text{phase advance}=90\alpha\ \text{degrees}$$
The gaps between the integers interpolate perfectly smoothly.
\(\alpha=1/2\) gives \(+3\) dB/oct and \(+45°\) of phase.
02Half-differentiation is real ── it is measured every day
"Differentiate by one half" sounds abstract, but in an electrochemistry lab it is routine.
The boundary impedance when a species diffuses into a semi-infinite medium
This is half-integration itself
It shows up in every battery, fuel-cell and corrosion measurement. The 45° straight line on a Nyquist plot is exactly this \(\alpha=-1/2\) (phase \(-45°\), as in the table above).
| Phenomenon | Order \(\alpha\) | Content | Status |
|---|---|---|---|
| Warburg diffusion | \(-1/2\) | Boundary condition for semi-infinite diffusion | ★ measured routinely |
| Anomalous diffusion | variable | \(\langle x^2\rangle\propto t^\alpha\), \(\alpha\ne1\) | biological, porous media |
| Viscoelasticity (springpot) | \(0\sim1\) | stress \(\propto D^\alpha(\text{strain})\) | rubber, tissue, asphalt |
| Cole–Cole relaxation | \(0\sim1\) | \(\varepsilon(\omega)=\varepsilon_\infty+\dfrac{\Delta\varepsilon}{1+(i\omega\tau)^\alpha}\) | dielectric spectroscopy |
They have something in common
Fractional orders appear at a boundary, or in a disordered medium.
Warburg is what you get after squeezing the whole medium's diffusion into a single boundary point. Viscoelasticity is what you get after eliminating a polymer's internal degrees of freedom.
── That observation leads to the hypothesis in the next section.
03Why is physics usually integer-order?
This is the deep part. Write the fractional derivative in the time domain and its nature becomes visible.
★ There is an integral in it ── it looks at the entire past
| Order | Character | Meaning | Memory kernel |
|---|---|---|---|
| Integer (\(\alpha=0,1,2\dots\)) | local | determined by the present value alone | none (Markovian) |
| Fractional (non-integer) | non-local | a weighted sum over the past | \((t-\tau)^{-\alpha}\) (power law) |
[Hypothesis] §03
Physical laws are nearly always integer-order because nature is local — it has no memory.
Put the other way round ── fractional orders appear only after degrees of freedom have been integrated out.
Warburg: the medium's diffusion pushed into a boundary. Viscoelasticity: the polymer's internal degrees of freedom erased.
※ This is our own conjecture. It is, however, in falsifiable form ──
"every fractional-order system should be explainable as the coarse-graining of hidden degrees of freedom".
04Noise spectra ── which order does nature use?
Generate white noise, integrate it to order \(\alpha\), and measure the spectral exponent \(\beta\) (\(S(f)\propto f^{-\beta}\)).
| \(\alpha\) | Measured \(\beta\) | Name |
|---|---|---|
| \(0\) | \(-0.032\) | white noise |
| \(-0.25\) | \(0.467\) | |
| \(-0.50\) | \(0.966\) | ★ pink noise (1/f) ＝ half-integration |
| \(-0.75\) | \(1.466\) | |
| \(-1.00\) | \(1.965\) | Brownian noise (1/f²) |
| \(-1.50\) | \(2.965\) | black noise (1/f³) |
Integer-order integration moves \(\beta\) in steps of 2
★ \(\beta=1\) (1/f noise) is unreachable with integer orders
05[Hypothesis] Is 1/f noise nature's half-integral?
1/f noise turns up everywhere.
And there is still no unified explanation for the origin of 1/f noise. White (\(\beta=0\)) and Brownian (\(\beta=2\)) are straightforward, yet exactly the midpoint \(\beta=1\) remains a puzzle.
[Hypothesis] §05
Is \(\beta=1\) special because it is the half-integral?
The standard explanation today is "a superposition of a broad distribution of relaxation times". If time constants \(\tau\) are distributed with density \(1/\tau\), 1/f does indeed come out ── we checked it numerically:
| Frequency [Hz] | Superposed \(S(f)\) | \(f\cdot S(f)\) (constant ⇒ 1/f) |
|---|---|---|
| 0.01 | \(1.567\times10^{2}\) | \(1.567\) |
| 0.1 | \(1.567\times10^{1}\) | \(1.567\) |
| 1 | \(1.567\) | \(1.567\) |
| 10 | \(1.566\times10^{-1}\) | \(1.566\) |
| 100 | \(1.531\times10^{-2}\) | \(1.531\) |
Being honest about it
\(f\cdot S(f)\) is nearly constant ── "log-uniform time constants give 1/f" is correct.
