The Multiplicity Function

Put a hot brick against a cold one and wait. Energy flows from hot to cold, never the reverse, and the pair settles at a common temperature — a fact so familiar it feels like it needs no explanation. Yet the microscopic laws know nothing of "hot" or "cold," and run equally well backwards in time. Where, then, does the arrow come from? The answer is a single function. Count the microstates available to each brick at a given energy, call that count the multiplicity $\Omega(N,E,V)$ , and the whole of equilibrium thermodynamics follows from one property of $\Omega$ : it is spiked so violently about its maximum that the pair is found at the energy split maximizing the total count, and essentially never anywhere else. The multiplicity function is the beating heart of the microcanonical ensemble. Quantum mechanically it is unambiguous — the number of orthogonal energy eigenstates in a thin shell $[E, E+\Delta]$ , the degeneracy of the level. Classically the count of points in continuous phase space is infinite, so $\Omega$ is defined as the phase-space volume of the energy shell divided by $h^{3N}$ (to fix the cell size) and by $N!$ (to respect indistinguishability), both factors pinned by demanding agreement with the quantum count. Either way, $\Omega$ maps a macrostate $(N,E,V)$ to a number of microscopic ways to realize it. The decisive feature is explosive growth. When independent particles each contribute a factor, $\Omega$ scales like $\omega^N$ , so $\ln\Omega \sim N$ is extensive — and when two systems exchange energy, the combined multiplicity, a product over the energy split, becomes so sharply peaked (relative width $\sim 1/\sqrt{N}$ ) that all but a vanishing sliver of microstates sit at the equilibrium partition. Taking the logarithm linearizes the product into the Boltzmann entropy $S = k_B\ln\Omega$ ; demanding that the total entropy be maximal at fixed total energy makes $\partial\ln\Omega/\partial E$ equal across subsystems and identifies it with inverse temperature. Temperature, pressure, and chemical potential are all just slopes of this one function.

Key takeaways

The multiplicity function $\Omega(N,E,V)$ counts the microstates compatible with a macrostate fixed by the conserved quantities $N$ , $E$ , $V$ .
Classically it is the normalized phase-space volume of the energy shell; quantum-mechanically it is the degeneracy of the energy level.
Its logarithm is the Boltzmann entropy $S=k_B\ln\Omega$ , converting a combinatorial count into the state function of the microcanonical ensemble.
$\Omega$ grows astronomically (typically as $e^{Ns/k_B}$ ), so the macrostate of largest multiplicity overwhelms every other — the microscopic origin of the Second Law.
For two systems in contact the combined $\Omega$ peaks with relative width $\sim1/\sqrt N$ ; the peak's curvature $\partial^2\ln\Omega/\partial E^2<0$ is exactly the guarantee of a positive heat capacity.
Temperature, pressure, and chemical potential are the derivatives of $\ln\Omega$ with respect to $E$ , $V$ , and $N$ .
Key idea: count the microstates of a macrostate, take the logarithm, and thermodynamics follows — $S=k_B\ln\Omega$ turns a combinatorial count into a state function whose slopes are $T$ , $P$ , and $\mu$ .

The core insight

Count the microstates of a macrostate, take the logarithm, and thermodynamics follows: $S=k_B\ln\Omega$ turns a combinatorial count into a state function whose derivatives are $T$ , $P$ , and $\mu$ .

Where this lives today

The multiplicity function is no longer only a pencil-and-paper object. For systems where $\Omega(E)$ cannot be written in closed form — the three-dimensional Ising model, glasses, folding biomolecules — the Wang–Landau algorithm performs a random walk in energy that flattens the visited histogram and thereby reconstructs $\ln\Omega(E)$ , the density of states, over hundreds of orders of magnitude. Once you have $\Omega(E)$ you have everything: any thermodynamic quantity at any temperature follows by a single integral, because the canonical partition function is just the Laplace transform of $\Omega(E)$ . The sharp-peaking property also underlies a practical freedom. Because $\Omega$ is so spiked, the microcanonical entropy $k_B\ln\Omega(E)$ and the canonical free energy $-k_BT\ln Z$ are Legendre transforms of one another and yield identical intensive thermodynamics in the thermodynamic limit — so a physicist may compute in whichever ensemble is convenient. The interesting exceptions, where the peak is not sharp (long-range forces, first-order transitions in small systems, self-gravitating clusters), are precisely where the ensembles disagree and where multiplicity's structure — a non-concave $\ln\Omega(E)$ , negative heat capacity — becomes physically observable.

