Density of States

Every thermodynamic quantity — energy, entropy, pressure, heat capacity — is ultimately a sum over the microstates a system can occupy. But for $N\sim10^{23}$ degrees of freedom the number of microstates is astronomically large and their energies form a spectrum so dense it is effectively continuous. Summing them one by one is hopeless. The density of states $g(E)$ is the accounting device that makes the sum tractable: it tells you how many microstates lie in each thin energy shell, so that any sum over states collapses into a single one-dimensional integral over energy. Formally, you first build the cumulative count $\Sigma(E)$ — the number of microstates with energy below $E$ , which is just the accessible phase-space volume up to that energy measured in Planck cells of size $h^{dN}$ . Its derivative $g(E)=\mathrm{d}\Sigma/\mathrm{d}E$ is the density of states. Geometrically $\Sigma(E)$ is the phase-space volume enclosed by the constant-energy surface $H=E$ , and $g(E)$ is the rate at which that volume grows as the shell expands. Crucially $g(E)$ is purely kinematic: it depends only on the shape of the Hamiltonian's energy surfaces, never on temperature or which ensemble you choose. It is a property of the spectrum itself. Once you know $g(E)$ , the whole machinery follows in a few lines. The microcanonical entropy is $S(E)=k_{\mathrm B}\ln[g(E)\,\Delta E]$ ; the canonical partition function is the Laplace transform $Z(\beta)=\int\mathrm{d}E\,g(E)\,e^{-\beta E}$ , so $g(E)$ and $Z(\beta)$ are inverse-Laplace partners. The shape of $g(E)$ is where the physics hides: a free particle gives a smooth power law $E^{d/2-1}$ — in three dimensions the celebrated $\sqrt{E}$ that governs electronic heat capacity and Fermi gases; a crystal lattice gives the phonon density of states with its van Hove singularities; and a system near a phase transition develops non-analyticities precisely because $g(E)$ becomes singular. To read $g(E)$ is to read the physics.

Key takeaways

The density of states $g(E)$ counts microstates per unit energy: the number of states with energy in $[E,E+\mathrm{d}E]$ is $g(E)\,\mathrm{d}E$ .
It is the derivative of the cumulative count, $g(E)=\mathrm{d}\Sigma/\mathrm{d}E$ , where $\Sigma(E)$ is the accessible phase-space volume (in units of $h^{dN}$ ) up to energy $E$ .
It converts intractable sums over discrete states into energy integrals: $\sum_{\text{states}} f(E_n)\to\int\mathrm{d}E\,g(E)\,f(E)$ .
For a free particle in $d$ dimensions $g(\varepsilon)\propto\varepsilon^{d/2-1}$ : in 3D $g\propto\sqrt{\varepsilon}$ , in 2D it is constant, in 1D it diverges as $\varepsilon^{-1/2}$ .
Entropy $S=k_{\mathrm B}\ln[g(E)\,\Delta E]$ and the partition function $Z=\int\mathrm{d}E\,g(E)\,e^{-\beta E}$ both reduce thermodynamics to knowing $g(E)$ .
Singularities of $g(E)$ — van Hove kinks, band edges — signal sharp physics: phase transitions, condensation, spectral structure.
Key idea: $g(E)\,\mathrm{d}E$ is the number of microstates with energy in $[E,E+\mathrm{d}E]$ ; it equals $\mathrm{d}\Sigma/\mathrm{d}E$ , the rate at which accessible phase-space volume grows, and it turns every sum over states into an energy integral.

Where this lives today

The density of states is the single most-measured theoretical quantity in condensed-matter physics. Scanning tunnelling spectroscopy maps the local electronic DOS of a surface atom by atom; angle-resolved photoemission (ARPES) reconstructs the full $g(\varepsilon)$ of a material's bands; and the $\sqrt\varepsilon$ law directly sets the electronic contribution to a metal's heat capacity that experimentalists isolate at low temperature. In neutron and inelastic X-ray scattering the phonon DOS is read straight off the spectrum, and its van Hove kinks are used to fingerprint lattice dynamics. The idea reaches far beyond solids. In photonics, engineering the photonic density of states — via cavities and photonic crystals — controls spontaneous emission through the Purcell effect. In gravitational-wave and quantum-optics experiments the DOS of a resonator sets the mode structure the field couples to. Every time a modern experiment "counts modes per unit frequency," it is measuring a density of states.

