Counting the places where "you may discard" fails ── a claim only acquires content once you count its exceptions
Four episodes of saying "you may discard." But if you may always discard, that isn't a claim, it's a slogan. A claim gets its content only when you count the places where you must not. Coarse-graining breaks in roughly three ways ── (1) all scales couple (critical points); (2) energy cascades between scales (turbulence); (3) the microscopic is amplified into the macroscopic (chaos). The third is especially brutal: improve your initial-condition accuracy a thousandfold and the horizon over which you can predict barely moves, because accuracy is trapped inside an exponent and only enters through a logarithm. And there is a speed limit on how fast things can break ── with black holes running right at it.
| Type | What happens | Examples | How coarse-graining fails |
|---|---|---|---|
| ① scales couple | the correlation length diverges and structures of every size matter at once | critical points, phase transitions | you cannot "drop the small end" (no small end exists that separates) |
| ② energy flows between scales | big eddies break into small eddies, small eddies become heat (a cascade) | turbulence | the macroscopic equations don't close (the discarded terms matter) |
| ③ microscopic amplified to macroscopic | tiny differences in initial conditions grow exponentially | chaos, weather, the three-body problem, life | the \(10^{-15}\) you discarded returns as \(10^{0}\) after a finite time |
What they share is that one of Episode 1's conditions is broken: scale separation. Coarse-graining works only while microscopic and macroscopic are orders apart and not talking to each other. Once the conversation starts, what you discarded comes back.
Chaos means two nearby initial conditions separate exponentially. With separation \(\delta\), \(\delta(t)\approx\varepsilon\,e^{\lambda t}\), where \(\lambda\) is the Lyapunov exponent.
Define "predictable" as "until the separation exceeds a practical tolerance \(\Delta\)." Then:
\(\varepsilon\) is your initial accuracy. The fatal part is the \(\ln\). Improve your measurements a thousandfold and you gain \(\ln 1000/\lambda\approx 6.9/\lambda\) ── only seven Lyapunov times. A millionfold gets you fourteen. Effort is crushed by a logarithm.
The atmosphere's Lyapunov time is roughly 1–2 days (\(\lambda\sim0.5\ \mathrm{day^{-1}}\)). Suppose we shrank today's initial-condition error a thousandfold by multiplying the observation network. The gain is
$$\Delta t=\frac{\ln 1000}{\lambda}=\frac{6.9}{0.5\ \mathrm{day^{-1}}}\approx 14\ \text{days}$$Two weeks, for a thousandfold effort. That is why the deterministic prediction limit is quoted as about two weeks. Launch a thousand times as many satellites and next month's weather is still out of reach. The failure of coarse-graining absorbs human effort in the shape of a logarithm.
The figure below runs the simplest chaotic system (the logistic map \(x\to 4x(1-x)\)) with two initial values that differ by a hair. The vertical axis is the separation of the two, on a logarithmic scale.
Use the slider to shrink the initial difference \(\varepsilon\). The curve just translates to the right; the slope never changes ── which is what "predictability grows only like \(\ln(1/\varepsilon)\)" looks like. Press "Improve accuracy 1000×" and see that an enormous observational effort buys roughly ten steps.
The other way to break is when scales exchange energy. In turbulence a big eddy breaks into smaller ones, which break into smaller ones, until it all becomes molecular heat (the energy cascade). Every scale is connected at once, and in one direction.
Kolmogorov scaling says the number of grid points needed to directly simulate a flow at Reynolds number \(\mathrm{Re}\) is
$$N\sim \mathrm{Re}^{9/4}$$Around an airliner \(\mathrm{Re}\sim10^{7}\), giving \(N\sim10^{15.75}\approx 6\times10^{15}\) points. Several numbers per point, plus time stepping ── no supercomputer today comes close. So engineering uses turbulence models (i.e. coarse-grained approximations), and those models are not exact; they contain empirical fitting, because we are forcibly coarse-graining a place where coarse-graining is broken. That is also the root of why the existence and smoothness of solutions to the Navier–Stokes equations is still a million-dollar Millennium Prize problem.