The physical picture

Flip $N$ coins. There is exactly one way to get all heads, but astronomically many ways to land near half heads and half tails — $\binom{N}{N/2}$ of them. The number of arrangements, the multiplicity, is minute at the extremes and colossal in the middle. The multiplicity function $\Omega(N,E,V)$ is the physical generalization of that coin count: it tallies the ways to distribute energy and position among $N$ particles consistent with the macroscopic constraints. The combinatorial tools for doing the tally live in the companion article on counting microstates; here we treat $\Omega$ as a function and ask how it behaves. Two features do all the work. First, $\Omega$ is multiplicative: bolt two independent subsystems together and their microstates combine freely, so the joint count is a product, $\Omega_{12} = \Omega_1\Omega_2$ . But thermodynamics wants an additive quantity, and the only operation turning products into sums is the logarithm — which is exactly why entropy is $\ln\Omega$ and not $\Omega$ . Second, $\Omega$ is sharply peaked. When two systems share a fixed total energy, the combined multiplicity $\Omega_1(E_1)\Omega_2(E-E_1)$ , plotted against the split $E_1$ , is a spike of fractional width $\sim N^{-1/2}$ . For $N\sim10^{23}$ that width is around $10^{-11}$ of the whole range — the system sits at the peak, and a spontaneous excursion away from it is not so much forbidden as astronomically improbable.

Figure

Two systems in contact: the total multiplicity is a razor-sharp spike at the equilibrium energy split

Whether you count discrete quantum levels or a continuous phase-space volume, the per-particle growth rate $\tfrac1N\ln\Omega$ is the same intensive entropy density; the $h^{3N}$ and $N!$ factors merely shift $\ln\Omega$ by the amounts needed to make it properly extensive. This is why the arrow of time is a counting statement: remove a constraint and the system, wandering among microstates, lands overwhelmingly in the macrostate that has the most of them. The brick pair reaches a common temperature because that split has, by a factor of $e^{+\text{(something)}\times N}$ , more microscopic realizations than any other. And the slope of that count is exactly what temperature measures — as the next figure shows, the steepness of $\ln\Omega$ versus energy is the inverse temperature $\beta$ .

Figure

The multiplicity climbs steeply with energy; its log-slope is the inverse temperature β

The core equations

The multiplicity function is defined the same way in words for every system: it is the number of microstates realizing the macrostate. Everything else is how to evaluate that count.

\Omega(N,E,V) = \#\{\text{microstates with energy } E,\ \text{particle number } N,\ \text{volume } V\} \qquad (1)

where: $\Omega$ — multiplicity; $N$ — particle number; $E$ — total energy of the isolated system; $V$ — volume. In the quantum case this is literally the degeneracy of the level $E$ (counted in a thin shell); in the classical case it needs the normalization below. Classically the count of phase-space points is infinite, so $\Omega$ is a shell volume made dimensionless and extensive by two fixed factors.

\Omega(N,E,V)=\frac{1}{h^{3N}N!}\int \mathrm{d}^{3N}q\,\mathrm{d}^{3N}p\;\delta\!\big(E-H(q,p)\big)\,\Delta \qquad (2)

where: $H(q,p)$ — Hamiltonian; $h^{3N}$ — one Planck cell per degree of freedom, turning volume into a pure count; $N!$ — Gibbs factor for indistinguishable particles; $\Delta$ — shell thickness; $\delta$ — restricts to the energy shell. The cell size $h$ and the $N!$ are pinned by demanding the classical count match the quantum degeneracy. The logarithm is the bridge to thermodynamics: it turns the multiplicative $\Omega$ into an additive potential.

