Thinking in Waves Episode 10 ／ Part IV · Constraints imposed on waves
Episode 2's half-order starts to pay off
The arrow of time lives
in the parity of the derivative order
Under \(t\to-t\), \(d/dt\to-d/dt\), so an \(n\)-th derivative picks up \((-1)^n\).
That alone classifies reversible from irreversible ── the dissipative term was always the odd one.
And fractional orders live continuously in between, as quantities that are actually measured.
Causality constrained real and imaginary parts; the bandwidth theorem constrained time and frequency widths.
This time a plainer question ── where is the direction of time written?
01What happens to a derivative under time reversal
| Order \(n\) | \((-1)^n\) | Under time reversal | Quantity |
|---|---|---|---|
| 0 | \(+1\) | unchanged | position \(x\) |
| 1 | \(-1\) | flips sign | velocity \(v\) |
| 2 | \(+1\) | unchanged | acceleration \(a\) |
| 3 | \(-1\) | flips sign | jerk |
Even orders are invariant, odd orders flip. That alone classifies reversible from irreversible.
02Score an equation of motion term by term
| Term | Order | Under time reversal | Character |
|---|---|---|---|
| \(m\ddot x\) (inertia) | 2 | unchanged | conservative |
| \(b\dot x\) (friction) | 1 | flips sign | ★ dissipative |
| \(kx\) (spring) | 0 | unchanged | conservative |
There is exactly one odd-order term ── and it is the only dissipative one. Let us check numerically: run forward, flip the velocity, run back the same time.
| \(b\) (friction) | \(x\) after forward run | \(x\) after flipping and returning | Back to start? |
|---|---|---|---|
| 0.00 | \(-0.839072\) | 1.000000 | ○ yes |
| 0.01 | \(-0.800801\) | 0.908981 | × no |
| 0.10 | \(-0.529209\) | 0.385025 | × no |
| 0.50 | \(-0.084776\) | 0.007654 | × no |
Conclusion of §02
Only with \(b=0\) does it return to exactly 1.000000.
With friction, flipping the velocity is not enough ── to return you would also have to flip the sign of the friction, which is a different physics.
03There are exactly two routes to reversibility
Order alone is not enough. The imaginary unit \(i\) matters too.
| Equation | Time derivative | Order | \(i\)? | T-symmetry |
|---|---|---|---|---|
| Wave equation | \(\partial^2u/\partial t^2\) | 2 | no | ○ reversible |
| Diffusion equation | \(\partial u/\partial t\) | 1 | no | × irreversible |
| Schrödinger | \(i\hbar\,\partial\psi/\partial t\) | 1 | ★ yes | ○ reversible |
| Damped oscillator (friction term) | \(b\dot x\) | 1 | no | × irreversible |
| Maxwell | \(\partial\mathbf{E}/\partial t\) | 1 | no | ○ reversible (as a pair) |
The heart of §03 ── exactly two routes
① be of even order (the wave equation)
② be of odd order but carry an \(i\) (Schrödinger)
How ② works: \(t\to-t\) sends \(i\hbar\partial\psi/\partial t\to-i\hbar\partial\psi/\partial t\), but taking \(\psi\to\psi^*\) at the same time flips the sign a second time and restores it.
★ Quantum mechanics is reversible thanks to the \(i\).
Maxwell's equations are first order too, but \(\mathbf{E}\) and \(\mathbf{B}\) come as a pair and \(\mathbf{B}\to-\mathbf{B}\) restores invariance ── the same mechanism in another guise.