Breaking is not always defeat. The star of Episode 4, the critical point, is itself a breakdown of coarse-graining ── the correlation length diverges and you can no longer separate off the small scales.
But at a critical point, the lost scale separation is replaced by scale invariance ── a new symmetry that says "the same at every magnification." So there is no need to despair; the renormalization group handles it. You cannot coarse-grain, but coarse-graining changes nothing.
There is an upper bound on the speed of chaos, \(\lambda\). Proved in 2016, it is the most beautiful theorem in this episode.
In a quantum system at temperature \(T\), initial-condition information cannot be scrambled faster than this (the Maldacena–Shenker–Stanford bound). And the thing that saturates the bound is a black hole ── which is why black holes are called the fastest scramblers in nature. The time for information dropped in to spread over the whole horizon is $$t_*\sim\frac{\beta}{2\pi}\ln S$$ ── growing only with the logarithm of the entropy \(S\) (the bit count). Extraordinarily fast.
Notice that \(\ln\) again. The predictability time of chaos is a logarithm; the scrambling time of a black hole is a logarithm. Whenever information gets scrambled, exponentials and logarithms show up as a pair. Episode 4 of the sister series "Black Holes That Click" is exactly this story told from the black-hole side.
Established: exponential trajectory separation in chaotic systems and \(t_{\text{pred}}=\lambda^{-1}\ln(\Delta/\varepsilon)\); the roughly two-week predictability limit of the atmosphere; that direct numerical simulation degrees of freedom grow as \(\mathrm{Re}^{9/4}\) from Kolmogorov theory; correlation-length divergence and scale invariance at critical points; the quantum chaos bound \(\lambda\le2\pi k_BT/\hbar\) (Maldacena–Shenker–Stanford 2016); and that black-hole scrambling time \(\sim(\beta/2\pi)\ln S\) saturates it (the fast-scrambler conjecture, Sekino–Susskind) ── all established theoretical results or standard understanding.
Caveats: (1) "Three ways to break" is an organising convenience, not an official taxonomy ── in reality they overlap (turbulence is also chaotic; chaos appears near critical points too). (2) Turbulence is an open problem: Kolmogorov scaling receives intermittency corrections, and existence and smoothness for Navier–Stokes remains a Millennium Prize problem. (3) The claim that black holes saturate the bound rests on calculations in specific models (AdS/CFT and the like) and on conjecture; there is no direct verification with real black holes. (4) The figure is the logistic map, an idealised one-dimensional map, not a reproduction of weather or turbulence (it is there to show the essential point that predictability grows only logarithmically).
Coarse-graining breaks in three ways ── critical points (the correlation length diverges and all scales couple), turbulence (energy flows between scales; directly simulating the air around an aircraft needs \(\mathrm{Re}^{9/4}\approx10^{16}\) grid points) and chaos (the microscopic amplified exponentially into the macroscopic). What is broken in all three is the premise that scales don't talk to each other.
Chaos in particular is merciless: predictability is \(t_{\text{pred}}=\lambda^{-1}\ln(\Delta/\varepsilon)\) ── accuracy is trapped inside a logarithm, so a thousandfold observational effort buys seven Lyapunov times. That is the two-week wall in weather forecasting. But breaking isn't always defeat: critical points hand you scale invariance instead (Episode 4), and even in chaos, going up one level of coarse-graining lets you talk statistics (weather vs climate). And the breaking has a speed limit, \(\lambda\le2\pi k_BT/\hbar\), saturated by black holes ── whenever information gets scrambled, exponentials and logarithms come as a pair.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, changing ε translates the curve without changing its slope. "Improve accuracy 1000×" shows how much effort a logarithm can absorb. "See the answer" opens each solution.