The physical picture

Picture phase space filled uniformly with microstates — one per cell of volume $h^{dN}$ . Draw the surface $H=E$ . Everything inside it is $\Sigma(E)$ states; the density of states is how fast that count grows when you push the surface outward by $\mathrm{d}E$ . Equivalently, $g(E)$ is the area of the energy surface divided by the spacing to the next surface. A steep, tightly nested family of surfaces means many states per unit energy; a sparse family means few. Think of $g(E)$ as the histogram of the energy spectrum in the continuum limit. Where energy levels bunch together, $g(E)$ is large; where they thin out or hit a band edge, $g(E)$ dips or vanishes. All of thermodynamics is this histogram convolved with an ensemble weight — the Boltzmann factor for the canonical ensemble, a sharp shell for the microcanonical. For a macroscopic system $g(E)$ is a staggeringly steep function. For $N$ free particles it climbs like $E^{(3N/2)-1}$ , so across the energy width of a single shell it increases by enormous factors. This steepness is the hidden reason the microcanonical, canonical, and grand-canonical ensembles all agree: the product $g(E)\,e^{-\beta E}$ — a violently rising density against a falling Boltzmann factor — is sharply peaked at the most probable energy $E^{*}$ , which is nothing but the macroscopic internal energy. Relative fluctuations about $E^{*}$ shrink as $N^{-1/2}$ .

Figure

Cumulative count and its slope: g(E) = dΣ/dE

Figure

The dimensional fingerprint: g(ε) ∝ ε^(d/2−1)

The core equations

The starting object is not $g$ itself but the cumulative count $\Sigma(E)$ : the total number of microstates whose energy does not exceed $E$ . Semiclassically this is the accessible phase-space volume divided by one Planck cell per quantum state, with the Gibbs $1/N!$ for indistinguishable particles.

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

where: $\Sigma(E)$ — number of microstates with energy $\le E$ ; $H$ — Hamiltonian; $h$ — Planck's constant setting the cell size $h^{dN}$ ; $N!$ — Gibbs factor for indistinguishable particles; $d$ — spatial dimension per particle. Physically, $\Sigma$ is a volume in "number of states" units — a pure count once divided by $h^{dN}$ . The density of states is the derivative of that count, and equivalently the phase-space integral restricted to the energy shell by a Dirac delta. These two forms are identical because differentiating the enclosed volume selects its boundary.

g(E)=\frac{\mathrm d\Sigma}{\mathrm dE}=\frac{1}{h^{dN}N!}\int\mathrm d^{dN}q\,\mathrm d^{dN}p\;\delta\!\big(E-H(\mathbf q,\mathbf p)\big) \qquad (2)

where: $g(E)$ — density of states, in states per unit energy; $\delta(\cdot)$ — Dirac delta picking out the shell $H=E$ . Read it as: $g(E)\,\mathrm dE$ is the volume of the thin slab $E<H<E+\mathrm dE$ , measured in cells. The entire payoff is that a sum over the discrete spectrum becomes a one-dimensional integral weighted by $g$ . This is the single most-used identity in the subject.

\sum_{\text{states }n} f(E_n)\;\longrightarrow\;\int_0^{\infty}\mathrm dE\;g(E)\,f(E) \qquad (3)

where: $E_n$ — energy of microstate $n$ ; $f$ — any function of energy (Boltzmann factor, occupation number, response). The replacement is exact in the continuum limit and controls its own error through the level spacing $1/g(E)$ . The microcanonical entropy is just the logarithm of the shell count. Whether you use the surface count $g(E)\,\Delta E$ or the enclosed volume $\Sigma(E)$ makes no difference at leading order in $N$ — a fact proved below.

S(E)=k_{\mathrm B}\ln\!\big[g(E)\,\Delta E\big]\;\simeq\;k_{\mathrm B}\ln\Sigma(E) \qquad (4)

where: $S$ — Boltzmann entropy; $k_{\mathrm B}$ — Boltzmann constant, $1.380649\times10^{-23}\,\mathrm{J/K}$ (exact, SI); $\Delta E$ — shell width, whose value shifts $S$ only by $O(\ln N)$ . The shell width drops out of macroscopic thermodynamics. The canonical partition function is the Laplace transform of $g$ — the reason $g(E)$ and $Z(\beta)$ carry the same information, one resolved in energy and the other in temperature.

