Cool it down and at some temperature the reaction refuses to stop ── and the 2π that appears there is the Unruh 2π
In chemistry you learn that reaction rates rise steeply with temperature ── the Arrhenius law \(k\propto e^{-E_a/k_BT}\), the probability of climbing over a wall thermally. But cool the system down and below some temperature the law stops working. The reaction should stop, and it does not. The rate becomes independent of temperature and settles at a constant value all the way toward absolute zero. It has given up climbing over and switched to slipping through. The switch happens at \(T_0=\hbar\omega_b/2\pi k_B\) ── and this \(2\pi\) is the same \(2\pi\), arriving for the same reason, as in the Unruh temperature \(T=\hbar a/2\pi ck_B\) from the finale of the sister series "Temperature That Clicks." Heat and tunneling were in the same arena from the start.
| Climb over thermally | Slip through | |
|---|---|---|
| Probability | \(e^{-E_b/k_BT}\) | \(e^{-2S_E/\hbar}\) |
| Temperature dependence | yes (cooling stops it) | none (cooling changes nothing) |
| What matters | only the wall's height | the wall's height, width and the particle's mass |
| Proposed by | Arrhenius (1889) | Gamow and others (1928) |
Set them side by side and one thing is immediate ── one changes with temperature and the other does not. So cooling far enough must flip which one wins. At high temperature heat wins; at low temperature tunneling wins.
Experimentally this shows up as a kink in the Arrhenius plot. Plot \(\log(\text{rate})\) against \(1/T\) and the thermal regime is a straight line of slope \(-E_b/k_B\), while the tunneling regime is horizontal. The position of the kink is \(T_0\).
The crossover comes from setting the two exponents equal. For that we need the tunneling exponent \(2S_E/\hbar\). The standard move is to approximate the top of the wall as a parabola ── call the (imaginary) frequency at the summit \(\omega_b\).
Take the wall to be \(V(x)=E_b-\frac12 m\omega_b^2x^2\) and the particle's energy to be \(0\), and feed it into Episode 1's formula. The turning points are at \(x_0=\sqrt{2E_b/m}\,/\,\omega_b\), and
$$S_E=\int_{-x_0}^{x_0}\!\!\sqrt{2m\big(V(x)\big)}\,dx =\sqrt{2mE_b}\;x_0\int_{-1}^{1}\!\sqrt{1-u^2}\,du=\frac{\pi E_b}{\omega_b}$$So the tunneling exponent is
$$\frac{2S_E}{\hbar}=\frac{2\pi E_b}{\hbar\omega_b}$$Here is the satisfying part. Read that in the same form as \(E_b/k_BT\) and ──
In other words ── the probability of tunneling is exactly the probability of getting over thermally at temperature \(T_0\). So for \(T>T_0\) heat is faster, and for \(T Put another way: however far you cool it, the system keeps behaving as though a heat bath at temperature \(T_0\) were present. Tunneling is also an apparent temperature that survives absolute zero.
Why did a \(2\pi\) appear? "It came out of the parabolic integral" is true as far as it goes, but there is a deeper reason.
| The "period" in the imaginary-time direction | What it fixes | |
|---|---|---|
| having a temperature ("Temperature That Clicks" Ep. 6) | \(\beta\hbar=\hbar/k_BT\) | the Boltzmann factor |
| tunneling (this episode) | \(2\pi/\omega_b\) | one round trip across the inverted valley |
| an accelerating observer ("Temperature That Clicks" Ep. 7) | \(2\pi c/a\) | the Unruh temperature |
As Episode 1 showed, tunneling is a round trip in imaginary time. The inverted valley (= the original wall) is parabolic near the summit, so the period of that round trip is the harmonic one, \(2\pi/\omega_b\).
Meanwhile, in a system at temperature \(T\) the imaginary-time direction is a circle of circumference \(\beta\hbar\). Whether the round trip fits on that circle decides the contest ── if \(\beta\hbar > 2\pi/\omega_b\) (cold, long circle) the round trip fits and tunneling happens; if \(\beta\hbar < 2\pi/\omega_b\) (hot, short circle) it does not, and the only way over is thermal. The boundary is
The figure plots the two lines just derived. Horizontal axis: temperature (log). Vertical: reaction rate (log). The line falling to the left is thermal, the flat line is tunneling, and they meet exactly at \(T_0\).
