LE COMPTE — Entropia I

The microstates

Nothing but a counting problem — and yet the whole arrow of time is in it.

Put $N$ particles in a box and ask only one thing: how many are on the left? That is the macrostate, a single number. But every macrostate is realised by a crowd of microstates — which particles exactly, one by one. There is exactly one microstate with everything on the left, and $\binom{N}{N/2}$ with half and half. Push $N$ up: the hump becomes a needle. By $N=100$ almost every microstate lies a hair away from half — not because a law demands it, but because there is almost nothing else to find.

The box

N particles20

microstates in all—

in the richest macrostate—

all on the left1

in the central window—

S max / k = ln W—

the draw gave—

each bar: one macrostate, its height the number of microstates — green: the window within ±5 % of half

Macrostate: what you can measure: $k$, how many are on the left
Microstate: the whole list: which particle is where. $2^N$ of them.
Multiplicity: $W(k)=\binom{N}{k}$, the number of microstates wearing the same macrostate

Boltzmann

An equation short enough to fit on a gravestone — and it is on one.

Boltzmann tied counting to measurable heat with a single line: $S=k\log W$. The logarithm is not decoration, it is compulsory. Put two boxes side by side: the microstates multiply, $W=W_1W_2$, while entropy, like mass or volume, must add. Only the logarithm turns one into the other. Drag the slider and watch the same curve as in the previous chapter, this time in units of $k$.

The state of the box

k on the left20

$S = k \log W$

Ludwig Boltzmann · 1844–1906 · Zentralfriedhof, Wien

W (k)—

S / k = ln W—

short of the maximum—

verdict—

doubling the volume of one mole costs R ln 2 = 5.76 J/K

S(k)/k against k — flat at the top, which is exactly why equilibrium is indifferent

Diffusion, and the return

The second law forbids nothing. It only makes certain things implausible.

All the particles begin on the left, then the door opens. They spread — not because they repel one another, but because almost every path leads to the mixture. Entropy climbs and starts trembling about its maximum. Now the point: with $N=6$, wait a moment and you will see the box come entirely back to the left, in front of you, several times a minute. The second law has just been violated and nothing broke. Switch to $N=200$ and the same return takes $2^{200}$ steps: longer than the age of the universe by a factor with no name. That is the whole difference between “impossible” and “improbable”.

The gas

steps elapsed0

on the left—

S / k = ln W—

S max / k—

full returns seen0

typical wait for a return—

the box above, S(t)/k below — the pale line is the maximum chance never long exceeds

Recurrence: Poincaré: a bounded, energy-conserving system returns arbitrarily close to any state it has visited. Given time.
The catch: the time. For $N$ particles the wait scales like $2^{N}$ — the theorem is true and useless in the same breath.

Gibbs’ paradox

Two nearly identical gases give the same mixing entropy as two opposite ones. At exact identity it drops to zero at once.

Take the partition away between two different gases and entropy rises by $2Nk\ln2$ — every particle now has twice the room. Now make the two gases more and more alike. The value does not move: $2Nk\ln2$, again and again, however close they get. Then, at exact identity, it falls to zero all at once, because removing a partition between a gas and itself changes nothing whatever. A step, not a slope. Gibbs saw a scandal in this; the answer is that identical particles are indistinguishable in the strong sense — swapping two of them does not make a new microstate. That is the $N!$ you must divide by, and it is quantum mechanics arriving early.

The two gases

δ how different they are1

δ—

distinguishable?—

ΔS / k of mixing—

verdict—

for N per side: ΔS = 2Nk ln 2 — or exactly zero

the box above, ΔS(δ) below: a step, and the isolated point at δ = 0

Sur la version quantique du paradoxe et ce qui se passe quand « identique » devient une question de degré : Nature Communications 12, 1745 (2021) — Mixing indistinguishable systems leads to a quantum Gibbs paradox.

Rubber

A rubber band pulls because it has more ways of being short than of being long. That is the entire mechanism.

A steel spring pulls because its bonds store energy. A rubber band does not: its force is almost entirely entropic. The chain is a random walk, and the number of conformations with end-to-end length $L$ is $W(L)\propto e^{-L^2/2Mb^2}$ — short, there are countless; stretched, almost none are left. From that comes $S(L)$, and from that the force $f=kTL/Mb^2$, proportional to temperature. The consequence is startling and testable at home: heat a stretched rubber band and it contracts instead of expanding like everything else. Stretch one quickly against your lip: it warms. Let it go: it cools. Gough noticed this in 1802, Joule measured it in the 1850s.

The chain

L extension0.30

T temperature300 K

segments M60

L / L max—

S(L) / k—

force f—

f at 200 K, same L—

verdict—

one conformation among countless — the curve below is S(L)/k, greatest at rest

L'effet Gough–Joule et une salle de travaux pratiques bâtie autour de lui : Stretching Rubber, Stretching Minds (arXiv:1508.00538).

The fluctuations

Small enough and brief enough, and the second law starts failing — in a perfectly regulated way.

A bead in an optical trap being dragged: a system of a few femtojoules. Every run gives a different work $W$, because the heat bath pushes at random. On average $\langle W\rangle\ge\Delta F$ — the second law holds. But the left tail of the histogram is entirely real: in a measurable fraction of runs the bath returns more energy than you put in, and the entropy of the world falls for a moment. Jarzynski showed those runs are not noise: they are exactly what is needed for $\langle e^{-W/kT}\rangle=e^{-\Delta F/kT}$ to hold as an equality. Slow the pull down and watch the mean fall toward $\Delta F$ while the tail grows fatter.

The pull

v trap speed1.0

runs0

⟨W⟩ / kT—

ΔF / kT0

⟨e^(−W/kT)⟩—

runs with W < 00

verdict—

histogram of work in units of kT — camellia: the share that gives energy back instead of taking it

Théorèmes de fluctuation, égalité de Jarzynski, et leur vérification sur une épingle d'ARN tirée une molécule à la fois : F. Ritort, Nonequilibrium fluctuations in small systems: from physics to biology.

Épreuves the page checks by itself—

Six commissions in the counting-house — and an endless examen. The page measures; the combinatorics never lie.

Pitfalls

Six ways entropy catches the careless.

Entropy is not disordersalt water freezing grows an ordered crystal and raises the total entropy anyway. Lambert spent twenty years getting the word out of the textbooks.

A microstate has no entropyentropy belongs to the macrostate — to the description, to what you chose not to measure. A single configuration has none at all.

“Tends toward disorder” is not a forcenothing pushes the particles to mix. They move at random; it is the number of paths that is lopsided.

The second law is statisticalit does not forbid the return, it makes it implausible. At six particles you watch it happen, in chapter III.

ln W has no units, S doesit is $k$ that carries the joules per kelvin. Without it you are only counting states.

A rubber band is not a springits force comes from counting, not from stored energy: it rises with temperature instead of falling.

Continued in L’INFORMATION, where the same formula changes science — and Maxwell's demon is finally handed the bill.