Boltzmann Entropy

Carve a single equation onto a tombstone and you have chosen the one idea you want to outlast you. Ludwig Boltzmann's headstone in Vienna reads $S = k \log W$ , and it is the most consequential three-symbol claim in physics: that the entropy of a chunk of matter is nothing but the logarithm of the number of ways its atoms can be arranged to look the same from the outside. Heat, the arrow of time, the impossibility of un-scrambling an egg — all of it reduces to counting. The number being counted is $\Omega$ , the multiplicity: how many microscopic configurations (microstates) are compatible with the macroscopic facts you can measure — energy $E$ , volume $V$ , particle number $N$ . A gas in a box has an astronomically larger $\Omega$ when spread uniformly than when crowded into one corner, and Boltzmann's relation $S = k_{\mathrm B}\ln\Omega$ says the uniform state therefore has more entropy. Equilibrium is simply the macrostate with the most microstates, and the second law is the statement that a system drifts toward it because there is overwhelmingly more room there. Why a logarithm? Because entropy must add when you place two systems side by side, while their microstate counts multiply — and the logarithm is the unique function that turns products into sums. That single requirement forces the form of the equation; we will prove it below as this article's owned result. The prefactor $k_{\mathrm B} = 1.380649\times10^{-23}\ \mathrm{J/K}$ is pure units — it calibrates counting to the historical Kelvin scale so that $1/T = (\partial S/\partial E)_{V,N}$ reproduces Clausius. Master this one relation and all of equilibrium thermodynamics follows by differentiating $S(E,V,N)$ .

Key takeaways

The Boltzmann entropy assigns to a macrostate the logarithm of its multiplicity: $S = k_{\mathrm B}\ln\Omega$ , converting a microstate count into a thermodynamic potential.
$\Omega(E,V,N)$ counts the accessible microstates on the energy shell; for a classical continuum it is a phase-space volume divided by $h^{3N}$ (to be dimensionless) and $N!$ (for indistinguishability).
The logarithm is forced, not chosen: independent systems have $\Omega = \Omega_1\Omega_2$ but entropy must be additive, and only $\ln$ turns products into sums.
All of equilibrium thermodynamics comes from derivatives of $S$ : $1/T = (\partial S/\partial E)$ , $p/T = (\partial S/\partial V)$ , $-\mu/T = (\partial S/\partial N)$ .
Because $\Omega \sim e^{cN}$ , entropy is sharply maximized at equilibrium — this is the microscopic content of the second law.
Key idea: entropy counts. It is $k_{\mathrm B}$ times the log of how many microscopic ways a macrostate can be realized, and maximizing that count selects equilibrium and yields all of thermodynamics by differentiation.

The core insight

Entropy counts. The Boltzmann relation $S = k_{\mathrm B}\ln\Omega$ says the entropy of a macrostate is $k_{\mathrm B}$ times the natural logarithm of the number of microstates that realize it. Maximizing that count — equivalently maximizing $\ln\Omega$ — selects the equilibrium macrostate and delivers every thermodynamic relation as a derivative of $S(E,V,N)$ .

Where this lives today

Boltzmann's relation is the load-bearing beam of modern computational science. A Monte Carlo simulation of a spin lattice or a protein does not compute $\Omega$ by brute enumeration — that is hopeless — but algorithms such as Wang–Landau and nested sampling estimate the density of states $\ln\Omega(E)$ directly, to many digits, for systems no one can solve by hand. Every reported entropy of a materials-science or biophysics simulation is, under the hood, a numerical evaluation of $k_{\mathrm B}\ln\Omega$ . The relation also anchors the physics of the very cold and the very informational. Adiabatic demagnetization cools paramagnetic salts to sub-kelvin temperatures by manipulating the spin entropy $S(B/T) = k_{\mathrm B}\ln\Omega_{\text{spin}}$ at constant total $S$ ; and Landauer's principle — that erasing one bit collapses two microstates into one and must dissipate at least $k_{\mathrm B}T\ln 2$ — is Boltzmann's counting applied to information, now measured directly in single-electron and colloidal experiments. Wherever "number of ways" matters, $S = k_{\mathrm B}\ln\Omega$ is the accounting.