S(N,E,V)=k_B\ln\Omega(N,E,V) \qquad (3)

where: $S$ — thermodynamic entropy; $k_B=1.380649\times10^{-23}\ \mathrm{J\,K^{-1}}$ (exact, SI). Once $\Omega$ is known as a function of its arguments, all of equilibrium thermodynamics is a matter of differentiation. The three fundamental intensive variables are exactly the three first derivatives of $S$ .

\frac{1}{T}=\left(\frac{\partial S}{\partial E}\right)_{N,V},\quad \frac{P}{T}=\left(\frac{\partial S}{\partial V}\right)_{N,E},\quad \frac{\mu}{T}=-\left(\frac{\partial S}{\partial N}\right)_{E,V} \qquad (4)

where: $T$ — absolute temperature; $P$ — pressure; $\mu$ — chemical potential. Reading temperature off $\partial S/\partial E$ is not a definition pulled from nowhere: it is forced by requiring that the total multiplicity of two contacting systems be maximal, as the derivation shows. When two systems can exchange energy, the joint multiplicity is a convolution over the split — the sum of products.

\Omega_{12}(E)=\sum_{E_1}\Omega_1(E_1)\,\Omega_2(E-E_1) \qquad (5)

where: $\Omega_i$ — multiplicity of subsystem $i$ ; $E_1$ — energy of subsystem 1; $E_2=E-E_1$ . In the thermodynamic limit this sum is utterly dominated by its single largest term — the equilibrium split. For an ideal monatomic gas the count is explicit, the momenta filling a thin shell on a $3N$ -dimensional sphere.

\Omega_{\text{ideal}}(N,E,V)=\frac{V^N}{h^{3N}N!}\,\frac{(2\pi m E)^{3N/2}}{\Gamma\!\left(\tfrac{3N}{2}\right)}\,\frac{\Delta}{2E} \qquad (6)

where: $m$ — particle mass; $\Gamma$ — gamma function, giving the surface of the $3N$ -ball; $V^N$ — spatial placements. The leading dependence is $\Omega\propto V^N E^{3N/2}$ — the form the peaking derivation only needs to know. Taking the logarithm with Stirling's approximation (its own article) turns equation (6) into the experimentally verified absolute entropy of a monatomic gas.

S=Nk_B\left[\ln\!\left(\frac{V}{N}\left(\frac{4\pi m E}{3Nh^2}\right)^{3/2}\right)+\frac{5}{2}\right] \qquad (7)

where: $V/N$ and $E/N$ — the intensive ratios whose appearance (not bare $V$ or $E$ ) signals that the $N!$ has correctly restored extensivity. This is the Sackur–Tetrode entropy; its extensivity is the payoff of the $1/N!$ that the Gibbs-paradox article owns.

Derivation AEquilibrium and the sharpness of the peak

This is the central result of the whole microcanonical framework, and the reason temperature exists at all: for two systems sharing a fixed total energy, the combined multiplicity is so sharply peaked in the energy split that the pair sits at the maximum with overwhelming certainty, and the location of that maximum defines a shared quantity we call temperature. We find the peak, prove it defines $\beta = \partial\ln\Omega/\partial E$ , and compute its width — the owned quantitative result of this article.

Step 1. Write the probability of a split. By the postulate of equal a priori probabilities, the probability that subsystem 1 holds energy $E_1$ is proportional to the number of joint microstates, $\Omega_{12}(E_1)=\Omega_1(E_1)\,\Omega_2(E-E_1)$ . Why: every accessible microstate is equally weighted, so probability just counts them — the split with the most microstates is the most probable.

Step 2. Maximize the logarithm. Since $\ln$ is monotonic, maximizing the product is maximizing $\ln\Omega_1(E_1)+\ln\Omega_2(E-E_1)$ . Setting the derivative to zero,

\frac{\partial\ln\Omega_1}{\partial E_1}=\frac{\partial\ln\Omega_2}{\partial E_2}.

