What Exactly Is Entropy? Boltzmann’s S = k log W, Explained Simply

Entropy is usually introduced as “disorder,” and then everyone spends years un-learning it. A better one-line answer: entropy measures how many microscopic arrangements are consistent with what you can see from the outside. Our Ludwig Boltzmann sticker puts it the way he might have: there are more ways to be a mess.

Where did the word entropy come from?

Entropy began as a quantity about heat, long before anyone could count atoms. Rudolf Clausius introduced it in an 1865 paper in Annalen der Physik (Clausius 1865^[1]). In his thermodynamic definition, when a small amount of heat δQ flows reversibly into a system at absolute temperature T, its entropy rises by

dS = δQ_rev / T

The division by T matters. The same joule of heat raises the entropy of a cold object more than a hot one. So when heat flows from hot to cold, the cold side gains more entropy than the hot side loses, and the total goes up. That is the second law of thermodynamics in its classic form: in an isolated system, total entropy does not decrease.

What does S = k log W mean?

Ludwig Boltzmann gave entropy a microscopic meaning. Describe a gas two ways. The macrostate is what you measure: pressure, volume, temperature. The microstate is the full list of where every molecule is and how it moves. Enormous numbers of microstates share the same macrostate. Boltzmann’s entropy is proportional to the logarithm of that number:

S = k log W

Here W is the number of microstates (more carefully, the volume of the region of possible states) compatible with the macrostate, and log is the natural logarithm. The Stanford Encyclopedia of Philosophy writes the same definition as S_B(M) = k log μ(X_M), where μ(X_M) is the measure of the macrostate’s region of state space and k is Boltzmann’s constant (SEP, Statistical Mechanics^[2]). Quanta Magazine summarises Boltzmann’s idea as entropy related to the number of equivalent ways a system’s particles can be arranged (Quanta Magazine^[3]).

How big is Boltzmann’s constant?

In the current SI, k has an exact defined value: k = 1.380 649 × 10⁻²³ J/K (NIST CODATA^[4]). It is tiny because it converts a pure count into the units of heat divided by temperature.

A worked example: 100 coins

Toss 100 coins. The macrostate “all heads” has exactly one arrangement, so W = 1 and log W = 0. The macrostate “50 heads, 50 tails” can be made in C(100, 50) ≈ 1.01 × 10²⁹ different ways. Nothing forbids all heads; it is simply outnumbered by about 10²⁹ to 1. Now replace 100 coins with the roughly 10²³ molecules in a mouthful of air and the odds against a spontaneous return to a special arrangement become absurd. That is why entropy “increases”: not because a force pushes it, but because high-entropy macrostates are overwhelmingly more numerous.

The Stanford Encyclopedia makes the same point carefully: the equilibrium macrostate is the one with the largest Boltzmann entropy, and the approach to equilibrium is explained as a move toward macrostates of higher Boltzmann entropy that is highly probable, not certain (SEP^[2]).

Is entropy the same as disorder?

Only loosely. “Disorder” works for a shuffled deck, but it misleads elsewhere. Oil and water separating looks more ordered, yet the total entropy (including the arrangement of water molecules) goes up. Crystals form from solution while releasing heat that raises the entropy of the surroundings. The safe word is multiplicity: how many ways.

Boltzmann vs Gibbs entropy

There is more than one formal definition. Alongside Boltzmann’s, physicists use the Gibbs entropy of a probability distribution ρ, S_G = −k ∫ ρ log ρ dx. The two coincide in important cases but are conceptually different, and the Stanford Encyclopedia discusses which one does what work (SEP^[2]).

Why does entropy point time forward?

The laws of motion for individual particles work equally well run backwards. The arrow comes from the starting point. In 1928 Arthur Eddington called the one-way direction of time “the arrow of time” and believed it was connected to entropy (Quanta Magazine, Arrows of Time^[5]). Many physicists explain the arrow by assuming the universe began in a very low-entropy state, an assumption philosophers call the Past Hypothesis (SEP, Thermodynamic Asymmetry in Time^[6]). Why the early universe was like that is not settled; Quanta describes it as happening “for reasons not fully understood or agreed on” (Quanta^[3]). Our pillar post Why Does Time Only Go Forward? goes deeper.

Is information a kind of entropy?

The two are closely linked. Rolf Landauer argued in 1961 that erasing information is a dissipative process with a minimum heat cost, and in 2012 Bérut and colleagues measured heat release approaching that limit when erasing a one-bit memory made of a trapped colloidal particle (Bérut et al. 2012^[7]). How general Landauer’s principle is, and how it should be derived, is still debated among physicists and philosophers. It is also the modern answer to Maxwell’s demon, the little sorter our Sortex sticker personifies.

Wear the second law

Entropy Warden enforces the second law; Entropy Echo, its heat-death sibling, is where it ends up. The full set lives in our thermodynamics tees. For the gentler intro to the laws, see Thermodynamics Made Simple.

FAQ

What is entropy in simple terms?

A measure of how many microscopic arrangements are consistent with a system’s large-scale state. More ways means higher entropy.

What do the letters in S = k log W stand for?

S is entropy, k is Boltzmann’s constant (1.380 649 × 10⁻²³ J/K, exact), and W is the number of microstates compatible with the macrostate.

Can entropy ever decrease?

Locally, yes, if entropy is exported elsewhere (a fridge, a growing crystal). For an isolated system, a decrease is not forbidden outright but is overwhelmingly improbable for large systems.

Is entropy disorder?

Only as a rough metaphor. “Number of ways” is the accurate version.

References

1. R. Clausius (1865), Annalen der Physik 201, 353 (the paper that introduced “entropy”). https://doi.org/10.1002/andp.18652010702
2. Stanford Encyclopedia of Philosophy, “Philosophy of Statistical Mechanics”. https://plato.stanford.edu/entries/statphys-statmech/
3. Philip Ball, “Physicists Trace the Rise in Entropy to Quantum Information,” Quanta Magazine (2022). https://www.quantamagazine.org/physicists-trace-the-rise-in-entropy-to-quantum-information-20220526/
4. NIST CODATA, Boltzmann constant. https://physics.nist.gov/cgi-bin/cuu/Value?k
5. Dan Falk, “Arrows of Time,” Quanta Magazine (2020). https://www.quantamagazine.org/what-is-time-a-history-of-physics-biology-clocks-and-culture-20200504/
6. Stanford Encyclopedia of Philosophy, “Thermodynamic Asymmetry in Time”. https://plato.stanford.edu/entries/time-thermo/
7. Bérut et al. (2012), “Experimental verification of Landauer’s principle linking information and thermodynamics,” Nature 483, 187. https://doi.org/10.1038/nature10872

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.