The physical picture

Toss $N$ coins and ask for the number of heads. The outcome "half heads" is realized by a colossal number of individual head/tail sequences, while "all heads" is realized by exactly one. The Boltzmann entropy measures the size of that realization set on a logarithmic scale: the macrostate with the most sequences is the one you almost always see. Nothing about any single sequence is preferred — the majority macrostate simply owns far more of them. The logarithm is the bookkeeping trick that makes this into thermodynamics. Two isolated boxes with $\Omega_1$ and $\Omega_2$ microstates form a joint system with $\Omega_1\Omega_2$ microstates, because any configuration of one pairs with any of the other. We insist entropy add like energy, so we take the log, turning the product into a sum. The figure below shows why this taming matters: $\Omega$ itself is unwriteable, but $\ln\Omega$ is a sane, extensive $\sim Nk_{\mathrm B}$ .

Figure

The multiplicity Ω explodes as eᶜᴺ while its logarithm stays extensive

Because $\Omega(E)$ grows like $E^{cN}$ for $N$ particles, $\ln\Omega$ is a mountain with exponentially steep slopes: a fractional shift of order $1/\sqrt N$ in energy or density corresponds to a relative change in $\Omega$ of order $e^{-N}$ . That is why thermodynamics — the science of the single most probable macrostate — is reproducible to the last decimal. The next figure makes the sharpness visible for a two-box gas, where the equilibrium macrostate is a needle whose relative width collapses as $1/\sqrt N$ .

Figure

The entropy landscape S(n) is a peak whose relative width vanishes as 1/√N

The point the second figure makes is the whole of the subject in one image: the equilibrium macrostate is not a delicate balance but a summit, and its dominance over every competitor grows exponentially with $N$ . That is why the second law feels absolute even though the underlying dynamics is perfectly reversible.

The core equations

The central definition ties a thermodynamic quantity to a pure count. It is deceptively short, but every symbol carries weight, and the rest of statistical mechanics is its consequences.

S = k_{\mathrm B}\ln\Omega \qquad (1)

where: $S$ — thermodynamic entropy of the macrostate; $k_{\mathrm B} = 1.380649\times10^{-23}\ \mathrm{J/K}$ (exact, 2019 SI); $\Omega$ — number of microstates (or dimensionless phase-space volume) compatible with the macrostate. Read it as a bridge: the left side is measured with a thermometer and a calorimeter, the right side is combinatorics. For a classical system, microstates form a continuum, so $\Omega$ is a phase-space volume — but a raw volume has dimensions and overcounts identical particles. Two corrections fix this and turn the volume into a legitimate count.

\Omega(E,V,N) = \frac{1}{h^{3N}N!}\int_{E \le H \le E+\delta E}\mathrm d^{3N}q\,\mathrm d^{3N}p \qquad (2)

where: $H(q,p)$ — the Hamiltonian; $h^{3N}$ — the volume of one quantum phase-space cell, making $\Omega$ dimensionless; $N!$ — the Gibbs factor for indistinguishable particles; $\delta E$ — a thin shell width that affects $S$ only at $O(\ln N)$ . Omit the $1/N!$ and entropy stops being extensive — that omission is the Gibbs paradox, so these factors are physics, not decoration. Every equilibrium thermodynamic relation is now a partial derivative of the single function $S(E,V,N)$ . This is the payoff: no separate laws for temperature, pressure, and chemical potential — all three are slopes of the entropy surface.

\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{V,N},\quad \frac{p}{T} = \left(\frac{\partial S}{\partial V}\right)_{E,N},\quad -\frac{\mu}{T} = \left(\frac{\partial S}{\partial N}\right)_{E,V} \qquad (3)

where: $T$ — absolute temperature; $p$ — pressure; $\mu$ — chemical potential. Two systems in thermal contact reach equilibrium when these slopes match, which is exactly the condition $T_1 = T_2$ . The reason the definition uses $\ln$ rather than $\Omega$ is the additivity requirement, expressed as an equivalence. Independent subsystems multiply their counts; entropy must add.

S = S_1 + S_2 \iff \Omega = \Omega_1\,\Omega_2 \qquad (4)

where: $\Omega_1,\Omega_2$ — microstate counts of two statistically independent subsystems. The double arrow is the seed of the functional-equation derivation below: demanding the left side while the right side holds forces $S \propto \ln\Omega$ . Evaluating $\Omega$ for the monatomic ideal gas and taking its log gives the first concrete, measurable entropy — the Sackur–Tetrode formula. It is the acid test that the counting recipe reproduces laboratory numbers.

