Thinking in Waves Episode 28 ／ Part XI · Settling the hypothesis

The last link in the chain ── where the exponentials themselves come from

Why is nature so full of exponentials? ──
waves that cannot travel, and independent events. That was all Tunnelling, the Yukawa potential, the skin effect ── all from one sign, $k^2<0$.
The Boltzmann factor and the decay law ── because only the exponential turns addition into multiplication.
★ And last episode's empirical law is reproduced from nuclear physics with a single adjustable parameter.

Prerequisites: Episodes 6, 11, 19, 26, 27Computation: kenshou/shisuu.js

Last time the answer was "log-uniform comes out of the exponential map automatically." Which raises the next question ── ★ why is nature so full of exponentials?
Seen as waves, there are only two answers.

01Source ① Waves that cannot travel ── Episode 6's evanescent waves

The spatial part of a wave equation always has this form $$\frac{d^2\psi}{dx^2}=-k^2\psi\qquad\Longrightarrow\qquad k^2<0\ \text{gives, with}\ \kappa^2=-k^2,\ \frac{d^2\psi}{dx^2}=+\kappa^2\psi$$

whose only solutions are $\psi\propto e^{\pm\kappa x}$

The heart of §01

The only functions whose second derivative is proportional to themselves are these.
★ So a wave that cannot travel is necessarily an exponential ── there is no choice in the matter.

Numerically ── WR-90 waveguide (broad wall 22.86 mm, cutoff $f_c=c/2a=6.5571$ GHz).

Frequency [GHz]	Sign of $k^2$	$\kappa$ [1/m]	Attenuation [dB/m]	What it is
1.0	$k^2<0$	135.8	1180	★ exponential decay
3.0	$k^2<0$	122.2	★ 1061	★ exponential decay
6.557	$k^2<0$	0.899	7.81	just below cutoff
8.0	$k^2>0$	──	0	a travelling wave

★ Below cutoff that is 1061 dB per metre at 3 GHz ── the ferocity of the exponential.
In Episode 6 we read this as "mass = cutoff frequency". The $e^{-r/\lambda}$ of the Yukawa potential comes from the same $k^2<0$.

02The same exponential becomes tunnelling

Inside a barrier $E<V$, so again $k^2<0$ ── the same story. A rectangular barrier can be solved exactly, so we compare it with the exponential-only approximation.

Exact, versus the exponential alone $$T=\frac{1}{1+\dfrac{V^2\sinh^2\kappa a}{4E(V-E)}}\qquad\text{vs}\qquad T\approx\frac{16E(V-E)}{V^2}\,e^{-2\kappa a}$$

Electron, $E=0.5$ eV, $V=1.0$ eV → $\kappa=3.6226$ /nm

Barrier thickness [nm]	Exact $T$	Exponential approx.	Ratio
0.1	8.79e−1	1.94e+0	2.204 (★ approximation broken)
0.5	1.01e−1	1.07e−1	1.054
1.0	2.85e−3	2.85e−3	1.0014
2.0	2.04e−6	2.04e−6	1.00000

The heart of §02

For thin barriers the approximation breaks ($T>1$ at 0.1 nm ── a meaningless value). It only applies for $\kappa a\gg1$.
★ Beyond that the ratio sticks at 1.00 ── the prefactor $16E(V-E)/V^2=4.00$ is of order 1, and everything else is decided by the exponential.
★★ A 10 % change in thickness drops $T$ by 2.06×, and one decade costs only 0.32 Å ── smaller than an atom. That is why a scanning tunnelling microscope resolves single atoms.

03★ Rebuilding last episode's Geiger–Nuttall law from the exponent

In Episode 27 we put $\alpha$-decay half-lives on a straight line in $Z/\sqrt Q$ ── an empirical fit. This time we build it from the language of waves alone.