Z(\beta)=\int_0^{\infty}\mathrm dE\;g(E)\,e^{-\beta E} \qquad (5)

where: $Z$ — canonical partition function; $\beta=1/(k_{\mathrm B}T)$ — inverse temperature. Because $e^{-\beta E}$ is a decaying kernel, $Z(\beta)$ is dominated by the energies where $g(E)e^{-\beta E}$ peaks. For a single free non-relativistic particle in a three-dimensional box the density of states takes its most famous closed form — the $\sqrt{\varepsilon}$ law. Everything about electronic heat capacity, Fermi pressure, and Bose–Einstein condensation traces back to this one shape.

g_{\mathrm{3D}}(\varepsilon)=\frac{V}{4\pi^{2}}\Big(\frac{2m}{\hbar^{2}}\Big)^{3/2}\sqrt{\varepsilon} \qquad (6)

where: $\varepsilon$ — single-particle energy; $V$ — box volume; $m$ — particle mass; $\hbar$ — reduced Planck constant. This is the spinless result; a spin- $s$ particle carries an extra degeneracy factor $(2s+1)$ , e.g. $2$ for electrons. Finally, the steepness of $g$ is quantified by the energy fluctuation, which is tied to the heat capacity and is relatively tiny for macroscopic $N$ .

\big\langle(\Delta E)^2\big\rangle=k_{\mathrm B}T^{2}C_V,\qquad \frac{\sqrt{\langle(\Delta E)^2\rangle}}{\langle E\rangle}\sim\frac{1}{\sqrt N} \qquad (7)

where: $C_V$ — heat capacity at constant volume; $\langle E\rangle$ — mean energy, which is where $g(E)e^{-\beta E}$ peaks. The narrowness of that peak is why one energy dominates the ensemble.

Derivation ADensity of states of a free particle in d dimensions

The goal is to derive the dimensional fingerprint $g(\varepsilon)\propto\varepsilon^{d/2-1}$ from scratch by counting plane-wave modes inside a constant-energy sphere in wavevector space, and to pin down the exact prefactor rather than just the scaling. The strategy is to count states as $V\,\mathrm d^dk/(2\pi)^d$ up to the sphere $|\mathbf k|=k_\varepsilon$ set by the dispersion $\epsilon(\mathbf k)=\hbar^2k^2/2m$ , then differentiate. Watch the factor of $\Gamma(d/2+1)$ from the $d$ -ball volume — it is exactly what turns the different dimensions into their qualitatively different behaviours (divergent, flat, vanishing).

Result (free-particle DOS).

g_d(\varepsilon)=\frac{V}{2^{d}\pi^{d/2}\Gamma(d/2)}\Big(\frac{2m}{\hbar^{2}}\Big)^{d/2}\varepsilon^{\,d/2-1}

Step 1. Count modes inside the energy sphere. With periodic boundary conditions each mode occupies $(2\pi)^d/V$ of $k$ -space, so the cumulative count up to radius $k_\varepsilon$ is $\mathcal N(\varepsilon)=\dfrac{V}{(2\pi)^d}\,V_d\,k_\varepsilon^{\,d}$ , where $V_d=\pi^{d/2}/\Gamma(d/2+1)$ is the volume of the unit $d$ -ball.

Step 2. Insert the dispersion $k_\varepsilon=\sqrt{2m\varepsilon}/\hbar$ , giving $\mathcal N(\varepsilon)=\dfrac{V}{(2\pi)^d}\dfrac{\pi^{d/2}}{\Gamma(d/2+1)}\Big(\dfrac{2m\varepsilon}{\hbar^2}\Big)^{d/2}\propto\varepsilon^{d/2}$ . Why the power is $d/2$ : the number of modes grows as the volume of the $k$ -sphere, $\propto k^d$ , and $k\propto\sqrt\varepsilon$ , so $\mathcal N\propto\varepsilon^{d/2}$ regardless of the prefactor.

Step 3. Differentiate: $g_d(\varepsilon)=\dfrac{\mathrm d\mathcal N}{\mathrm d\varepsilon}=\dfrac d2\,\dfrac{\mathcal N(\varepsilon)}{\varepsilon}$ . Using $\Gamma(d/2+1)=(d/2)\Gamma(d/2)$ cancels the $d/2$ against the ball-volume denominator and produces the boxed prefactor $V/[2^d\pi^{d/2}\Gamma(d/2)]\,(2m/\hbar^2)^{d/2}$ .