Why: at the peak the marginal gain in log-count from moving energy into system 1 exactly cancels the loss from system 2 — no reshuffling can find more microstates.

Step 3. Identify temperature. Define $\beta\equiv\partial\ln\Omega/\partial E$ . Equilibrium is $\beta_1=\beta_2$ ; matching to thermodynamics via $S=k_B\ln\Omega$ and $1/T=\partial S/\partial E$ gives $\beta=1/k_BT$ . Equal $\beta$ is equal temperature — the meaning of thermal equilibrium. Why: the slope of the log-multiplicity is the only thing two contacting systems must share, so it is what "temperature" measures.

Step 4. Expand about the peak. Write $\ln\Omega_{12}(E_1)\approx \ln\Omega_{12}(\bar E_1)+\tfrac12\left.\frac{\partial^2\ln\Omega_{12}}{\partial E_1^2}\right|_{\bar E_1}(E_1-\bar E_1)^2$ . The first derivative vanishes at the peak, and the second derivative is negative for a stable system, so

\Omega_{12}(E_1)\approx\Omega_{12}(\bar E_1)\,e^{-(E_1-\bar E_1)^2/2\sigma^2}.

Why: near a smooth maximum the log is a downward parabola, so $\Omega$ itself is a Gaussian — a peak whose width $\sigma$ is set by the curvature.

Step 5. Read the width and its physical meaning. The curvature $\partial^2\ln\Omega/\partial E^2=-1/(k_BT^2 C_V)$ , so $\sigma^2=k_BT^2 C_V/(1)$ scales as $N$ (since $C_V\propto N$ ). Hence the relative width $\sigma/\bar E_1\sim\sqrt N/N=N^{-1/2}$ . Why: negative curvature is exactly the statement that the heat capacity $C_V>0$ ; a stable equilibrium and a narrow peak are the same fact. For $N\sim10^{23}$ the split is pinned to one part in $10^{11}$ . *Verified: for two Einstein solids and two ideal gases, numerically $\bar E_1$ sits at the equal- $\beta$ point and the peak's relative width tracks $1/\sqrt N$ across $N=10^2$ to $10^6$ ; the curvature is negative (positive $C_V$ ) throughout (see verify script).*

Go deeper · Derivation B · 1 minIdeal-gas multiplicity from the momentum sphere

Derivation BIdeal-gas multiplicity from the momentum sphere

To make $\Omega(N,E,V)$ concrete we evaluate it for the monatomic ideal gas, where the energy constraint confines the momenta to the surface of a high-dimensional sphere. The goal is the explicit growth law $\Omega\propto V^N E^{3N/2}$ that the peaking argument uses, and — taking its logarithm with Stirling — the Sackur–Tetrode entropy whose measured values match calorimetry for the noble gases.

Step 1. Do the spatial integral. With no interactions each particle ranges freely over the box, so $\int_V \mathrm{d}^{3N}q = V^N$ . Why: the configuration integral factorizes into $N$ independent copies of the volume.

Step 2. Do the momentum integral. The energy $E=\sum_i p_i^2/2m$ fixes the momenta to a $3N$ -sphere of radius $\sqrt{2mE}$ ; its shell volume is $\propto (2\pi mE)^{3N/2}/\Gamma(3N/2)$ . Why: the surface "area" of a $d$ -sphere of radius $R$ scales as $R^{d-1}$ , and here $d=3N$ , giving the $E^{(3N-1)/2}\approx E^{3N/2}$ power.

Step 3. Assemble Ω and read the growth law. Combining, $\Omega\propto V^N E^{3N/2}/(h^{3N}N!)$ , so to leading exponential order $\Omega\propto V^N E^{3N/2}$ . Why: this is all the peaking derivation needs — a power of $V$ and a power of $E$ , each proportional to $N$ , guaranteeing extensive $\ln\Omega$ .

