Equivalence of the Surface and Volume Density of States

When you count microstates to get the entropy $S=k_B\ln\Omega$ , there is a nagging ambiguity: which $\Omega$ ? Do you count all the microstates with energy up to $E$ — the phase-space volume $\Sigma(E)$ ? Or only those in a thin shell at exactly $E$ — the surface density of states $\omega(E)=d\Sigma/dE$ ? Or the number in a small energy band $\Gamma(E)=\omega(E)\,\Delta E$ ? These three quantities differ enormously in magnitude, and one might fear the entropy — and hence all of thermodynamics — depends on an arbitrary choice. The reassuring answer, and the subject of this article, is that in the thermodynamic limit it does not matter: the three definitions give the same entropy to leading order in $N$ . Their logarithms differ only by terms of order $\ln N$ , which are utterly negligible against the bulk entropy of order $N$ . So $\ln\Sigma(E)$ , $\ln\omega(E)$ , and $\ln\Gamma(E)$ all yield the same $S/k_B$ per particle — the microcanonical entropy is well-defined despite the freedom of definition. The reason is a striking feature of high-dimensional geometry: almost all the volume of a high-dimensional region sits in a thin skin near its boundary. For $N\sim10^{23}$ particles the phase space is $\sim10^{23}$ -dimensional, and the volume enclosed by the energy surface is dominated by an infinitesimally thin shell right at the surface. So "everything up to $E$ " and "just at $E$ " count essentially the same states. This article makes the equivalence precise for the ideal gas — where $\Sigma\propto E^{3N/2}$ , $\omega\propto E^{3N/2-1}$ , and their logs differ by exactly $\ln(3N/2)=O(\ln N)$ — proves it geometrically via the concentration of measure, and shows why the microcanonical entropy is robustly defined.

Key takeaways

The microcanonical entropy can be defined from the phase-space volume $\Sigma(E)$ , the surface density of states $\omega(E)=d\Sigma/dE$ , or a thin shell $\Gamma(E)=\omega\,\Delta E$ .
These differ hugely in magnitude but their logarithms agree to leading order: $\ln\Sigma$ , $\ln\omega$ , $\ln\Gamma$ differ only by $O(\ln N)\ll N$ .
For the ideal gas, $\Sigma\propto E^{3N/2}$ and $\omega\propto E^{3N/2-1}$ , so $\ln\Sigma - \ln(E\omega) = \ln(3N/2) = O(\ln N)$ — negligible per particle.
The deep reason is concentration of measure: in high dimensions nearly all the volume lies in a thin skin at the surface, so "up to $E$ " ≈ "at $E$ ."
Consequently $S=k_B\ln\Omega$ is well-defined independently of which count you use, and the microcanonical entropy is robust.
Key idea: the volume, surface, and shell densities of states give the same entropy to leading order in $N$ , because in high dimensions the enclosed volume is concentrated in a thin surface shell.

The core insight

The three microcanonical counts — phase-space volume $\Sigma(E)$ , surface density of states $\omega(E)=d\Sigma/dE$ , and shell $\Gamma(E)=\omega\,\Delta E$ — differ in magnitude but yield the same entropy to leading order, since their logarithms differ only by $O(\ln N)$ . This is the geometric fact that in high dimensions almost all volume lies at the surface.

Where this lives today

The equivalence of the density-of-states definitions is the quiet foundation that lets microcanonical calculations proceed without fuss — every derivation of the ideal-gas entropy, every molecular-dynamics estimate of a microcanonical temperature, silently relies on it. In computational statistical mechanics, methods like Wang–Landau sampling compute $\omega(E)$ directly across the whole energy range; the fact that $\ln\omega$ , $\ln\Sigma$ , and $\ln\Gamma$ agree to leading order is what makes the resulting entropy meaningful regardless of the sampling details. The breakdown at finite $N$ , and the subtle Gibbs-versus-Boltzmann distinction, are live research topics. As experiments probe genuinely small systems — a few trapped ions, a handful of ultracold atoms, single molecules — the $O(\ln N)$ differences among the entropy definitions become measurable, and one must choose carefully which entropy to report. The most contested arena is negative absolute temperature: systems with a bounded energy spectrum (nuclear spins, cold atoms in optical lattices with capped kinetic energy) can be driven into population inversion, and whether one says their temperature is "negative" (Boltzmann surface entropy) or always positive (Gibbs volume entropy) hinges precisely on which density of states defines $S$ . Landmark cold-atom experiments realizing negative-temperature states have made this once-academic distinction experimentally sharp. The concentration of measure that makes the definitions equivalent for a mole of gas is exactly what fails for a handful of spins — and that failure is now in the laboratory.