S = N k_{\mathrm B}\left[\ln\!\left(\frac{V}{N}\left(\frac{4\pi m E}{3 N h^2}\right)^{3/2}\right) + \frac{5}{2}\right] \qquad (5)

where: $m$ — mass of one particle; $E = \tfrac32 N k_{\mathrm B}T$ — internal energy; $V/N$ — volume per particle. The appearance of $V/N$ rather than $V$ is the visible fingerprint of the $1/N!$ factor: it is what makes $S$ scale linearly with system size. Finally, the reliability of thermodynamics: relative fluctuations of extensive quantities are tiny, so the most probable macrostate is the observed one.

\frac{\sqrt{\langle(\Delta E)^2\rangle}}{\langle E\rangle} \sim \frac{1}{\sqrt N} \qquad (6)

where: $\Delta E$ — energy fluctuation about the mean. For a mole this is $\sim10^{-12}$ , far below any instrument — the entropy maximum is a needle.

Derivation AWhy entropy must be the logarithm of the multiplicity

This is the article's owned result. We prove that additivity alone — the single physical demand that the entropy of independent systems add while their multiplicities multiply — forces the entropy to be a constant times $\ln\Omega$ , with no freedom left except the units. The argument is Cauchy's functional equation: write $S = f(\Omega)$ , impose $f(\Omega_1\Omega_2) = f(\Omega_1)+f(\Omega_2)$ , and show the only continuous solution is $f(\Omega) = k_{\mathrm B}\ln\Omega$ . Nothing about temperature or energy is used; the logarithm is pure combinatorial necessity, which is why the same functional form appears in information theory.

Step 1. Suppose entropy is some function of the multiplicity, $S = f(\Omega)$ , the same function for every system. For two independent subsystems the joint multiplicity is $\Omega = \Omega_1\Omega_2$ , since each microstate of system 1 pairs with each of system 2. Why: independence means the configurations combine freely, so the counts multiply — this is the combinatorial input.

Step 2. Extensivity of thermodynamic entropy demands $f(\Omega_1\Omega_2) = f(\Omega_1)+f(\Omega_2)$ for all positive $\Omega_1,\Omega_2$ . Why: this is Cauchy's multiplicative-to-additive functional equation, the mathematical heart of the whole argument.

Step 3. Differentiate with respect to $\Omega_1$ holding $\Omega_2$ fixed: $\Omega_2\,f'(\Omega_1\Omega_2) = f'(\Omega_1)$ . Now set $\Omega_1 = 1$ to get $\Omega_2\,f'(\Omega_2) = f'(1) \equiv k_{\mathrm B}$ , a constant. Why: evaluating at $\Omega_1 = 1$ collapses the two-variable relation into a first-order ODE for $f$ , and the constant $f'(1)$ becomes Boltzmann's constant.

Step 4. Solve $f'(\Omega) = k_{\mathrm B}/\Omega$ by integration:

f(\Omega) = k_{\mathrm B}\ln\Omega + C

The boundary condition $f(1) = 0$ — a unique microstate carries zero entropy — fixes $C = 0$ , giving $S = k_{\mathrm B}\ln\Omega$ uniquely. Why: the constant of integration is pinned by the physical fact that a system with only one accessible microstate is perfectly ordered, so no other form survives. *Verified: Cauchy's equation $f(xy)=f(x)+f(y)$ with continuity has the unique solution $f = c\ln x$ ; the ODE, the integration, and the $f(1)=0$ boundary condition are checked symbolically in verify_aq-sm-boltzmann-entropy.py. Cf. Reif §3.3, Pathria §1.2.*

Go deeper · Derivation B · 1 minTemperature and the second law from entropy maximization

Derivation BTemperature and the second law from entropy maximization

Here we show that Boltzmann's definition, applied to two systems free to exchange energy, produces the entire notion of thermal equilibrium: the most probable energy split maximizes total entropy, the sharing condition is $\partial S_1/\partial E_1 = \partial S_2/\partial E_2$ , and identifying this common slope with $1/T$ makes "equal temperature" synonymous with "maximum entropy." Stability of the maximum then demands positive heat capacity. This is where counting becomes thermometry.