The preset buttons switch to real systems. Check that the same logic covers six orders of magnitude in temperature, from chemical reactions through hydrogen in metals to superconducting circuits.
| System | typical ω_b | T₀ | Meaning |
|---|---|---|---|
| a chemical reaction moving a hydrogen | 10¹⁴ s⁻¹ | about 120 K | tunneling takes over above liquid nitrogen |
| a reaction moving a heavy atom (carbon etc.) | 10¹³ s⁻¹ | about 12 K | invisible in an ordinary laboratory |
| hydrogen diffusing in a metal | 2×10¹³ s⁻¹ | about 25 K | diffusion becomes temperature-independent at low T |
| a Josephson junction (macroscopic quantum tunneling) | 10¹¹ s⁻¹ | about 0.1 K | dilution-refrigerator territory. On to Episode 4 |
The lighter the particle, the larger \(\omega_b\) and the higher \(T_0\) (\(\omega_b\propto1/\sqrt m\)). So "tunneling matters" applies first to hydrogen, then deuterium, then… That mass dependence becomes the tool for detecting tunneling inside enzymes in bonus ⑤ (swap hydrogen for deuterium; if the rate drops by orders, it was tunneling).
To be honest: \(T_0\) is not a clean "phase transition." In reality both routes are open at once, and the fastest path is often "climb partway thermally, then tunnel the rest" (thermally assisted tunneling). The kink in the figure is, in practice, rounded rather than sharp.
Even so, \(T_0\) is a good landmark. Above or below it, the kind of behaviour changes. And that the boundary can be written in the tidy form \(\hbar\omega_b/2\pi k_B\) is itself evidence that heat and tunneling are written in the same language.
Established: the Arrhenius law \(k\propto e^{-E_a/k_BT}\); that Arrhenius plots bend and become tunneling-dominated at low temperature (observed in many systems); the Euclidean action of a parabolic barrier \(S_E=\pi E_b/\omega_b\); the crossover temperature \(T_0=\hbar\omega_b/2\pi k_B\) (Langer 1967, Affleck 1981, within the Callan–Coleman framework); that finite-temperature tunneling is decided by whether the bounce solution fits on the imaginary-time circle \(\beta\hbar\); the observation of macroscopic quantum tunneling in Josephson junctions (Voss–Webb 1981, Devoret–Martinis–Clarke 1985 and others); and the isomerisation of methylhydroxycarbene at 11 K with tunneling control (Schreiner et al. 2011, Science). All standard results.
Points to note: (1) \(T_0\) is a crossover, not a sharp transition. Near it, thermally assisted tunneling dominates and the kink is rounded (for some barrier shapes it can be sharper, "first-order-like"; classifying this is itself a research topic). (2) \(S_E=\pi E_b/\omega_b\) is a parabolic approximation and deviates for real barrier shapes; the low-temperature magnitude in particular is sensitive to shape. (3) Real systems have dissipation (coupling to the environment), which lowers \(T_0\) and suppresses tunneling (Caldeira–Leggett theory). The figure idealises this away. (4) The "rate" in the figure is a relative value with the prefactor normalised to 1, not an absolute one. (5) The \(\omega_b\) and \(T_0\) in the table are order-of-magnitude guides and move by factors of a few from system to system.
There are two ways over a wall. Climbing over, \(e^{-E_b/k_BT}\), depends on temperature; slipping through, \(e^{-2S_E/\hbar}\), does not. So cooling must eventually flip the winner, and the Arrhenius plot bends.
For a parabolic wall, \(2S_E/\hbar=2\pi E_b/\hbar\omega_b\). Read that as \(E_b/k_BT_0\) and you get \(T_0=\hbar\omega_b/2\pi k_B\) ── which also means tunneling is an apparent temperature \(T_0\) that survives absolute zero.
And this \(2\pi\) is the Unruh \(2\pi\). Both come from the line "the period in the imaginary-time direction is \(2\pi/(\text{an angular frequency})\)" ── temperature being the imaginary-time circle \(\beta\hbar\), tunneling the round trip \(2\pi/\omega_b\) across the valley, and the crossover being where the two periods coincide. About 120 K for hydrogen, about 0.1 K for a Josephson junction: systems six orders apart, lined up by one formula.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the barrier height and summit frequency shifts the position of the kink T₀. The preset buttons travel six orders of magnitude in temperature, from chemical reactions to Josephson junctions. "See the answer" opens each solution.