The physical picture

The puzzle is best seen with a sphere. In two dimensions, the area of a disk of radius $R$ is $\pi R^2$ , and the circumference is $2\pi R$ — very different objects. But their logarithms grow the same way with $R$ : $\ln(\pi R^2)=2\ln R + \text{const}$ and $\ln(2\pi R)=\ln R+\text{const}$ , differing by just $\ln R$ . In statistical mechanics the "radius" is set by the energy and the "dimension" is $\sim 3N$ , so the difference between the enclosed volume and the bounding surface is $\sim\ln(3N)$ — a whisker against the bulk.

Figure

Three ways to count microstates at energy E: the volume Σ(E) enclosed, the surface ω(E) at E, and a thin shell Γ(E) — all give the same entropy to leading order.

The physics that makes this work is concentration of measure: in a space of dimension $d\sim3N$ , essentially all of the volume of a ball sits in a shell of relative thickness $\sim1/d$ at the surface. If you shrink the radius by even a tiny fraction $\delta$ , the enclosed volume drops by a factor $(1-\delta)^d$ , which for large $d$ is nearly zero. So the interior is empty compared to the skin — the volume $\Sigma(E)$ is dominated by states with energy right at $E$ , which is exactly the surface $\omega(E)$ counts. "Almost all the states below $E$ are almost exactly at $E$ ." The punchline is quantitative and reassuring: whichever count you pick, the entropy per particle is the same up to a correction that vanishes as $\ln N/N$ . The plot below shows the three log-counts as functions of energy for the ideal gas — three parallel curves separated by only $O(\ln N)$ , indistinguishable once divided by $N$ . The microcanonical entropy is defined without ambiguity.

Figure

Log-counts vs energy for the ideal gas: ln Σ, ln(E ω), ln Γ are three nearly-coincident lines separated by only O(ln N) — the same entropy per particle.

The core equations

For a classical system the three counts are built from the phase-space measure. The volume below energy $E$ and the surface density of states are:

\Sigma(E)=\int_{H(p,q)\le E}\frac{\mathrm{d}^{3N}p\,\mathrm{d}^{3N}q}{N!\,h^{3N}},\qquad \omega(E)=\frac{\mathrm{d}\Sigma}{\mathrm{d}E} \qquad (1)

where: $\Sigma(E)$ — dimensionless number of microstates with energy $\le E$ ; $\omega(E)$ — density of states (states per unit energy) at $E$ ; the $N!\,h^{3N}$ makes the count dimensionless and indistinguishability-correct. The shell count over a band $\Delta E$ is $\Gamma(E)=\omega(E)\,\Delta E$ . Three candidate entropies follow, and the claim is that they coincide to leading order:

S_\Sigma=k_B\ln\Sigma(E),\quad S_\omega=k_B\ln[\omega(E)\,\varepsilon_0],\quad S_\Gamma=k_B\ln\Gamma(E);\qquad S_\Sigma \simeq S_\omega \simeq S_\Gamma \quad (\text{to } O(\ln N)) \qquad (2)

where: $\varepsilon_0$ — an arbitrary reference energy width making $\omega\varepsilon_0$ dimensionless. The differences among the three are $O(\ln N)$ , dwarfed by the leading $O(N)$ entropy. For the ideal gas these are explicit. The energy surface $\sum p_i^2/2m=E$ is a $3N$ -sphere of radius $\sqrt{2mE}$ , so the enclosed momentum volume is that of a $3N$ -ball:

\Sigma(E)\propto V^N E^{3N/2},\qquad \omega(E)=\frac{\mathrm{d}\Sigma}{\mathrm{d}E}\propto V^N\,\frac{3N}{2}\,E^{3N/2-1} \qquad (3)

where: the $E^{3N/2}$ is the volume of a $3N$ -dimensional ball of radius $\propto\sqrt{E}$ ; differentiating drops the power by one and brings down the factor $3N/2$ . The key comparison is the log-difference between the volume and the (energy-weighted) surface:

\ln\Sigma(E) - \ln[E\,\omega(E)] = -\ln\frac{3N}{2} = O(\ln N) \qquad (4)

where: $E\omega(E)$ is the shell count with band width $\Delta E=E$ ; the difference from $\ln\Sigma$ is exactly $\ln(3N/2)$ , which grows like $\ln N$ — negligible against the entropy $\ln\Sigma\sim\tfrac32 N\ln E$ which grows like $N$ . Divided by $N$ , the difference vanishes.

Derivation AThe three counts give the same entropy to O(ln N) — worked for the ideal gas

The goal is to compute $\ln\Sigma$ , $\ln\omega$ , and $\ln\Gamma$ explicitly for the ideal gas and show their differences are all of order $\ln N$ , so the entropy per particle is independent of the choice. The strategy is to use the $3N$ -ball volume for $\Sigma$ , differentiate for $\omega$ , form a shell for $\Gamma$ , and subtract the logarithms.

Step 1. Volume. The momentum part of $\Sigma$ is the volume of a $3N$ -ball of radius $R=\sqrt{2mE}$ : $V_{3N}(R)=\frac{\pi^{3N/2}}{\Gamma(3N/2+1)}R^{3N}$ . Times $V^N$ (position) and constants, $\Sigma(E)\propto V^N E^{3N/2}$ . Why: fixing total kinetic energy $\le E$ confines the $3N$ momenta to a ball whose radius grows as $\sqrt E$ , so the volume grows as $E^{3N/2}$ .

Step 2. Surface. Differentiate: $\omega=d\Sigma/dE\propto V^N\cdot\frac{3N}{2}E^{3N/2-1}$ . So $E\omega\propto V^N\cdot\frac{3N}{2}E^{3N/2}$ , the same power of $E$ as $\Sigma$ but with an extra factor $3N/2$ . Why: differentiating $E^{3N/2}$ brings down $3N/2$ and lowers the power by one; multiplying by $E$ restores the power, isolating the $3N/2$ prefactor as the whole difference.

Step 3. Subtract logarithms: $\ln\Sigma - \ln(E\omega) = \ln\frac{\Sigma}{E\omega} = -\ln\frac{3N}{2}$ . This is $O(\ln N)$ . Meanwhile $\ln\Sigma = N\ln V + \frac{3N}{2}\ln E + \text{const}$ is $O(N)$ . Why: the entire discrepancy between counting the volume and counting the surface shell is the single factor $3N/2$ — logarithmically small compared to the extensive entropy.

Step 4. Divide by $N$ : the entropies per particle are $s_\Sigma = s_\omega + O(\ln N/N) \to s_\omega$ as $N\to\infty$ . A shell of any reasonable width $\Delta E$ (even $\Delta E=E/N$ ) shifts the count by at most another $O(\ln N)$ , so all three per-particle entropies coincide. Why: the $\ln N$ difference, spread over $N$ particles, contributes $\ln N/N\to0$ per particle — invisible in the thermodynamic limit. *Verified: symbolically $\omega=\frac{3N}{2}CV^NE^{3N/2-1}$ , $\ln\Sigma-\ln(E\omega)=-\ln(3N/2)$ , and $(\ln\Sigma-\ln\Gamma)/N\to0$ ; numerically at $N=10^6$ the per-particle discrepancy is $<10^{-3}$ , in verify_aq-sm-surface-volume-density-of-states-equivalence.py.*

Go deeper · Derivation B · 2 minConcentration of measure: why the volume lives in the surface shell — the owned result

Derivation BConcentration of measure: why the volume lives in the surface shell — the owned result

This is the article's owned structural result: the equivalence is not an algebraic accident of the ideal gas but a geometric theorem — in high dimensions almost all of a ball's volume lies in a vanishingly thin shell at its surface, so counting the interior and counting the surface are the same. The goal is to prove the concentration of measure and connect it directly to the density-of-states equivalence. The strategy is to compute the fraction of a high-dimensional ball's volume that lies within a thin outer shell.