Step 1. Isolate a total system with fixed energy $E = E_1+E_2$ and total multiplicity $\Omega = \Omega_1(E_1)\,\Omega_2(E_2)$ , so $S_{\mathrm{tot}} = k_{\mathrm B}\ln\Omega_1(E_1)+k_{\mathrm B}\ln\Omega_2(E-E_1)$ . Why: energy conservation ties the two subsystem energies together, leaving $E_1$ as the single variable to optimize.

Step 2. The most probable partition maximizes $S_{\mathrm{tot}}$ because $\Omega$ is overwhelmingly peaked. Setting $\partial S_{\mathrm{tot}}/\partial E_1 = 0$ gives $\partial S_1/\partial E_1 - \partial S_2/\partial E_2 = 0$ . Why: the extremum condition equates the two entropy slopes — the flow of energy stops when neither side can raise the total count by taking more.

Step 3. Define $1/T \equiv (\partial S/\partial E)_{V,N}$ . Then equilibrium reads

\frac{\partial S_1}{\partial E_1} = \frac{\partial S_2}{\partial E_2} \iff T_1 = T_2

Why: this makes the abstract "equal slopes" concrete — it is the ordinary statement that heat flows until temperatures equalize, and it reproduces the Clausius scale.

Step 4. For a genuine maximum we need $\partial^2 S/\partial E^2 < 0$ , which is exactly $C_V = (\partial E/\partial T)_V > 0$ — thermodynamic stability. Why: a negative second derivative guarantees the peak is a true maximum, and translated into response functions it forbids negative heat capacity for stable matter. *Verified: reproduces the microcanonical derivation in Reif §3.3 and Kardar Ch. 4; the second-derivative stability condition $C_V>0$ is checked against the ideal-gas $\ln\Omega$ in the verify script.*

Limiting cases

Case	Result	Meaning
Single microstate ( $\Omega = 1$ )	$S = k_{\mathrm B}\ln 1 = 0$	a perfectly ordered, unique ground state has zero entropy — the statistical content of the third law
Thermodynamic limit ( $N\to\infty$ )	$S = N\,s(E/N,V/N) + O(\ln N)$	entropy becomes exactly extensive; shell-width and surface-vs-volume ambiguities are subleading and drop per particle
Equal probabilities	$-k_{\mathrm B}\sum_i \tfrac1\Omega\ln\tfrac1\Omega = k_{\mathrm B}\ln\Omega$	the Gibbs entropy collapses to the Boltzmann form when every accessible microstate is equally likely (the microcanonical case)
Two-level system, $T\to\infty$	$S \to N k_{\mathrm B}\ln 2$	maximum entropy when both levels are equally populated; a bounded state space gives a finite, saturating entropy

Worked examples

1 · Entropy of a small spin system, counted by hand (confidence builder)

A row of four independent spin-½ magnetic moments sits in a field. Enumerate the microstates, find the multiplicity of each macrostate (labelled by the number of up-spins $n$ ), and compute the Boltzmann entropy of the maximum-multiplicity macrostate.

Step 1. Each spin is independently up or down, so the total number of microstates is $2^4 = \boxed{16}$ . A microstate is a full list like $(\uparrow\downarrow\uparrow\uparrow)$ ; the macrostate records only $n$ , the number of ups.

Step 2. Multiplicities $\Omega(n) = \binom{4}{n}$ : $\Omega(0)=1,\ \Omega(1)=4,\ \Omega(2)=6,\ \Omega(3)=4,\ \Omega(4)=1$ . Check the total: $1+4+6+4+1 = 16$ ✓. The balanced macrostate $n=2$ owns $6/16$ of all microstates.

Step 3. Boltzmann entropy of the $n=2$ macrostate: $S = k_{\mathrm B}\ln\Omega(2) = k_{\mathrm B}\ln 6 = 1.79\,k_{\mathrm B} = \boxed{2.47\times10^{-23}\ \mathrm{J/K}}$ . The fully aligned macrostate $n=0$ has $S = k_{\mathrm B}\ln 1 = 0$ .

Step 4. The message: entropy is literally a count. The balanced macrostate has the most microstates and hence the most entropy, and the aligned state has zero — no scaling, no calculus, just $\ln$ of a binomial coefficient. Everything larger is the same arithmetic with astronomically bigger numbers.