But that only moves the question to "why is the distribution \(1/\tau\)?"
Our position is modest: adding the half-integral viewpoint at least changes the shape of the question. We do not claim to have solved it.
06[Hypothesis] There is an upper bound on the order
Line up the mechanical quantities by their derivative order relative to position.
| Order \(\alpha\) | Name | Note |
|---|---|---|
| \(-1\) | absement | it has a name, but is almost never used |
| \(0\) | position \(x\) | |
| \(1\) | velocity \(v\) | |
| \(2\) | acceleration \(a\) | ★ Newton's law lives here |
| \(3\) | jerk | a ride-comfort metric; genuinely used |
| \(4\) | snap | almost never used |
Physics essentially confines itself to \(\alpha=0,1,2\). Why stop at 2? ── there is an answer.
The heart of §06
The answer to "why \(F=ma\) and not \(F=m\cdot\text{jerk}\)" is: third order and above is unstable and cannot exist.
The spectrum of derivative orders has a ceiling.
※ Ostrogradsky's theorem is a rigorous result. The inference "hence nature stops at second order" is our own framing; there are other ways to argue against higher-order theories, and degenerate higher-order theories are known to evade the theorem.
Exercises
- In one line, why is the phase of half-integration \(-45°\)?
Show answer
The argument of \((i\omega)^{-1/2}\) is \(-\frac12\cdot90°=\mathbf{-45°}\).
That is exactly why it looks like a 45° line on a Nyquist plot. - Derive \(\beta=-2\alpha\).
Show answer
The amplitude is multiplied by \(|\omega|^\alpha\), so the power (amplitude squared) is multiplied by \(|\omega|^{2\alpha}\).
White noise has \(S\propto f^0\), so after the operation \(S\propto f^{2\alpha}\). Since \(\beta\) is the exponent in \(S\propto f^{-\beta}\), \(\beta=-2\alpha\).
\(\alpha=-1/2\) gives \(\beta=1\) ── exactly 1/f. - Why does "fractional order has memory" so rarely show up in physics?
Show answer
Because the fundamental laws are local. Even in field theory the Lagrangian is written using the field at a point and its derivatives at that point (locality).
Memory appears only after degrees of freedom are eliminated — pushing a medium into a boundary, coarse-graining internal structure, and so on.
So this episode's hypothesis says: when you see a fractional order, go looking for what was eliminated.
Summary The derivative order is another axis
Sweeping \(\alpha\) continuously gives dB/oct \(=6\alpha\) and phase \(=90\alpha\) degrees ── measured, and a clean interpolation.
And \(\alpha=-1/2\) (half-integration) is real ── the Warburg impedance \(Z\propto(i\omega)^{-1/2}\), measured daily in battery labs and visible as the 45° line on a Nyquist plot.
Fractional orders always appeared at a boundary or in a disordered medium. Written in the time domain they carry an integral kernel ── fractional orders have memory. Integer orders are local. [Hypothesis] Physics is nearly integer-order because nature is local.
For noise, \(\beta=-2\alpha\) holds exactly, and \(\beta=1\) (1/f noise) is unreachable with integer orders. [Hypothesis] perhaps 1/f has stayed a puzzle because it is the half-integral ── though we also verified the standard explanation (log-uniform time constants), and we claim no solution.
The lower bound of the order is \(-1/2\) (the diffusion boundary), the upper bound is \(2\) (Ostrogradsky).
Physics is written inside that band.
Remember that one line in the Episode 1 code ──
if (ω === 0) { output = 0; } ── integration cannot determine the \(\omega=0\) component.That is called the constant of integration. And perhaps the physical constants live there too.
This document is Episode 2 of the series "Thinking in Waves". The computation is in kenshou/bunsuu.js. On the numbers in this episode: the fractional derivative is implemented spectrally (multiply by \((i\omega)^\alpha\) in the FFT), which assumes periodic boundaries. It differs from the time-domain Riemann–Liouville definition in how initial conditions are handled (on a finite interval, edge effects appear). The noise exponents come from 16384 points of a linear-congruential pseudo-random sequence, averaged in logarithmic bins and least-squares fitted, and scatter by about ±0.05 between runs. The Warburg impedance, Cole–Cole relaxation and Ostrogradsky instability are all established results, not derived here. The "hypotheses" in §03 and §05 are our own conjectures, not claims from the literature ── the origin of 1/f noise in particular is an open problem, and this episode offers no new explanation. ── To print, use your browser's Print → Save as PDF.