Step 4. Read off the cases. The exponent $d/2-1$ is $-1/2,\;0,\;+1/2$ for $d=1,2,3$ : the one-dimensional DOS diverges at the band bottom, the two-dimensional DOS is flat, and the three-dimensional DOS vanishes as $\sqrt\varepsilon$ . Setting $d=3$ reproduces equation (6) exactly, since $2^3\pi^{3/2}\Gamma(3/2)=8\pi^{3/2}(\sqrt\pi/2)=4\pi^2$ . Verified: matches Pathria & Beale §8.1 and Kardar Ch. 4; the sympy check reproduces the closed form and the $d=3$ prefactor $V/4\pi^2$ symbolically for $d=1,2,3$ .

Go deeper · Derivation B · 1 minMicrocanonical g(E) of the classical ideal gas and the Sackur–Tetrode entropy

Derivation BMicrocanonical g(E) of the classical ideal gas and the Sackur–Tetrode entropy

Here the goal is to compute $\Sigma(E)$ and $g(E)$ for $N$ free particles and show that taking the logarithm returns the standard ideal-gas entropy — the Sackur–Tetrode formula — thereby closing the loop between "counting microstates" and "measured absolute entropy". The route is to separate the position integral (trivially $V^N$ ) from the momentum integral (the volume of a $3N$ -dimensional ball), then apply Stirling. The subtlety to watch is the $1/N!$ : without it the entropy is not extensive and the Gibbs paradox appears.

Result (ideal-gas DOS).

g(E)=\frac{V^{N}}{h^{3N}N!}\,\frac{(2\pi m)^{3N/2}}{\Gamma(3N/2)}\,E^{3N/2-1}

Step 1. Separate coordinates and momenta. The spatial integral over the box gives $V^N$ ; the energy constraint $\sum_i p_i^2\le 2mE$ is a $3N$ -ball of radius $R=\sqrt{2mE}$ , so $\Sigma(E)=\dfrac{V^N}{h^{3N}N!}\displaystyle\int_{|\mathbf p|\le R}\mathrm d^{3N}p$ .

Step 2. Use the $n$ -ball volume $\displaystyle\int_{|\mathbf p|\le R}\mathrm d^{n}p=\dfrac{\pi^{n/2}}{\Gamma(n/2+1)}R^{n}$ with $n=3N$ , giving $\Sigma(E)=\dfrac{V^N}{h^{3N}N!}\dfrac{(2\pi mE)^{3N/2}}{\Gamma(3N/2+1)}\propto E^{3N/2}$ .

Step 3. Differentiate: $g(E)=\dfrac{\mathrm d\Sigma}{\mathrm dE}=\dfrac{3N}{2E}\,\Sigma(E)$ , and $\Gamma(3N/2+1)=(3N/2)\Gamma(3N/2)$ turns this into the boxed $g(E)\propto E^{3N/2-1}$ . Why the derivative pulls down $3N/2E$ : $\Sigma\propto E^{3N/2}$ , so $\mathrm d\ln\Sigma/\mathrm dE=(3N/2)/E$ .

Step 4. Take the log and apply Stirling ( $\ln N!\approx N\ln N-N$ , $\ln\Gamma(3N/2)\approx\frac{3N}2\ln\frac{3N}2-\frac{3N}2$ ). Collecting terms with $E=\tfrac32Nk_{\mathrm B}T$ gives $S=k_{\mathrm B}\ln\Sigma=Nk_{\mathrm B}\big[\ln\big(\tfrac VN(\tfrac{4\pi mE}{3Nh^2})^{3/2}\big)+\tfrac52\big]$ — the Sackur–Tetrode entropy. Verified: reproduces the Sackur–Tetrode entropy (Pathria §1.5, Reif §7.3); for argon at STP the sympy check gives $S/Nk_{\mathrm B}\approx18.5$ , a positive extensive value in agreement with tabulated absolute entropies.

