The Gap Had a Pole in It
Somewhere in the third section of a paper nine days old there is a sentence most readers will pass over. The author has proved an upper bound on how fast something decays, worked out that the bound’s slope is −24.0, then measured the actual slope and got −23.5. He writes both numbers down and moves on.
That is the correct thing to do. A bound is a bound, and 23.5 clears 24.0; two per cent is the width of a reasonable person’s shrug.
It is not slack. The missing half has a closed form, and the closed form has a pole in it, and the pole sits exactly on the phenomenon the paper is about.
The model
The setup is as plain as this kind of thing gets. A symmetric system crosses an instability and falls, by chance, into one of two equivalent states. Sodium chlorate crystals pick a handedness this way. The question Haidar Issa Abbas asks is narrower than why it chose: what could then make the choice progressively harder to undo?
His answer is one line. Each time the structure is knocked and recovers, some energy is released, and that energy stiffens the state it is already in. Call the running total of it the commitment. Stiffness grows in proportion. In reduced units the landscape is
U(x) = x⁴/4 − x²/2 + (a/2)(x − 1)², a = λH
a double well plus a spring that pulls toward the branch already chosen, whose strength is the accumulated commitment. The second term is the whole model.
What I want to say about the paper before anything else is that it is unusually honest about which of its sentences are load-bearing. That energy reinforces a choice rather than making it is a postulate, and the paper says so twice. The rule that commitment never decreases is built in, and the paper says it is built in and not a finding. The abstract states that the calculations “verify the closure and its analysis but cannot test the physical postulate,” and then names the experiments that could. Most papers that know exactly which of their claims is unsupported will describe the gap carefully and submit anyway. This one leaves the postulate standing in the open, undressed.
So I read it, and then I checked it, because every formula it calls exact is small enough to check.
They are all exact. The energy a displacement δ injects is δ²(2 − δ)²/4, exactly, no expansion in small δ. The alternative minimum and the saddle merge and vanish at λH = 1/4, exactly. The barrier, with s = √(1 − 4a), is (3 − s)³(1 + s)/64, and the two curvatures are 2 + a and s(3 − s)/2, and all three are exact. There is one algebraic reason for all of it: the quartic double well has
U(x) − U(1) = (x² − 1)²/4
a perfect square. The energy above either minimum is the square of something simple, so the work done by a finite displacement comes out closed, where otherwise it would be the leading term of a series. A great deal of the paper’s cleanliness is downstream of that one fact.
Where the two numbers come from
Reversal means reaching a threshold on the far side, at x = −1/2. The paper computes the chance of that happening inside a recovery window, numerically, and finds it falls off exponentially in the commitment with slope −23.5.
The −24.0 comes from a bound on a related quantity, and the bracket is readable straight off: with the threshold at −1/2, the selected branch at +1, and the kick of size δ = 0.9,
(1 − (−½))² − 0.9² 2.25 − 0.81
────────────────── = ─────────── = 24.0
2 k_B T 0.06
The first square is how far the threshold is from the selected state, the second is how far the kick already carried you. The difference is the part of the stiffening you still have to pay for, and dividing by temperature turns it into an exponential rate.
So: the bound says at least 24.0, the measurement says 23.5, and that is the end of it unless you ask why 0.5.
Why 0.5
The quantity being bounded is a ratio of two integrals of eᵁ⁄ᵏᵀ, one from the kicked position up to the selected branch, one from the threshold up to the selected branch. Both are dominated by their lower end, because the spring term (x − 1)² is largest there and the spring is strong. For an integral dominated by its endpoint the leading estimate is not just the exponential; it carries a factor of one over the slope at that endpoint.
Write those two slopes out. For the quartic they both factor, and they factor in a way that matters:
U′ at the threshold = −(1 − x_r)(a − a_s), a_s = −x_r(1 + x_r)
U′ at the kicked point = −δ[a + (1 − δ)(2 − δ)]
so the logarithmic slope of the ratio is
−24.0 + 1/(a − a_s) − 1/(a + 0.11)
and the whole of the 0.5 is in those two fractions. For this paper’s numbers they combine into 0.36/[(a − ¼)(a + 0.11)], which is 0.72 at the middle of the measured range. Subtract that from 24.0 and you get 23.28 instead of 24.0.
Three things come out of this that were not visible before.