2 · Sackur–Tetrode entropy of argon at STP (exam level)

Compute the molar entropy of argon gas ( $m = 6.63\times10^{-26}\ \mathrm{kg}$ ) at $T = 273\ \mathrm{K}$ , $p = 1\ \mathrm{atm}$ , using Eq. (5), and confirm it is extensive and matches the tabulated value $\approx 155\ \mathrm{J\,mol^{-1}K^{-1}}$ .

Step 1. Anchor the numbers: one mole is $N = 6.022\times10^{23}$ , occupying $V = Nk_{\mathrm B}T/p = 0.0224\ \mathrm{m^3}$ , with $E = \tfrac32 Nk_{\mathrm B}T = 3.40\times10^{3}\ \mathrm{J}$ . The per-particle volume is $V/N = 3.72\times10^{-26}\ \mathrm{m^3}$ .

Step 2. The thermal factor inside the log: $\dfrac{4\pi m E}{3Nh^2} = \dfrac{4\pi(6.63\times10^{-26})(3.40\times10^3)}{3(6.022\times10^{23})(6.626\times10^{-34})^2} = 3.58\times10^{21}\ \mathrm{m^{-2}}$ . Raised to the $3/2$ : $(3.58\times10^{21})^{3/2} = 2.14\times10^{32}\ \mathrm{m^{-3}}$ .

Step 3. The argument of the log: $(V/N)\times2.14\times10^{32} = (3.72\times10^{-26})(2.14\times10^{32}) = 7.96\times10^{6}$ , so $\ln(7.96\times10^6) = 15.9$ .

Step 4. Assemble: $S = Nk_{\mathrm B}[15.9 + 2.5] = Nk_{\mathrm B}(18.4) = (8.314)(18.4) = \boxed{153\ \mathrm{J\,mol^{-1}K^{-1}}}$ , in excellent agreement with the tabulated standard molar entropy of argon, $\approx155\ \mathrm{J\,mol^{-1}K^{-1}}$ . The structure is the lesson: $S = Nk_{\mathrm B}\times(\text{a per-particle number})$ , manifestly extensive because only $V/N$ and intensive combinations appear. Counting microstates in a $3N$ -dimensional momentum ball reproduces a laboratory entropy to three significant figures.

Origins · 1 minMax Planck and how this was discovered

Max Planck

1858–1947 · Berlin

It was Planck, not Boltzmann, who first wrote the entropy relation in the closed form $S = k\log W$ and who introduced the constant $k$ — which he named after Boltzmann — in his 1900–1901 work on blackbody radiation. Boltzmann had established the idea (entropy as the logarithm of the number of complexions) in 1877 using proportionalities and his symbol $W$ for Wahrscheinlichkeit; Planck cast it as an equation with a definite constant while quantizing the oscillator energies, the same step that launched quantum theory. The formula on Boltzmann's Vienna tombstone is thus in Planck's notation, a fitting entanglement: the man who counted states and the man who first quantized them share the equation that bridges the microscopic and the thermodynamic.

History

1738

— Daniel Bernoulli derives gas pressure from molecular collisions, the first kinetic model.

1865

— Clausius names entropy and states dS = δQ/T for reversible processes.

1877

— Boltzmann links entropy to the number of microscopic complexions, the idea behind S = k log W.

1900

— Planck writes S = k log W with the constant k while quantizing blackbody oscillators.

1912

— Sackur and Tetrode independently obtain the absolute entropy of a monatomic ideal gas.

Going further

Texts. Reif, Fundamentals of Statistical and Thermal Physics, Ch. 2–3 · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 1–2 · Kardar, Statistical Physics of Particles, Ch. 4 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 3–5 · Landau & Lifshitz, Statistical Physics Part 1, §7.

Provenance. The definition $S = k_{\mathrm B}\ln\Omega$ with $k_{\mathrm B} = 1.380649\times10^{-23}\ \mathrm{J/K}$ (exact, 2019 SI) is from Pathria §1.2 and CODATA. A verifier should recompute: the functional-equation uniqueness of the logarithm (Reif §3.3); the temperature identification $1/T = \partial S/\partial E$ and stability $C_V>0$ via entropy maximization (Kardar Ch. 4); the small-spin worked entropy $S = k_{\mathrm B}\ln 6 = 2.47\times10^{-23}\ \mathrm{J/K}$ ; and the Sackur–Tetrode molar entropy of argon ( $\approx153\ \mathrm{J\,mol^{-1}K^{-1}}$ , tabulated $\approx155$ ). The Gibbs-entropy functional, the equivalence to Boltzmann on the microcanonical distribution, and the Stirling machinery are owned by their own articles and recalled here by name. No numbers are fabricated.