The WKB exponent for a Coulomb barrier (the Gamow factor) $$G=4\eta\Big[\arccos\sqrt x-\sqrt{x(1-x)}\Big],\quad x=\frac{R}{b},\quad \eta=\frac{Z_dZ_\alpha\alpha}{v/c},\quad b=\frac{Z_dZ_\alpha\hbar c\,\alpha}{Q}$$

The only adjustable parameter is $r_0$ (the nuclear radius coefficient). We use $r_0=1.20$ fm

Nuclide	$\eta$	$x=R/b$	$G$	$\log_{10}T_{1/2}$	$G/\ln10$
$^{232}$Th	27.44	0.1487	89.91	17.647	39.047
$^{226}$Ra	24.55	0.1804	73.43	10.703	31.888
$^{222}$Rn	22.39	0.2109	61.40	5.519	26.665
$^{214}$Po	18.46	0.2998	39.37	−3.784	17.098
$^{212}$Po	17.27	0.3418	32.60	−6.524	14.158

Least squares over 9 nuclides ── the predicted slope is 1 $$\log_{10}T_{1/2}=0.9961\cdot\frac{G}{\ln10}-20.768,\qquad R^2=0.9934$$

The heart of §03

★★ Slope 0.996 ── essentially the predicted 1.

Episode 27's empirical $1.5065\,(Z/\sqrt Q)$ turns out to have been the contents of the exponent itself.

The intercept means "how often the barrier is struck" ($T_{1/2}=\ln2/(\nu e^{-G})$). Compare it with the round-trip frequency inside the nucleus, $v/2R$.

Nuclide	$\nu$ from the intercept [1/s]	$v/2R$ [1/s]	Ratio
$^{232}$Th	1.74e+21	7.60e+20	2.29
$^{226}$Ra	1.06e+21	8.36e+20	1.27
$^{222}$Rn	9.70e+20	9.00e+20	1.08
$^{212}$Po	3.34e+20	1.15e+21	0.29
$^{235}$U	7.04e+18	8.10e+20	★ 0.009

§03 continued ── keeping the outlier

For 8 of the 9 nuclides $\nu=3.3\times10^{20}$–$1.7\times10^{21}$ /s ── a range of only 5×, and within 3× of the geometric estimate $v/2R$.
★★ Half-lives spanning 24 decades are accounted for by one exponent and one round-trip speed.

★ But $^{235}$U is off by 200×. This is a known effect called a hindered transition ── the $\alpha$ cannot connect the two ground states directly, and angular momentum or nuclear structure adds an extra barrier.
We keep the outlier rather than delete it ── ★ it marks the boundary: the exponent explains the barrier, and the prefactor contains different physics.

04Source ② Turning addition into multiplication ── the Boltzmann factor

The second source has nothing to do with waves. It comes from the structure of multiplication.

Fact	Form
Probability of two independent events together	multiplies: $P(A\wedge B)=P(A)P(B)$
Energy of two systems combined	adds: $E=E_A+E_B$
★ The function connecting them	$f(x+y)=f(x)f(y)$

Its only continuous solutions are $f(x)=e^{cx}$. Checked numerically ── from $f(1)=0.5$ and $f(x+y)=f(x)f(y)$ alone, build $f(1/2^n)$ by repeated square roots and assemble any $x$.

$x$	$f(x)$ built from the rule	$2^{-x}$	Rel. error
0.5	0.707106781187	0.707106781187	1.6e−16
$\sqrt2$	0.375214227246	0.375214227246	7.4e−16
3.7	0.076946525834	0.076946525834	7.2e−16

The heart of §04

It agrees even at irrational points ── the rule had determined the function completely.
★ So the Boltzmann factor $e^{-E/kT}$ is not something to be derived but the only form that connects addition to multiplication.
★ The same argument applies to decay ── memorylessness (whether you have survived so far does not affect what happens next) $\Leftrightarrow S(t+s)=S(t)S(s)\Leftrightarrow S(t)=e^{-\lambda t}$. The probabilistic counterpart of Episode 19's "a decay constant is an imaginary frequency".

05Where the two sources meet ── heat or tunnelling

There are two ways over a barrier ── climb it thermally (source ②, $e^{-E_b/kT}$, temperature dependent) or dig through it (source ①, $e^{-G}$, temperature independent). At low temperature tunnelling always wins.

The crossover temperature $$T_c=\frac{\hbar\omega_b}{2\pi k_B}$$

$\omega_b$ is the curvature at the top of the barrier

$\hbar\omega_b$ [eV]	$T_c$ [K]	Example
0.010	18.5	heavy-atom hopping
0.050	92.3	hydrogen diffusion
0.100	184.7	molecular rotation
0.400	738.8	C–H stretch

The heart of §05

★ Tunnelling visible at room temperature (300 K) needs $\hbar\omega_b\gtrsim0.16$ eV ── only light particles. Which is why hydrogen and electrons tunnel and heavy atoms do not.
★★ And the Arrhenius line ($\log k$ vs $1/T$) bends flat below $T_c$ ── that is measurable, and it is evidence that the source of the exponential has switched.