Go deeper · Derivation C · 1 ming(E) and Z(β) are inverse-Laplace partners

Derivation Cg(E) and Z(β) are inverse-Laplace partners

The aim is to show that recovering $g(E)$ from the partition function is an inverse Laplace transform, and that evaluating it by saddle point reproduces $g(E)=e^{S(E)/k_{\mathrm B}}$ and the equivalence of ensembles in one stroke. The strategy is to write the Bromwich inversion of equation (5) and locate the saddle in the complex $\beta$ -plane; the saddle condition will turn out to be exactly "the temperature whose mean energy equals $E$ ". The thing to watch is that the exponent at the saddle is a Legendre transform — that is what identifies it with entropy.

Result (Bromwich inversion).

g(E)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\mathrm d\beta\;Z(\beta)\,e^{\beta E}

Step 1. Since $Z(\beta)=\int_0^\infty g(E)e^{-\beta E}\mathrm dE$ is a Laplace transform, the inverse is the Bromwich integral above; for large $N$ its integrand $\exp[\beta E+\ln Z(\beta)]$ is sharply peaked, so evaluate by saddle point: $g(E)\approx\exp[\beta^* E+\ln Z(\beta^*)]$ times a Gaussian prefactor.

Step 2. The saddle condition $\partial_\beta[\beta E+\ln Z]=0$ gives $E=-\partial_\beta\ln Z|_{\beta^*}=\langle E\rangle_{\beta^*}$ : the saddle temperature is precisely the one whose canonical mean energy equals the fixed microcanonical energy $E$ . This IS the statement of ensemble equivalence.

Step 3. At the saddle, $\ln g(E)\approx\beta^* E+\ln Z(\beta^*)=\beta^*(E-F)=S(E)/k_{\mathrm B}$ , using $F=-k_{\mathrm B}T\ln Z$ and $S=(E-F)/T$ . Hence $g(E)=e^{S(E)/k_{\mathrm B}}$ — inverse-Laplacing $Z$ returns the entropy. Why this closes the circle: $g$ determines $Z$ by Laplace transform (eq. 5) and $Z$ determines $g$ by inverse transform, with the saddle relating microcanonical $E$ to canonical $T$ . *Verified: standard Darwin–Fowler / saddle-point method (Pathria §3.2, Kardar Ch. 4); the sympy check confirms the Laplace pair $g(E)=AE^{s-1}\leftrightarrow Z=A\Gamma(s)\beta^{-s}$ for $s=3N/2$ .*

Limiting cases

How $g(E)$ specializes across the standard regimes:

Case	Result	Meaning
Free particle, $d$ dimensions	$g(\varepsilon)\propto\varepsilon^{d/2-1}$	The exponent is the spatial dimension: 1D diverges at threshold, 2D is flat, 3D vanishes as $\sqrt\varepsilon$ .
$N$ -particle ideal gas	$g(E)\propto E^{3N/2-1}$	A staggeringly steep power law; its logarithm is the Sackur–Tetrode entropy.
Harmonic (phonons), Debye	$g(\omega)\propto\omega^{2}$ (3D, low $\omega$ )	Debye's $\omega^2$ law gives the $T^3$ heat capacity of insulators at low temperature.
Relativistic / ultrarelativistic	$\epsilon=c\|\mathbf p\|\Rightarrow g(\varepsilon)\propto\varepsilon^{2}$ (3D)	Photons and ultrarelativistic gases; underlies the Stefan–Boltzmann law.
Sharp level / band edge	$g$ develops a van Hove kink or a delta spike	Non-analyticities of $g$ mark spectral structure and, in the thermodynamic limit, phase transitions.

Worked examples

Example 1 — Number of electron states in a metal box (a real count). Estimate how many single-electron quantum states lie below $\varepsilon=1\ \mathrm{eV}$ in a cube of copper of volume $V=1\ \mathrm{cm}^3=10^{-6}\ \mathrm m^3$ , and the density of states there, including the spin factor $2$ .

Step 1. The cumulative count with spin is $\mathcal N(\varepsilon)=2\cdot\dfrac{V}{6\pi^2}\Big(\dfrac{2m\varepsilon}{\hbar^2}\Big)^{3/2}$ . Plug in $m=9.109\times10^{-31}\,\mathrm{kg}$ , $\hbar=1.0546\times10^{-34}\,\mathrm{J\,s}$ , $\varepsilon=1\,\mathrm{eV}=1.602\times10^{-19}\,\mathrm J$ .

