Entropy and disorder — Boltzmann S = k·ln(W): Visual walkthrough

If you have not met the words microstate and macrostate yet, read them once in Microstates and Macrostates — but we will re-picture them below anyway.

Step 1 — What are we counting? (Microstates vs macrostates)

WHAT. Imagine four coins in a row. The thing you see from outside is "how many heads." The thing you don't see is which particular coins are heads. The entropy $S$ we just named will end up being built out of this hidden count.

WHY. Everything that follows is about a single number: how many hidden arrangements produce the same visible result. Get this picture rock-solid and the rest is bookkeeping.

PICTURE. In the figure, the top box is one macrostate: "2 heads out of 4." Below it sit all the different hidden arrangements — the microstates — that all look like "2 heads" from outside.

Figure — Entropy and disorder — Boltzmann S = k·ln(W)

Here $W$ is just a plain counting number. No logarithm yet — we have not needed one.

Step 2 — The fundamental rule: every microstate is equally likely

WHAT. Nature does not weight the hidden arrangements. Coin-pattern HTHT is exactly as likely as HHTT. Each individual microstate has the same tiny probability.

WHY. This is the one physical assumption of the whole theory (the equal a-priori probability postulate). Without it, "counting" would not predict anything.

PICTURE. Every small tile below has the same size — same probability. A macrostate's probability is then just how many tiles it owns.

So far our measure of "how likely / how many ways" is $W$ itself. Now watch why $W$ alone is the wrong quantity to call entropy.

Step 3 — Two systems: counting multiplies

WHAT. Put system $A$ (multiplicity $W_{A}$ ) next to an independent system $B$ (multiplicity $W_{B}$ ). How many ways can the combined system be arranged?

WHY. Real thermodynamics is about big systems built from small ones. We must know how the count behaves when we glue systems together.

PICTURE. A grid: each row is one arrangement of $A$ , each column one arrangement of $B$ . Every cell is a valid combined microstate. Rows $\times$ columns $=$ total cells.

Key observation for later: the combined count grows by a product, not a sum.

Step 4 — But the quantity we want must add

WHAT. We want an "amount of entropy" $S$ . Energy, mass, volume are extensive: two identical boxes side by side have double the amount. We demand entropy behave the same way.

WHY. Entropy is supposed to be a bulk "amount" like heat content — it should scale with how much stuff you have. Two identical gas boxes ⇒ twice the entropy.

PICTURE. Two identical boxes. Their volumes add ( $V + V = 2 V$ ), their entropies must add ( $S + S = 2 S$ ). But from Step 3 their counts multiply ( $W \times W = W^{2}$ ). There is a clash we must resolve.

Step 5 — What function turns $\times$ into $+$ ? The logarithm

WHAT. Write $S = f (W)$ for some unknown function $f$ . Steps 3 and 4 together demand: $f (W_{A} W_{B}) = f (W_{A}) + f (W_{B})$

WHY. This single equation pins down $f$ — but only once we add two mild physical demands that entropy obviously has to obey:

Smoothness (continuity). If the count $W$ changes only slightly, the entropy $S$ should change only slightly too — no jumps. Physically, a tiny change in how many arrangements are available cannot make entropy leap discontinuously. This is what lets us pick out the logarithm rather than some pathological product-to-sum function.
Increasing ( $d S / d W > 0$ ). More available microstates should mean more entropy, never less. So $f$ must rise as $W$ rises. This is the demand that forces the scaling constant to be positive in the next step.

With those two demands, the continuous increasing solution of $f (x y) = f (x) + f (y)$ is unique — it is the logarithm.

PICTURE. The graph shows what a log does: the horizontal axis stretches multiplicatively ( $1, 2, 4, 8, \dots$ equally spaced), and the log turns those into an evenly rising ladder ( $0, ln 2, 2 ln 2, 3 ln 2$ ). Notice the curve is smooth (no jumps) and always climbing (increasing) — matching both demands. Multiplying the input adds a fixed step to the output — exactly the rule we need.

Recall Why

the logarithm and not just some function? Requirements ::: $f (x y) = f (x) + f (y)$ for all positive $x, y$ , with $f$ continuous (smooth, no jumps) and increasing ( $d S / d W > 0$ ). Unique solution ::: $f (x) = c ln x$ with $c > 0$ — no other continuous, increasing function does this.

Step 6 — Fixing the units: the constant $k$

WHAT. $ln W$ is a pure number (no units). Real entropy is measured in joules per kelvin (J/K). We multiply by a constant $k$ to bridge them. Because $f$ had to be increasing (Step 5), this constant must be positive: $k > 0$ .

WHY. The abstract "count" logic cannot know about joules or kelvins — those come from the thermometer-and-heat world of Clausius. The constant $k$ is the exchange rate between "nats of counting" and "J/K of physics," and its positivity guarantees more microstates ⇒ more entropy.

PICTURE. Two dials — a pure-number dial ( $ln W$ ) and a physical J/K dial — connected by a gear labelled $k = 1.38 \times 1 0^{- 23}$ J/K. Turning one turns the other in fixed proportion.

Step 7 — Does it reproduce real thermodynamics? (Free expansion check)

WHAT. Let $N$ gas molecules, each in volume $V$ , suddenly access $2 V$ (a partition is removed). Each molecule independently gets twice as many places to be.

WHY. A formula is only trustworthy if it matches a known result. Free expansion has a classical answer, so it is the perfect test. See Free Expansion of an Ideal Gas.

PICTURE. Left: molecules crowded in volume $V$ . Right: same molecules spread over $2 V$ . Each molecule's "position-choices" double, and choices multiply across molecules.

Step 8 — The edge cases (never leave a gap)

WHAT. Check the extreme inputs so no reader hits an untested scenario.

WHY. A derivation you can trust must survive its own boundaries: the smallest possible count, and the way "not doubling doubles S" behaves.

PICTURE. A number line of $ln W$ : at $W = 1$ it sits exactly at $0$ ; it can never dip below; and doubling $W$ nudges it up by only $k ln 2$ , not by a factor of two.

The one-picture summary

Everything above in a single flow: count → log → scale, tested against reality.

Recall Feynman one-liner — the single sentence to keep

Count the hidden arrangements ( $W$ ), take the log so two systems' scores add instead of multiply, scale by $k$ to reach joules-per-kelvin — and the result predicts real experiments like free expansion. That is $S = k ln W$ .

Connections

Parent: Entropy and disorder — the topic this page dives into.
Microstates and Macrostates — Step 1's bookkeeping.
Second Law of Thermodynamics — Step 2's "biggest pile wins."
Free Expansion of an Ideal Gas — Step 7's test case.
Clausius Entropy dS = dQ_rev / T — the thermodynamic twin $S = k ln W$ reproduces.
Boltzmann Distribution — how energy spreads across the microstates we counted.
Information Entropy (Shannon) — the same $ln (count)$ idea in bits.
Third Law of Thermodynamics — Step 8's $W \to 1 \Rightarrow S \to 0$ .