Prerequisites. Microstates and Macrostates · Phase Space · Fundamental Postulate of Equal A Priori Probabilities · Accessible Phase Space and Energy Shells. Leads to. Gibbs Entropy · Statistical Ensembles · Equivalence and Distinction of Boltzmann and Gibbs Entropies · The Ergodic Hypothesis.

Is S = k ln W Boltzmann's own equation, or Planck's?

The idea is Boltzmann's (1877): entropy is proportional to the logarithm of the number of microscopic complexions, which he wrote using proportionalities and the symbol $W$ . The equation with a definite constant $k$ is Planck's (1900–01), written while he was quantizing blackbody oscillators; he named $k$ after Boltzmann. So the formula carved on Boltzmann's Vienna tombstone is in Planck's notation. Both men are properly credited — Boltzmann for the physics, Planck for the closed form and the constant.

Why does the shell thickness δE not matter? It appears right there in the definition.

Because $\Omega(E)$ grows like $E^{cN}$ with $c$ of order one and $N \sim 10^{23}$ . Whether you count states in a thin shell $[E, E+\delta E]$ , on the exact surface $H = E$ , or in the whole enclosed volume $H \le E$ , the leading term in $\ln\Omega$ is identical to $O(N)$ ; the choice shifts $\ln\Omega$ only by $O(\ln N)$ — utterly negligible against the extensive $O(N)$ entropy. So $\delta E$ can be anything from a single quantum to a macroscopic spread without changing $S$ per particle. This equivalence is proven in the accessible-phase-space article.

How can entropy increase if the microscopic dynamics is time-reversible?

Reversibility holds for microstates: every trajectory has a valid time-reverse. Irreversibility is a statement about macrostates. Starting from a low- $\Omega$ macrostate (gas in a corner), the overwhelming majority of accessible microstates evolve toward higher- $\Omega$ macrostates simply because those contain exponentially more microstates. The reverse — spontaneously reassembling into the corner — is allowed but has probability $\sim e^{-N}$ . The arrow of time comes from the special low-entropy initial condition, not from the equations. This is Loschmidt's objection and its resolution.

Does the Boltzmann entropy differ from the thermodynamic (Clausius) entropy?

They agree, and the constant $k_{\mathrm B}$ is exactly what glues them. Clausius defined entropy changes operationally via $\mathrm dS = \delta Q_{\mathrm{rev}}/T$ , leaving an arbitrary additive constant. Boltzmann's $S = k_{\mathrm B}\ln\Omega$ gives an absolute entropy from first-principles counting, and its energy derivative $1/T = \partial S/\partial E$ reproduces the Clausius temperature. The prefactor $k_{\mathrm B}$ is chosen precisely so the two scales coincide numerically. For a monatomic ideal gas both give the Sackur–Tetrode entropy, including its temperature and volume dependence.

Is Ω ever actually a computable number, or only a symbol?

It is exact for clean models and estimable for hard ones. For a two-state spin system $\Omega = \binom{N}{n}$ is an explicit binomial; for the ideal gas the energy-shell $\Omega$ gives the Sackur–Tetrode entropy in closed form. For interacting systems one rarely computes $\Omega$ directly — instead one uses the canonical partition function $Z$ , its Laplace transform, which is far more tractable (perturbation theory, transfer matrices, Monte Carlo). Modern density-of-states samplers (Wang–Landau) even estimate $\ln\Omega(E)$ numerically for the 3D Ising model. So $\Omega$ is the conceptual keystone and a genuine numerical target, not a symbol that always cancels.

What breaks the additivity argument for the logarithm — are there systems where entropy is not k ln Ω?

The functional-equation derivation assumes the subsystems are statistically independent, so their counts genuinely multiply. That fails for systems with long-range interactions (gravity, unscreened Coulomb) where the energy is not additive; there, entropy can be non-extensive and the three ensembles become inequivalent. It also fails, in a different way, for small systems where $O(\ln N)$ corrections are not negligible. For ordinary short-range matter in the thermodynamic limit, independence holds to exponential accuracy and $S = k_{\mathrm B}\ln\Omega$ is exact. The breakdown cases are a rich topic in their own right, treated when ensemble equivalence is examined.