06★★ The synthesis ── stack exponentials and you get a power law

An exponential is a function with a scale (one time constant $\tau$). A power law is a function without one (self-similar). They look like opposites. And yet ──

★ An exact relation $$\int_0^\infty e^{-\lambda t}\,\lambda^{a-1}\,d\lambda=\Gamma(a)\,t^{-a}$$

$a$	$t$	Superposed exponentials	$\Gamma(a)t^{-a}$	Rel. error
0.5	1	1.77245185	1.77245385	1.1e−6
1.5	1	0.886226925	0.886226925	2.1e−14
1.5	10	0.0280249561	0.0280249561	2.1e−14

★★ A power law is a continuum of exponentials. And Episode 26's spectral representation $g(\tau)=\sin(\alpha\pi)/\pi\cdot\tau^{\alpha-1}$ is exactly this $\lambda^{a-1}$ rewritten in $\tau$ ── the same theorem.

As a special case, if the rates are log-uniform ($g(\lambda)\propto1/\lambda$, i.e. $a\to0$), you get not a power law but a logarithm.

$t$	Log-uniform superposition	$\ln(1/\lambda_{\min}t)-\gamma$	Difference
1	20.14605017	20.14605017	1.0e−9
100	15.54088009	15.54087999	1.0e−7
1000	13.23829589	13.23829489	1.0e−6

The heart of §06

Logarithmic relaxation ── the "forever, a little at a time" seen in glasses and residual polarisation.
★ It says, on the time side, the same thing Episode 27 said with "1/f comes from log-uniform".

07Listing where the exponentials come from

Where it appears	Form	Source	Example
Evanescent waves (Ep. 6)	$e^{-\kappa x}$	★ ① $k^2<0$	waveguides, total reflection
Yukawa potential (Ep. 6)	$e^{-r/\lambda}$	★ ① mass = cutoff	nuclear force
Tunnelling	$e^{-2\kappa a}$	★ ①	STM, $\alpha$ decay
Skin effect (Ep. 11)	$e^{-x/\delta}$	★ ① imaginary $k$	copper skin
Boltzmann factor	$e^{-E/kT}$	★ ② add ⇒ multiply	thermal activation
Decay (Ep. 19)	$e^{-\lambda t}$	★ ② memorylessness	radioactivity
Gain and loss	$e^{gz}$	② product of small factors	lasers

The heart of §07

The exponentials of physics come down to these two.
★ Source ① because the wave equation is second order (with $k^2<0$ an exponential is the only solution).
★ Source ② because probability multiplies while energy adds.
── And either exponential becomes log-uniform when its contents are spread out (Episode 27), and log-uniform stacked up becomes a fractional order (Episode 26).

Exercises

Using this episode's numbers, why can a scanning tunnelling microscope resolve the bumps of single atoms?
Show the answer
Because the current is exponential, so the distance resolution is the reciprocal of the exponent. With $T\propto e^{-2\kappa a}$ and $\kappa\approx3.6$/nm, one decade of current costs $\ln10/2\kappa=0.32$ Å.
★ Far smaller than an atomic radius (~1 Å). A 10 % height difference shows up as a factor of 2 in current.
★ "An exponential makes an instrument" ── if you want sensitivity, put the quantity inside an exponent. The same structure appeared with $\alpha$ decay in Episode 27.
What does $^{235}$U's 200× smaller $\nu$ mean for the framework used here?
Show the answer
It marks the boundary: the exponent (the barrier) is explained, but the prefactor (the striking rate) contains other physics. The Gamow factor only describes going straight through a Coulomb barrier; it says nothing about how the $\alpha$ assembles inside the nucleus (the preformation factor) or about an angular-momentum barrier.
$^{235}$U's $\alpha$ decay is a hindered transition that does not connect the two ground states directly, so those effects bite.
★ The claim here is "all 24 decades of variation are explained by the exponent", not "the prefactor is explained too". The outlier is kept in order to show that distinction.
If an exponential has a scale and a power law does not, why is it not a contradiction that stacking exponentials gives a power law?
Show the answer
Because no single scale is picked out. Each $e^{-\lambda t}$ has the scale $\lambda$, but when $\lambda$ itself is spread over every decade, which $\lambda$ dominates depends on $t$ ── multiply $t$ by 10 and the dominant $\lambda$ divides by 10, leaving the shape unchanged.
★ That is how scale invariance is manufactured. The $\lambda^{a-1}$ in $\int e^{-\lambda t}\lambda^{a-1}d\lambda=\Gamma(a)t^{-a}$ is precisely "equal weight in every decade".
★★ Scale invariance was built out of things that each have a scale ── the conclusion of Part XI.