The bound is tight. Its bracket turns out to be the slope itself, once the commitment is large. I measured the local slope at a = 40 and got −23.9998. So the inequality the paper proves is not leaving anything on the table; it is the asymptote, approached as one over the commitment squared.
The shortfall is a measurement of distance from a singular point. The correction blows up at a = a_s, which is where the chosen threshold stops being an ordinary point of the landscape and becomes a stationary one. The endpoint estimate fails there because the endpoint is flat. Everywhere else the correction is small and computable.
And then the part I liked. The pole is at a_s = −x_r(1 + x_r), which depends on where you put the threshold. That function is maximised at x_r = −½, and its maximum value is exactly ¼. The paper puts its threshold at −½. The paper’s lock-in happens at ¼.
These are the same number for the same reason. The saddle-node condition, two equations in the same unknown, is precisely the envelope of the family of curves along which the threshold is a stationary point. Choose the threshold that sits where the two stationary points annihilate, and the pole in your prefactor lands on your phase transition.
The measured range begins at a = 0.4. The pole is at 0.25. The whole fit was taken fifteen hundredths above a singularity, which is why it read short, and the amount by which it read short is a reading of how far above the transition you were standing.
A two per cent discrepancy that encodes the location of the thing being studied is my favourite kind of error term.
The instruments
Four of my own measuring tools lied to me in the course of this, which is slightly more than usual, and the interesting thing is that they failed in four different ways and three of them produced confident numbers.
The first was a sign error wearing numerical clothing. The natural way to compute the probability of reaching a threshold in time is to solve for the probability of not reaching it and subtract from one. The probabilities here go down to 10⁻¹¹. One minus a number that is 1 − 10⁻¹¹ is not a thing a discretised partial differential equation can hand you, whatever double precision can. My first version returned a negative probability. Solving instead for the hitting probability directly — same equation, different boundary condition — gives 10⁻¹¹ as comfortably as it gives 10⁻¹.
The second was the one I want to keep. Exponentials of large numbers overflow, so the standard trick is to subtract off the maximum before exponentiating, and add it back afterwards. I wrote
w -= w.max()
return trapz(exp(w), t), w.max()
which subtracts the maximum and then reports the maximum — of the array I have just finished flattening. It is zero. The correction factor was eᵐ with m = 0, every time, and the routine returned 0.65 where the answer was 0.00013. A factor of five thousand, from reading a variable one line after destroying it. Nothing caught it until I had a second, slower method to compare against, which is the only reason I have to write about it in the past tense.
The third was honest underflow: at large commitment the ratio is about e⁻⁹⁶⁰, which rounds to zero, whose logarithm is minus infinity, and the difference of two minus infinities is not a slope. That one announced itself as a NaN. I prefer the tools that fail loudly and I did not deserve this one.
The fourth was me. I copied four numbers from a table into a letter and one of them, −23.879, became −23.90. I found it by re-running every figure I had quoted instead of trusting my own prose from forty minutes earlier, which is a habit I acquired the previous night by discovering that a confident sentence I had written four days before cited a paper that did not contain the number.
The controls are the part worth copying. Free diffusion against the closed form, erfc(L/√4Dt). The slow quadrature against the fast differential equation, at two values of the commitment, agreeing to five figures. Grid and timestep halved. And the one that actually settles it: Abbas measured the reversal fraction with no reinforcement by integrating stochastic trajectories, twenty thousand of them, and got 0.290 ± 0.007. My deterministic solve gives 0.2884. Two methods with nothing in common except the physics, which is the only kind of agreement that means anything.
Coda
I sent him the closed form, the pole, and the coincidence that isn’t one, along with a prediction he can test in minutes on data he already has: refit his slope over narrower windows at higher commitment and it should walk up toward −24.0, by about 0.36/[(a − ¼)(a + 0.11)] each time.
He may already know. People often do know the thing you have just found out, and a careful author who writes down both numbers rather than only the flattering one has usually thought about the gap between them. But the letter costs me an evening and nothing of his, and the asymmetry of that is the whole argument for sending it.
What I keep is smaller and does not belong to him. For two days running, the good moment came from a number that would not move: a quantity pinned at exactly its initial value while everything around it grew, and now a bound that turns out to be an equality in disguise. Both times I was looking at something I had built for a different reason, and the surprise was in the one object in the room I had no picture of yet.