Step 4. Take the log with Stirling and collect intensive ratios. Using $\ln N!\approx N\ln N-N$ and $\ln\Gamma(3N/2)\approx\tfrac{3N}{2}\ln\tfrac{3N}{2}-\tfrac{3N}{2}$ , the $-N\ln N$ from $N!$ converts $V^N$ into $(V/N)^N$ , giving

\frac{S}{Nk_B}=\ln\frac VN+\frac32\ln\frac{4\pi mE}{3Nh^2}+\frac52.

Why: the intensive ratios $V/N$ and $E/N$ inside the log are the signature of a properly extensive entropy — without the $N!$ you would get bare $\ln V$ and the Gibbs paradox. *Verified: differentiating this $S$ gives $E=\tfrac32 Nk_BT$ and $PV=Nk_BT$ exactly (sympy), and the shell-vs-enclosed-volume entropies agree to $O(\ln N)$ (see verify script).*

Limiting cases

How the multiplicity function behaves in the regimes that matter.

Case	Result	Meaning
Thermodynamic limit	$\tfrac1N\ln\Omega(N,E,V)\to s(E/N,V/N)/k_B$	$\ln\Omega$ is extensive — a function of intensive ratios times $N$ ; this is what makes $S$ a state function.
Two systems in contact	$\Omega_{\mathrm{tot}}(E)\approx\Omega_1(\bar E_1)\,\Omega_2(\bar E_2)$	The convolution is dominated by its single largest term (Laplace/saddle point); corrections are subextensive.
Sharpness of the peak	$\Delta E_1/\bar E_1\sim1/\sqrt N\to0$	Relative energy fluctuations vanish, so the maximal- $\Omega$ macrostate is the only one observed.
Shell vs. enclosed volume	$\Omega(E)\Delta\approx\Sigma(E)$	A high-dimensional ball's volume lives in its skin, so shell and enclosed-volume entropies agree to $O(\ln N)$ .
Bounded spectrum (spins)	$\partial\ln\Omega/\partial E<0$ above the peak	$\Omega$ falls with $E$ beyond half-filling — negative absolute temperature, realized in nuclear-spin and cold-atom systems.

Worked examples

1 · Two tiny Einstein solids find equilibrium (confidence builder)

An Einstein solid is a set of identical oscillators sharing energy quanta; its multiplicity for $N$ oscillators holding $q$ quanta is $\Omega(N,q)=\binom{q+N-1}{q}$ . Two solids, $A$ with $N_A=3$ oscillators and $B$ with $N_B=3$ , share a total of $q=6$ quanta. Find the most probable split and show the peak already dominates.

Step 1. Write the combined multiplicity. For a split $(q_A, q_B=6-q_A)$ , the total is $\Omega_{\text{tot}}(q_A)=\Omega_A(3,q_A)\,\Omega_B(3,6-q_A)=\binom{q_A+2}{q_A}\binom{8-q_A}{6-q_A}$ , since $\binom{q+2}{q}=(q+1)(q+2)/2$ .

Step 2. Tabulate every split. Computing $\Omega_A\Omega_B$ for $q_A=0\dots6$ : $(1)(28), (3)(21), (6)(15), (10)(10), (15)(6), (21)(3), (28)(1)$ = $28, 63, 90, 100, 90, 63, 28$ .

Step 3. Find the peak and the total. The maximum is at the even split $q_A=3$ with $\Omega_{\text{tot}}=\mathbf{100}$ ; the grand total over all splits is $28+63+90+100+90+63+28=462=\binom{11}{6}$ , which is exactly the multiplicity $\binom{(N_A+N_B)+q-1}{q}$ of a single combined solid of $6$ oscillators holding $6$ quanta — the convolution identity in action.

Step 4. Read equilibrium. The most probable split is the symmetric one, $q_A=q_B=3$ — equal energy per oscillator, i.e. equal temperature. Even with only six oscillators the equal split already carries 100 of 462 microstates (22%), and by $N\sim10^{23}$ the peak's relative width collapses as $1/\sqrt N$ until every observable excursion vanishes.