Summary　Exponentials come from only two places

We answered the question left last time ── "why is nature so full of exponentials?" There are only two answers.

★ Source ①: waves that cannot travel. The only solutions of $d^2\psi/dx^2=+\kappa^2\psi$ are $e^{\pm\kappa x}$. Below cutoff a WR-90 waveguide attenuates 1061 dB per metre at 3 GHz. The same sign drives tunnelling ── the exact rectangular-barrier solution and the exponential-only approximation coincide once $\kappa a\gg1$, with a prefactor of order 1. One decade of current costs just 0.32 Å (the origin of STM sensitivity).

★★ And Episode 27's empirical law was reproduced from nuclear physics. Computing the Gamow factor $G=4\eta[\arccos\sqrt x-\sqrt{x(1-x)}]$ and comparing with $\log_{10}T_{1/2}$ gives slope 0.996 (predicted 1), $R^2=0.9934$. The striking rate from the intercept is $3.3\times10^{20}$–$1.7\times10^{21}$ /s for 8 of 9 nuclides ── within 3× of the round-trip frequency $v/2R$. Half-lives over 24 decades follow from one exponent and one round-trip speed. ($^{235}$U is off by 200× because it is a hindered transition ── marking where the prefactor's own physics begins.)

★ Source ②: the function that turns addition into multiplication. The continuous solutions of $f(x+y)=f(x)f(y)$ are exponentials only, and values built from $f(1)=0.5$ plus the rule agree to 15 digits even at irrational points. The Boltzmann factor and the decay law are not derived but forced. The two sources meet at the crossover $T_c=\hbar\omega_b/2\pi k_B$ (184.7 K at 0.1 eV) ── which is why hydrogen and electrons tunnel and heavy atoms do not.

★★ And spreading the rates and stacking the exponentials gives a power law.
$\int_0^\infty e^{-\lambda t}\lambda^{a-1}d\lambda=\Gamma(a)t^{-a}$ (confirmed to 14 digits)

★ Three episodes of Part XI complete a chain ── Episode 26 "fractional order ⇐ scale-invariant degrees of freedom", Episode 27 "scale invariance ⇐ the exponential map", Episode 28 "the exponential ⇐ waves that cannot travel + independence". ★★ Scale invariance was built out of things that each have a scale.

Next episode Part XI closed the chain, but what remains at the very bottom is that the wave equation is second order.
★ Next time we dig there ── why second? What happens at first order, and what breaks at third? Episode 12's pre-acceleration (causality breaking at third order) and Episode 15's "narrow rectangle $\alpha\in[-1/2,2]$" meet head-on here.

This document is Episode 28 of the "Thinking in Waves" series. The computations are in kenshou/shisuu.js. On the numbers here: the waveguide attenuation, the rectangular-barrier transmission, the Gamow factors and least squares, the functional-equation construction and the superposition integrals were all computed for this article. The $Q$ values, half-lives and mass numbers of the 9 $\alpha$ emitters are literature values. The Gamow factor formula, $T_c=\hbar\omega_b/2\pi k_B$, the fact that the only continuous solutions of $f(x+y)=f(x)f(y)$ are exponentials, and $\int e^{-\lambda t}\lambda^{a-1}d\lambda=\Gamma(a)t^{-a}$ are all well-known results and not discoveries of this article ── what this article did was line them up as "exponentials come from only two places" and check numerically that last episode's empirical law follows from the exponent. The tunnelling rate in §05 is a model (set so that it equals the thermal rate exactly at $T_c$); the real shape of the crossover depends on the system. The nuclear radius coefficient $r_0=1.20$ fm is a representative value, and moving it between 1.1 and 1.3 shifts $\nu$ by 2–3 decades ($4.9\times10^{22}\to7.3\times10^{19}$ for $^{232}$Th) ── so no precision beyond "of order $10^{21}$" is claimed for $\nu$. The classification in §07 is this article's own arrangement. ── To print, use your browser's Print → Save as PDF.