04So what about fractional orders ── the main point
A non-integer \(\alpha\) is neither even nor odd. As measured in Episode 1, the phase of \((i\omega)^\alpha\) is \(90\alpha\) degrees.
| \(\alpha\) | Phase lag \(\delta\) | \(\tan\delta\) | Meaning |
|---|---|---|---|
| 0 | \(0°\) | 0 | purely conservative (spring) |
| 0.1 | \(9°\) | 0.1584 | |
| 0.3 | \(27°\) | 0.5095 | |
| 0.5 | \(45°\) | 1.0000 | ★ exactly halfway |
| 0.9 | \(81°\) | 6.3138 | |
| 1.0 | \(90°\) | \(\infty\) | purely dissipative (dashpot) |
The heart of §04
Fractional orders live continuously between reversible and irreversible.
And \(\tan\delta\) (the loss tangent) is a quantity genuinely measured in viscoelasticity.
05Where do real materials sit?
The loss tangent is a basic material measurement. Invert it: \(\alpha=(2/\pi)\arctan(\tan\delta)\).
| Material | \(\tan\delta\) | Effective order \(\alpha\) | Character |
|---|---|---|---|
| Fused silica | \(10^{-5}\) | \(6.4\times10^{-6}\) | tuning forks, resonators; almost a perfect spring |
| Steel | \(10^{-4}\) | \(6.4\times10^{-5}\) | structural |
| Glass | \(10^{-3}\) | \(6.4\times10^{-4}\) | |
| Hard rubber | 0.1 | 0.0635 | |
| Biological tissue | 0.2 | 0.1257 | fractional models are actually used |
| Asphalt | 1.0 | 0.5000 | ★ exactly \(\alpha=1/2\) |
Materials line up along the axis of derivative order
Fused silica sits at \(\alpha\approx6\times10^{-6}\) ── almost perfectly "even-order" (conservative), which is why it is used for resonators and tuning forks.
Asphalt sits at \(\alpha\approx1/2\) ── exactly halfway between reversible and irreversible, so it eats vibration.
★ Choosing a material is choosing a derivative order.
06Numerical check ── dissipation is proportional to \(\sin(\alpha\pi/2)\)
Apply a sinusoidal strain \(\varepsilon=\sin\omega t\) and integrate the dissipation per cycle \(\oint\sigma\,d\varepsilon\) for the stress \(\sigma=KD^\alpha\varepsilon\).
| \(\alpha\) | Numerical integral | Theory \(\pi K\omega^\alpha\sin(\alpha\pi/2)\) | Relative difference |
|---|---|---|---|
| 0.00 | 0.000000 | 0.000000 | \(1.8\times10^{-16}\) |
| 0.25 | 1.202235 | 1.202235 | \(2.1\times10^{-14}\) |
| 0.50 | 2.221441 | 2.221441 | \(7.8\times10^{-15}\) |
| 1.00 | 3.141593 | 3.141593 | \(3.5\times10^{-14}\) |
Conclusion of §06
Strength of the arrow of time ＝ \(\sin(\alpha\pi/2)\)
Zero at \(\alpha=0\), maximal at \(\alpha=1\), \(1/\sqrt2\) of that at \(\alpha=1/2\).
"How irreversible" is written as a trigonometric function of the order.
07[Hypothesis] Why does physics stay in \(\alpha\in[0,2]\)?
| Order | \(\sin(\alpha\pi/2)\) | Meaning |
|---|---|---|
| \(\alpha=0\) | 0 | fully reversible; position, spring |
| \(\alpha=1/2\) | 0.707 | halfway; the diffusion boundary, asphalt |
| \(\alpha=1\) | 1 | fully irreversible; friction, diffusion |
| \(\alpha=2\) | 0 | ★ reversible again; inertia |
| \(\alpha=3\) | \(-1\) | irreversible with the opposite sign (＝ a backward arrow) |
§07 ── as a hypothesis
In Episode 2 we saw that physics keeps to \(\alpha\in[0,2]\) (lower bound at the diffusion boundary \(-1/2\), upper bound from the Ostrogradsky instability).
[Hypothesis] Perhaps that range is exactly the half-period over which the arrow of time goes \(0\to1\to0\).