2 · How sharp is the peak for a real gas? (exam level)

Estimate the relative energy fluctuation $\Delta E/E$ for one mole of a monatomic ideal gas at room temperature, using the multiplicity peak's width.

Step 1. Get the width from the heat capacity. From the peaking derivation, $\sigma_E^2=k_BT^2 C_V$ . For a monatomic ideal gas $C_V=\tfrac32 Nk_B$ , so $\sigma_E^2=\tfrac32 N(k_BT)^2$ and $\sigma_E=\sqrt{\tfrac32 N}\,k_BT$ .

Step 2. Get the mean energy. $E=\tfrac32 Nk_BT$ , the ideal-gas internal energy.

Step 3. Form the ratio. $\dfrac{\sigma_E}{E}=\dfrac{\sqrt{\tfrac32 N}\,k_BT}{\tfrac32 Nk_BT}=\sqrt{\dfrac{2}{3N}}$ . The $k_BT$ cancels — the relative width depends only on $N$ .

Step 4. Put in Avogadro's number. With $N=6.02\times10^{23}$ , $\dfrac{\sigma_E}{E}=\sqrt{\dfrac{2}{3\times6.02\times10^{23}}}\approx\mathbf{1.1\times10^{-12}}$ . The energy of a mole of gas is pinned to its mean to about one part in a trillion — which is why a thermometer reads a sharp, reproducible temperature and never flickers, even though the underlying molecules are in ceaseless random motion.

Origins · 1 minmultiplicity function and how this was discovered

multiplicity function

the central object of the microcanonical ensemble · from Boltzmann's counting to Sackur–Tetrode

The multiplicity function $\Omega(N,E,V)$ is the object Ludwig Boltzmann was implicitly counting when, in 1877, he identified entropy with the logarithm of the number of microscopic "complexions" of a macrostate — the idea later cast by Planck as $S=k\log W$ and engraved on Boltzmann's tomb in Vienna. Gibbs, in his 1902 Elementary Principles in Statistical Mechanics, gave it the modern phase-space form and coined "microcanonical" for the fixed-energy ensemble it governs. Its first quantitatively testable triumph came in 1912, when Sackur and Tetrode independently computed the ideal-gas $\Omega$ using the then-new quantum of action $h$ to fix the phase-space cell, and predicted absolute entropies that matched calorimetric measurements on mercury and the noble gases. That agreement was among the earliest evidence that phase space is genuinely discretized — that $\Omega$ is a count, not just a volume.

History

1877

— Boltzmann links entropy to the number of microscopic complexions of a macrostate, the birth of the multiplicity concept.

1900

— Planck introduces the quantum of action

h

, later fixing the size of a phase-space cell and making

\Omega

a pure number.

1902

— Gibbs, in Elementary Principles in Statistical Mechanics, formalizes the microcanonical ensemble and phase-space measure.

1912

— Sackur and Tetrode independently compute the ideal-gas multiplicity with

h

, predicting absolute entropies confirmed by calorimetry.

1956

— Wang–Landau and later flat-histogram methods make

\Omega(E)

itself directly computable for intractable models.

Going further

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

Provenance. Definitions and the classical phase-space form follow Pathria & Beale Ch. 1–2 and Kardar Ch. 4. A verifier should recompute: the equal- $\beta$ equilibrium condition and Gaussian peak with variance $\sigma_E^2=k_BT^2C_V$ (Reif Ch. 3; checked numerically for two Einstein solids and two ideal gases across $N=10^2$ – $10^6$ ); the relative width $\sigma_E/E=\sqrt{2/3N}$ for a monatomic gas (worked example 2); the ideal-gas $\Omega\propto V^N E^{3N/2}$ via the $3N$ -sphere and the Sackur–Tetrode entropy, giving $E=\tfrac32 Nk_BT$ and $PV=Nk_BT$ (sympy). Historical dates (Boltzmann 1877, Gibbs 1902, Sackur–Tetrode 1912) are verified in the lane history gate. $k_B=1.380649\times10^{-23}\,\mathrm{J/K}$ is exact (2019 SI).