Step 1. In dimension $d$ , the volume of a ball scales as $R^d$ , so the fraction of the volume lying inside radius $(1-\delta)R$ is $(1-\delta)^d$ . The fraction in the outer shell of relative thickness $\delta$ is therefore $1-(1-\delta)^d$ . Why: volume in $d$ dimensions scales as the $d$ -th power of the radius, so shrinking the radius even slightly removes a factor $(1-\delta)^d$ of the volume.

Step 2. For large $d$ , $(1-\delta)^d = e^{d\ln(1-\delta)}\approx e^{-d\delta}\to 0$ for any fixed $\delta>0$ . So the shell fraction $1-(1-\delta)^d\to 1$ : all the volume concentrates in an arbitrarily thin skin. With $d\sim3N\sim10^{23}$ , a shell of relative thickness $10^{-20}$ already contains essentially all the volume. Why: the exponential $e^{-d\delta}$ crushes the interior fraction to zero once $d\delta\gg1$ , which happens for astronomically thin shells when $d$ is macroscopic.

Step 3. Translate to energy. The energy surface $H=E$ is the boundary of the accessible ball; concentration of measure says the enclosed phase volume $\Sigma(E)$ is dominated by states with energy within a fraction $\sim1/N$ of $E$ — i.e. right at the surface. Hence $\Sigma(E)$ and the surface count $\omega(E)$ measure essentially the same states. Why: the states "below $E$ " are overwhelmingly "at $E$ ," so the volume and surface counts cannot differ by more than the thin-shell factor, which is $O(\ln N)$ in the log.

Step 4. Conclude the robustness. Because the interior is negligible, the choice of $\Sigma$ vs $\omega$ vs $\Gamma$ affects only sub-leading $O(\ln N)$ terms, and $S=k_B\ln\Omega$ is well-defined by any of them. This is why textbooks freely switch between "number of states up to $E$ " and "density of states at $E$ ." Why: the entropy inherits the concentration of measure — the leading behaviour is fixed by the surface, and everything else is a logarithmic afterthought. *Verified: the shell fraction $1-(1-\delta)^d\to1$ (at $d=3000,\delta=10^{-3}$ it exceeds $0.95$ ); Stirling's $\ln\Gamma(n/2+1)$ leading term is accurate to $O(\ln n)$ , in the verify script.*

Limiting cases

The equivalence and its boundaries:

Case	Result	Meaning
Thermodynamic limit	$s_\Sigma=s_\omega=s_\Gamma$	All three entropy definitions coincide per particle.
Finite $N$	differ by $O(\ln N/N)$	Small discrepancies; matter for tiny systems (few particles).
Shell width choice	irrelevant to leading order	Any $\Delta E$ from $\varepsilon_0$ to $E$ gives the same $S/N$ .
Volume-entropy (Gibbs)	$S_G=k_B\ln\Sigma$	Guarantees the equipartition theorem and $T>0$ even for bounded spectra.
Surface-entropy (Boltzmann)	$S_B=k_B\ln\omega$	The usual textbook definition; can give negative $T$ for bounded spectra.

Common trap“Volume entropy and surface entropy are interchangeable in every respect”

To leading order in $N$ they give the same thermodynamics — that is the whole point of this article. But there is a genuine and much-debated subtlety at the level of definitions: the Gibbs (volume) entropy $S_G=k_B\ln\Sigma(E)$ and the Boltzmann (surface) entropy $S_B=k_B\ln\omega(E)$ can disagree about the sign of the temperature for systems with a bounded energy spectrum (like spin systems that admit negative absolute temperatures). The Gibbs entropy is monotonically increasing in $E$ , so $T_G>0$ always; the Boltzmann entropy can decrease, giving $T_B<0$ above the population-inversion point. For ordinary systems with unbounded kinetic energy the two agree and the distinction never bites — the difference is $O(\ln N)$ . But do not conclude the two definitions are identical in principle: for small systems, bounded spectra, and questions about negative temperature, the choice matters, and which is "correct" is still argued in the literature.

