Seeds and noise

The construction seeds its pulses with exponentially small forces. A fluid at temperature seeds them with noise. Same amplifier, different input.

This page is a back of the envelope, not a theorem. It compares two numbers: the relative amplitude at which a pulse of the construction starts, and the relative amplitude at which thermal noise would start the same mode. Both are then multiplied by the same growth factor.

The designed seed

On the pulses page the envelope of a pulse satisfies $P(v)\le e^{-c(v-L_s/2)^2/L_s}$, normalized to one at the midpoint. The seed at $v=0$ is therefore at most $e^{-cL_s/4}$. The slot length $L_s$ is of the order of $S_*=\ell^2$ where $\ell$ is the dyadic band index, $q\asymp 2^{-\ell}$. So the seed, relative to the pulse's own peak, is

$$\frac{\text{seed}}{\text{peak}} \lesssim \exp\!\big(-c'\,(\log_2 1/q)^2\big),$$

smaller than any power of the concentration scale $q$. The growth factor $G=1/P(0)$ is the reciprocal. At the peak the pulse amplitude is $q^{h/2}$ times the background speed, close to the background itself since $h<1/100$.

The thermal seed

Thermal fluctuations in a volume $\ell^3$ of fluid at temperature $T$ have velocity variance $k_BT/\rho\ell^3$ per component. Relative to the core speed $\nu/\ell$ at the same scale,

$$\epsilon_T(\ell) = \frac{1}{\nu}\sqrt{\frac{k_BT}{\rho\,\ell}},$$

which is $1.2\times10^{-4}$ for air at a millimetre and $1.5\times10^{-2}$ at the mean free path. It is a power of $\ell$, hence a power of $q$, hence larger than the designed seed at every scale the construction reaches. Only the projection of the noise onto the growing direction of the pulse equation is amplified, but a random field has an order-one projection onto any fixed mode, at a random phase.

What to look for: the designed pulse, seeded at $1/G$, peaks at the intended amplitude. The thermally seeded mode follows the same envelope from a higher start and peaks $G\epsilon_T$ times higher. Wherever the orange line crosses the background level, the linear description of the annulus has failed and the pulse cannot deliver its stress as designed. With the default values that happens; drag the thermal seed below $-\log_{10}G$ and it stops. The construction's own seeds sit far below that line at every stage, which is the point.

What this does and does not show

It shows that the constructed solution is not a solution of the thermally fluctuating equation with the same force, and that the difference is not small: the unmodelled input to the amplifier exceeds the modelled one by a factor $G\epsilon_T$ that grows without bound as the singular time approaches. It does not show that the thermal equation has no singularity under this or any force. A turbulent annulus might still concentrate; it would just not do so by this construction. And it takes the thermal noise at scale $\ell$ to be present at the start of each pulse, which is the fluctuation-dissipation relation at work: viscosity regenerates thermal fluctuations at scale $\ell$ on the time $\ell^2/\nu\asymp\tau$, which is also the duration of a pulse.

The seed comparison also clarifies what "fine-tuned" means for this proof. Every stage depends on a quantity, the seed, that is chosen smaller than any polynomial in the scale, and every physical perturbation is polynomial in the scale. The proof is right about the equation. The equation is a limit in which those seeds can be heard.

A test that could be run

Integrate the two-dimensional amplitude equation of the pulses page for one band with (a) the designed initial condition and (b) an ensemble of random initial conditions of relative size $\epsilon_T$, and record the stage at which (b) first exceeds the background. Then compare that stage with the stage at which the cumulative stress of (a) has cancelled the leading residual to the accuracy of one correction cycle, $q^{h/10}$. The first quantity depends on constants the paper fixes but does not evaluate. A careful reading of Sections 6 and 7 would extract them.

Back to the fluctuation question.