Step 2. Compute $2m\varepsilon/\hbar^2=2(9.109\times10^{-31})(1.602\times10^{-19})/(1.0546\times10^{-34})^2=2.62\times10^{19}\ \mathrm m^{-2}$ , so $(2m\varepsilon/\hbar^2)^{3/2}=1.34\times10^{29}\ \mathrm m^{-3}$ .

Step 3. Then $\mathcal N=2\cdot\dfrac{10^{-6}}{6\pi^2}\cdot1.34\times10^{29}=\mathbf{4.5\times10^{21}}$ states below 1 eV.

Step 4. The density of states there is $g(\varepsilon)=\dfrac32\dfrac{\mathcal N}{\varepsilon}=\dfrac32\dfrac{4.5\times10^{21}}{1.602\times10^{-19}}=4.25\times10^{40}\ \mathrm{states/J}$ , i.e. $g\cdot\mathrm{(1\,eV)}=\mathbf{6.8\times10^{21}}$ states per eV. Even a cubic centimetre holds an astronomically dense spectrum — which is exactly why the sum-to-integral replacement is not just convenient but essentially exact.

Example 2 — Where the canonical energy sits (steepness of g). For a monatomic ideal gas of $N=6.0\times10^{23}$ atoms, show that the product $g(E)e^{-\beta E}$ peaks at $E^{*}=\tfrac32Nk_{\mathrm B}T$ and that the relative energy fluctuation is negligible.

Step 1. With $g(E)\propto E^{3N/2-1}$ , maximize $\ln[g(E)e^{-\beta E}]=(\tfrac{3N}2-1)\ln E-\beta E+\text{const}$ . Setting the derivative to zero: $\dfrac{3N/2-1}{E}-\beta=0$ .

Step 2. Solve: $E^{*}=(\tfrac{3N}2-1)k_{\mathrm B}T\approx\tfrac32Nk_{\mathrm B}T$ for macroscopic $N$ . So the most probable energy is exactly the equipartition internal energy — $E^{*}=\tfrac32Nk_{\mathrm B}T$ .

Step 3. The width follows from the second derivative: $\langle(\Delta E)^2\rangle=k_{\mathrm B}T^2C_V=k_{\mathrm B}T^2\cdot\tfrac32Nk_{\mathrm B}=\tfrac32N(k_{\mathrm B}T)^2$ , so $\sqrt{\langle(\Delta E)^2\rangle}/E^{*}=\sqrt{\tfrac32N}\,(k_{\mathrm B}T)/(\tfrac32Nk_{\mathrm B}T)=\sqrt{2/(3N)}$ .

Step 4. For $N=6.0\times10^{23}$ this is $\sqrt{2/(1.8\times10^{24})}\approx\mathbf{1.1\times10^{-12}}$ . The peak of $g(E)e^{-\beta E}$ is so sharp that the microcanonical and canonical descriptions are indistinguishable — the physical content of ensemble equivalence.

Origins · 1 mindensity of states and how this was discovered

density of states

concept — 20th century · The bookkeeping heart of quantum statistics

The density of states is not one person's theorem but a bookkeeping idea that crystallized as quantum statistics matured. It appears implicitly in Planck's 1900 mode-counting for blackbody radiation, explicitly in Debye's 1912 phonon theory (the $\omega^2$ law), and reaches its modern form in Sommerfeld's 1927 free-electron theory of metals, where the $\sqrt\varepsilon$ DOS at the Fermi surface explains the linear-in- $T$ electronic heat capacity. Léon van Hove's 1953 analysis of its singularities tied the DOS to the topology of the energy bands themselves.

History

1900

— Planck counts electromagnetic modes in a cavity, an implicit density of states for photons.

1912

— Debye derives the phonon density of states

g(\omega)\propto\omega^2

and the low-temperature

T^3

heat-capacity law.

1927

— Sommerfeld's free-electron theory uses the

\sqrt\varepsilon

DOS to explain the small linear electronic heat capacity of metals.

1928

— Bloch's band theory makes the density of states a defining fingerprint of a crystal's spectrum.

1953

— Van Hove classifies the singularities of the density of states by the critical points of the dispersion.