Worked examples

1 · How thin is the shell that holds all the states? (a real number)

For $N=10^{22}$ ideal-gas particles ( $d=3N=3\times10^{22}$ momentum dimensions), find the relative energy shell thickness $\delta$ such that the shell at energy $E$ contains $99\%$ of the phase-space volume $\Sigma(E)$ .

Step 1. The interior fraction below relative radius $(1-\delta_r)$ (in radius) is $(1-\delta_r)^d$ . We want the shell to hold $99\%$ , so the interior holds $1\%$ : $(1-\delta_r)^d = 0.01$ .

Step 2. Solve: $d\ln(1-\delta_r)=\ln0.01=-4.605$ , so $\ln(1-\delta_r)=-4.605/d=-1.5\times10^{-22}$ , giving $\delta_r\approx1.5\times10^{-22}$ in radius.

Step 3. Convert to energy. Since $E\propto R^2$ , $\delta_E \approx 2\delta_r = 3\times10^{-22}$ in relative energy. So $99\%$ of all the microstates with energy $\le E$ lie in an energy shell of relative width $\mathbf{3\times10^{-22}}$ — thinner than any conceivable measurement.

Step 4. Interpret. The entire phase-space volume is packed into an unimaginably thin skin at the energy surface. This is why "count everything up to $E$ " and "count only the surface at $E$ " give the same entropy — there is essentially nothing in the interior.

2 · The log-difference between the three counts (exam level)

For a monatomic ideal gas of $N$ particles, compute $\ln\Sigma(E)-\ln[E\,\omega(E)]$ exactly, evaluate it for $N=6\times10^{23}$ , and compare it to the total entropy $\ln\Sigma\sim\tfrac32 N$ (taking $\ln E\sim\mathcal O(1)$ units).

Step 1. From $\Sigma\propto E^{3N/2}$ and $E\omega\propto\frac{3N}{2}E^{3N/2}$ : the ratio is $\Sigma/(E\omega)=2/(3N)$ , so $\ln\Sigma-\ln(E\omega)=-\ln(3N/2)$ .

Step 2. Evaluate for $N=6\times10^{23}$ : $3N/2 = 9\times10^{23}$ , so $\ln(3N/2)=\ln(9\times10^{23})=\mathbf{55.2}$ . The three log-counts differ by about $55$ .

Step 3. Compare to the entropy. The bulk entropy is $\ln\Sigma\sim\tfrac32 N\ln E + N\ln V\sim\mathcal O(N)=\mathcal O(10^{23})$ . The difference $55$ against $10^{23}$ is a ratio of $\sim10^{-22}$ .

Step 4. Per particle: the discrepancy is $\ln(3N/2)/N = 55.2/(6\times10^{23})\approx\mathbf{9\times10^{-23}}$ — completely negligible. The choice of density-of-states definition shifts the entropy per particle by one part in $10^{22}$ ; the microcanonical entropy is robustly defined.

Origins · 1 minhypersphere and how this was discovered

hypersphere

high-dimensional geometry · —

The $n$ -dimensional hypersphere is the geometric object at the heart of the density-of-states equivalence. Its volume, $V_n(R)=\frac{\pi^{n/2}}{\Gamma(n/2+1)}R^n$ , and its surface area, $A_n(R)=\frac{2\pi^{n/2}}{\Gamma(n/2)}R^{n-1}$ , differ dramatically in size, yet their logarithms track together up to $O(\ln n)$ — exactly the volume-versus-surface statement for the ideal gas, whose energy surface is a $3N$ -dimensional sphere of radius $\sqrt{2mE}$ . High-dimensional spheres have counterintuitive properties that make statistical mechanics work: nearly all their volume lies in a thin shell at the surface (concentration of measure), and nearly all lies near any equatorial slice. The peculiar factor $\Gamma(n/2+1)$ in the volume, whose logarithm Stirling's approximation handles, is what supplies the entropy's leading $N\ln N$ behaviour. The geometry of the hypersphere is, quite literally, the geometry of an ideal gas's phase space, and its concentration of measure is why the microcanonical entropy does not care how you count.