For \(\alpha>2\), \(\sin(\alpha\pi/2)<0\) ── a backward arrow ── and the sign changes at the same \(\alpha=2\) as the Ostrogradsky instability.
※ This is our conjecture. That the two coincide at \(\alpha=2\) is suggestive, but we give no rigorous argument connecting them.
Exercises
- What is wrong with "even a system with friction can be time-reversed in its equation"?
Show answer
Substituting \(t\to-t\) gives \(m\ddot x-b\dot x+kx=0\), which is a system with negative friction ── i.e. different physics.
Time-reversal symmetry means going back with the same equation. The numerical experiment in §02 is exactly that: with \(b\ne0\), flipping the velocity does not return you. - Why is the diffusion equation irreversible while Schrödinger's is reversible?
Show answer
Both are first order in time, but Schrödinger's has an \(i\).
Doing \(t\to-t\) and \(\psi\to\psi^*\) together, the sign flip of \(i\) cancels the sign flip of the time derivative (anti-unitary time reversal).
The diffusion equation has no \(i\), hence no escape route ── and indeed solving it backwards blows up numerically. - What \(\alpha\) do you want in a vibration-damping rubber?
Show answer
As large as possible (you want to eat vibrational energy). But \(\alpha\to1\) is a pure dashpot and cannot support a static load.
Practical damping materials sit around \(\alpha\approx0.3\) because that is the compromise between supporting (\(\alpha=0\)-like) and absorbing (\(\alpha=1\)-like).
Resonators — tuning forks, quartz, LIGO's suspension fibres — push \(\alpha\) as low as it will go. Design is choosing a position on this axis.
Summary The arrow of time was written in the order
Under \(t\to-t\), an \(n\)-th derivative picks up \((-1)^n\). Even orders are invariant, odd orders flip ── that alone classifies reversible from irreversible. In \(m\ddot x+b\dot x+kx=0\) there is exactly one odd-order term, and it is the only dissipative one (confirmed numerically: only \(b=0\) returns to 1.000000).
There are exactly two routes to reversibility ── be even-order, or be odd-order with an \(i\). Quantum mechanics is reversible thanks to the \(i\); the diffusion equation has no such escape.
And fractional orders live continuously in between: \(\tan\delta=\tan(\alpha\pi/2)\), a quantity actually measured in viscoelasticity. Fused silica sits at \(\alpha\approx6\times10^{-6}\), asphalt at exactly \(\alpha=1/2\). Choosing a material is choosing a derivative order.
Dissipation proportional to \(\sin(\alpha\pi/2)\) was confirmed numerically (agreement to \(10^{-14}\)). "How irreversible" is a trigonometric function of the order.
Strength of the arrow of time ＝ \(\sin(\alpha\pi/2)\).
And physics' range \(\alpha\in[0,2]\) is where the arrow makes one round trip (hypothesis).
From here we look at individual physical quantities with that eye. First, heat.
The diffusion equation's dispersion relation \(\omega=-iDk^2\) is purely imaginary ── heat does not propagate, it merely decays. And the decay length is set by frequency.
This document is Episode 10 of the series "Thinking in Waves". The computation is in kenshou/yajirushi.js. On this episode: the classification "even = reversible / odd = irreversible", the complex modulus \(E^*=K(i\omega)^\alpha\) and the springpot model are all standard results, not derived here ── what we did is confirm them numerically and set them side by side. The \(\tan\delta\) values in §05 are order-of-magnitude representatives, and real values depend strongly on frequency and temperature (viscoelastic measurements are normally reported as frequency–temperature pairs). An "effective order \(\alpha\)" can be read uniquely from \(\tan\delta\) only if the material is describable by a single springpot. The treatment of Maxwell's equations in §03 is simplified; strictly one must include the transformation of charges and currents. §07 is our hypothesis; we give no rigorous argument linking the Ostrogradsky instability to the sign change of \(\sin(\alpha\pi/2)\). ── To print, use your browser's Print → Save as PDF.