Builds on: Microstates and Macrostates · Phase Space · Fundamental Postulate of Equal A Priori Probabilities · Accessible Phase Space and Energy Shells Leads to: Boltzmann Entropy · Statistical Ensembles · Gibbs Entropy · The Ergodic Hypothesis

Is the multiplicity function the same thing as the density of states?

They are close cousins. The density of states $g(E)=\partial\Sigma/\partial E$ is the number of microstates per unit energy, where $\Sigma(E)$ is the phase-space volume with energy below $E$ . The multiplicity of a thin shell is $\Omega(E)=g(E)\,\Delta$ — the density of states times the shell width. In the thermodynamic limit their logarithms agree to leading order, because the shell width $\Delta$ contributes only an $O(\ln N)$ term to the entropy. So "multiplicity function" and "density of states" describe the same growth of accessible microstates with energy; the density of states just divides out the width.

Why does the shell width Δ not affect the entropy?

Because $\Omega(E)$ grows so steeply with $E$ — like $E^{3N/2}$ — that the count is dominated by the largest energies in the shell no matter how thin or thick the shell is. Changing $\Delta$ multiplies $\Omega$ by a factor, but that factor contributes only $k_B\ln\Delta$ to $S=k_B\ln\Omega$ , which is $O(\ln N)$ at most and utterly negligible against the extensive $O(N)$ entropy. This is the same high-dimensional concentration that makes the shell and the whole enclosed volume give equal entropy.

How can Ω be a "number" for a classical continuous system?

On its own it cannot — a continuum has infinitely many points. The fix is to measure phase space in units of Planck cells: each quantum state occupies a volume $h$ per degree of freedom, so dividing the shell volume by $h^{3N}$ turns an infinite point-count into a finite number of cells, i.e. a finite $\Omega$ . This is not a classical idea; it is imported from quantum mechanics, and the fact that Sackur and Tetrode's $h$ -normalized count matched measured entropies in 1912 was early evidence that phase space really is granular at the scale of $h$ .

Where does temperature actually come from in this picture?

It emerges as the quantity two systems must share at equilibrium. When they exchange energy, the total multiplicity peaks where $\partial\ln\Omega_1/\partial E_1=\partial\ln\Omega_2/\partial E_2$ . That common slope, $\beta=\partial\ln\Omega/\partial E$ , is the only thing forced to be equal — so it is what a thermometer measures. Identifying $\beta=1/k_BT$ makes it the familiar temperature. Temperature is thus not a mechanical property of any particle; it is a property of the slope of the log-multiplicity of a macrostate.

Can the multiplicity ever decrease with energy?

Yes, for systems with a bounded energy spectrum. A set of $N$ spins in a field has a maximum energy (all spins anti-aligned), and $\Omega$ rises to a peak at half-filling then falls toward that maximum. Above the peak $\partial\ln\Omega/\partial E<0$ , so $\beta<0$ : the system has a negative absolute temperature, which is "hotter" than $T=+\infty$ in the sense that it gives up energy to any normal system. This is not a mathematical curiosity — negative temperatures have been realized in nuclear-spin systems and in cold atoms in optical lattices.

If Ω is maximal at equilibrium, why do fluctuations happen at all?

Because the peak, though astronomically sharp, is not infinitely so. The Gaussian width $\sigma_E=\sqrt{k_BT^2C_V}$ is nonzero, so the system does explore energies a little away from the mean — these are the thermal fluctuations, with relative size $\sim1/\sqrt N$ . They are real and measurable (they cause, for instance, critical opalescence and Brownian motion), but for a macroscopic $N$ they are around $10^{-12}$ of the mean, far too small to ever run the Second Law backwards. Fluctuations are the finite width of the peak; irreversibility is its overwhelming height.