History

1867

— the volume formula for the n-ball, involving Γ(n/2+1), is well established in geometry.

1902

— Gibbs's Elementary Principles in Statistical Mechanics carefully distinguishes the volume and surface phase measures.

1910

s — Sackur and Tetrode compute the ideal-gas entropy from the n-sphere phase volume, obtaining the correct extensive form.

1930

s — concentration-of-measure phenomena in high dimensions are studied (Lévy), later central to probability.

2014

— Dunkel and Hilbert reignite the debate over Gibbs (volume) versus Boltzmann (surface) entropy and negative temperature.

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., §1.4, §2.4 · Kardar, Statistical Physics of Particles, §4.3 · Reif, Fundamentals of Statistical and Thermal Physics, §2.5, Appendix · Landau & Lifshitz, Statistical Physics Part 1, §7 · Dunkel & Hilbert, "Consistent thermostatistics forbids negative absolute temperatures," Nature Physics (2014) for the Gibbs/Boltzmann debate.

Provenance. The equivalence of the phase-volume, surface, and shell entropies is standard (Pathria §1.4, §2.4; Reif §2.5). A verifier should reproduce: for the ideal gas $\Sigma\propto E^{3N/2}$ , $\omega\propto E^{3N/2-1}$ , and $\ln\Sigma-\ln(E\omega)=-\ln(3N/2)=O(\ln N)$ ; the shell/volume log-difference divided by $N\to0$ ; and the concentration of measure $1-(1-\delta)^d\to1$ . Numerical instances (per-particle discrepancy $<10^{-3}$ at $N=10^6$ ; $\ln(3N/2)=55.2$ at $N=6\times10^{23}$ ; shell thickness $3\times10^{-22}$ for $99\%$ of volume at $N=10^{22}$ ) are recomputed in verify_aq-sm-surface-volume-density-of-states-equivalence.py.

Builds on: Second Law of Thermodynamics and Entropy · Thermodynamic Potentials and the Fundamental Relation · Zeroth Law of Thermodynamics and Empirical Temperature. Leads to: Euler Relation and Gibbs–Duhem Equation · Helmholtz Free Energy, Enthalpy, and Gibbs Free Energy · The Microcanonical Ensemble · Density of States.

If Σ(E) and ω(E) differ by a huge factor, how can they give the same entropy?

Because entropy is the logarithm of the count, and the logarithm crushes the huge factor to a tiny additive term. For the ideal gas $\Sigma$ and $E\omega$ differ by the factor $3N/2\sim10^{23}$ — an astronomically large ratio. But the entropy is $k_B\ln\Sigma$ , and $\ln$ of a ratio of $10^{23}$ is only about $55$ . Meanwhile the entropy itself is $k_B\ln\Sigma\sim k_B\times10^{23}$ . So the difference between the two entropies is $55$ out of $10^{23}$ — one part in $10^{22}$ , completely negligible. The magnitudes differ enormously, but the logarithms (which is what entropy is) differ only by $O(\ln N)$ . This is a recurring theme: extensive thermodynamics is about $\ln\Omega$ , and multiplicative factors in $\Omega$ become negligible additive shifts in $\ln\Omega$ .

What does "concentration of measure" actually mean, physically?

It means that in a very high-dimensional space, "typical" and "extreme" almost coincide — nearly all of the volume sits in a thin shell right at the boundary, and nearly all of it is close to any given slice. For statistical mechanics the relevant statement is about the ball of accessible phase space: essentially all of its volume lies in an infinitesimally thin shell at the energy surface. Concretely, for $N\sim10^{23}$ particles the momentum space is $\sim10^{23}$ -dimensional, and if you shrink the energy by even one part in $10^{20}$ , you lose essentially all the microstates. So the states "with energy up to $E$ " are overwhelmingly states "with energy essentially equal to $E$ ." Physically, this is why the microcanonical (fixed- $E$ surface) and the "all states below $E$ " pictures give the same answer, and it is closely related to why the energy in the canonical ensemble is so sharply peaked — high-dimensional volumes concentrate.

Is the Gibbs (volume) entropy or the Boltzmann (surface) entropy the "correct" one?

For ordinary large systems it does not matter — they agree to $O(\ln N)$ and give identical thermodynamics. The controversy is about small systems and systems with a bounded energy spectrum. The Gibbs entropy $S_G=k_B\ln\Sigma(E)$ counts all states up to $E$ ; since $\Sigma$ always increases with $E$ , $S_G$ is monotonic and gives $T_G>0$ always, and it makes the equipartition theorem exact for finite systems. The Boltzmann entropy $S_B=k_B\ln\omega(E)$ counts the surface; for a bounded spectrum $\omega$ can decrease past the mid-point, giving $S_B$ a maximum and hence negative temperatures above it. Both are used; Dunkel and Hilbert (2014) argued the Gibbs form is more consistent (forbidding negative $T$ ), while others defend the Boltzmann form and the reality of negative temperatures demonstrated in cold-atom experiments. The honest answer is that it is definitional and context-dependent — but for a mole of gas the question never arises.

Does the width ΔE of the energy shell affect the entropy?

Not to leading order, which is exactly why the microcanonical ensemble is well-defined despite the apparent arbitrariness of $\Delta E$ . If you widen or narrow the shell by any reasonable factor — from a tiny $\varepsilon_0$ up to $\Delta E\sim E$ — the count $\Gamma=\omega\,\Delta E$ changes by that factor, so $\ln\Gamma$ changes by $\ln(\Delta E)$ , an $O(1)$ or at most $O(\ln N)$ term. Against the extensive entropy $\sim N$ , that shift is negligible, and the entropy per particle is unaffected. This is a relief, because there is no physically preferred shell width — the energy is only known to some resolution — and thermodynamics had better not depend on it. The insensitivity is another face of concentration of measure: since almost all states are at the surface anyway, how thick you make the shell barely changes how many you enclose.

How is this equivalence related to ensemble equivalence?

They are cousins, both powered by the sharp concentration of probability in large systems, but they are distinct statements. This article's equivalence is within the microcanonical ensemble: the three ways of counting microstates at a fixed energy (volume, surface, shell) give the same entropy. Ensemble equivalence is between ensembles: the microcanonical (fixed $E$ ), canonical (fixed $T$ ), and grand canonical (fixed $\mu$ ) descriptions give the same thermodynamics in the thermodynamic limit. Both rest on the same high-dimensional fact — that distributions become extremely sharply peaked as $N\to\infty$ , so different-looking prescriptions pick out essentially the same states. The density-of-states equivalence ensures the microcanonical entropy is unambiguous; ensemble equivalence then ensures that entropy matches the one you would compute from the canonical partition function. Together they make the choice of ensemble and the choice of counting both irrelevant for bulk matter.

Why does the factor Γ(3N/2+1) show up, and where does it go?

It comes from the volume of the $3N$ -dimensional momentum ball, $V_{3N}(R)=\frac{\pi^{3N/2}}{\Gamma(3N/2+1)}R^{3N}$ — the $\Gamma$ function is the higher-dimensional generalization of the factorial that normalizes the ball's volume. When you take the logarithm to get the entropy, $\ln\Gamma(3N/2+1)$ is handled by Stirling's approximation, $\ln\Gamma(3N/2+1)\approx\frac{3N}{2}\ln\frac{3N}{2}-\frac{3N}{2}$ , which contributes the crucial $N\ln N$ and $-N$ pieces to the entropy. In fact this $\Gamma$ function is where the indistinguishability-corrected, extensive Sackur–Tetrode entropy gets its shape: the $N\ln N$ from $\Gamma(3N/2+1)$ combines with the $-N\ln N$ from the $1/N!$ Gibbs factor to leave the correct $\ln(V/N)$ dependence. So $\Gamma(3N/2+1)$ is not a nuisance — it is the source of the extensive entropy's leading behaviour, and it drops out of the volume-versus-surface difference because both counts share it.