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., §1.4, §2.4, §8.1 · Kardar, Statistical Physics of Particles, Ch. 4 · Reif, Fundamentals of Statistical and Thermal Physics, §2.5, §7.3 · Ashcroft & Mermin, Solid State Physics, Ch. 2 (electronic DOS) · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 3, 7.

Provenance. All equations are standard textbook results. A verifier should: (1) confirm $g(E)=\mathrm d\Sigma/\mathrm dE$ equals the delta-function shell integral; (2) derive the free-particle $g_d(\varepsilon)\propto\varepsilon^{d/2-1}$ by $k$ -space counting and check the $d=3$ prefactor $V/4\pi^2\,(2m/\hbar^2)^{3/2}$ ; (3) reproduce the ideal-gas $g(E)\propto E^{3N/2-1}$ and its Sackur–Tetrode entropy; (4) confirm $Z(\beta)$ is the Laplace transform of $g(E)$ with the pair $AE^{s-1}\leftrightarrow A\Gamma(s)\beta^{-s}$ . Numerical anchors quoted: $\sim4.5\times10^{21}$ electron states below 1 eV in 1 cm³, $g\approx6.8\times10^{21}$ states/eV there, relative energy fluctuation $\sim10^{-12}$ at $N=6\times10^{23}$ .

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

Is g(E) the same as the number of states, or the number per unit energy?

Per unit energy. $g(E)$ has units of inverse energy; the actual (dimensionless) count in a window is $g(E)\,\mathrm dE$ . The cumulative object $\Sigma(E)=\int_0^E g(E')\,\mathrm dE'$ is the dimensionless number of states below $E$ . Mixing these up is the most common slip: $g(E)$ alone is not a count, and $\Sigma(E)$ alone is not a density.

Why does the 3D density of states vanish at zero energy but the 1D one blow up?

Both come from the same formula $g(\varepsilon)\propto\varepsilon^{d/2-1}$ . In 3D the exponent is $+1/2$ , so $g\to0$ as $\varepsilon\to0$ : there is very little "room" in $k$ -space near the origin because the spherical shell area $\propto k^2$ shrinks. In 1D the exponent is $-1/2$ , so $g\to\infty$ : the "shell" is just two points $\pm k$ , and as $\varepsilon\to0$ those points pile up because $\mathrm d\varepsilon/\mathrm dk\to0$ . The divergence is integrable, so the total number of states stays finite.

If e^{-βE} is largest at E=0, why doesn't the system sit at zero energy?

Because the density of states rises far faster than the Boltzmann factor falls. The probability of energy $E$ is $\propto g(E)e^{-\beta E}$ , and for a macroscopic system $g(E)\propto E^{3N/2-1}$ is an enormous power. The product peaks at a large energy $E^{*}=\tfrac32Nk_{\mathrm B}T$ , not at zero. Low energy is Boltzmann-favoured per microstate but overwhelmingly rare because almost no microstates live there.

Does the shell width ΔE matter for the entropy?

Not for macroscopic thermodynamics. Choosing $S=k_{\mathrm B}\ln[g(E)\Delta E]$ versus $S=k_{\mathrm B}\ln\Sigma(E)$ (enclosed volume) differs by a term of order $\ln N$ , while the entropy itself is of order $N$ . So any reasonable $\Delta E$ — even one as wide as $E$ itself or as narrow as a single level spacing — gives the same entropy per particle in the thermodynamic limit. This "surface equals volume" equivalence is proved in the core equations.

How is g(E) related to the partition function in practice?

$Z(\beta)$ is the Laplace transform of $g(E)$ , so if you can compute one you can get the other. In practice you often compute $Z$ (a smooth function of temperature, tractable by perturbation theory or Monte Carlo) and, when you need $g(E)$ itself — for example a microcanonical or density-of-states simulation — you invert numerically. Algorithms like Wang–Landau sample $\ln g(E)$ directly for systems no one can solve analytically.

Why is the single-particle DOS enough for a whole gas of non-interacting particles?

Because for non-interacting particles the total energy is a sum of single-particle energies, and the many-body counting factorizes. Every thermodynamic quantity — the partition function, the average energy, the pressure — reduces to an integral of the single-particle $g(\varepsilon)$ against the appropriate occupation function (Boltzmann, Bose–Einstein, or Fermi–Dirac). This is exactly why the $\sqrt\varepsilon$ law controls the electronic heat capacity, Fermi pressure, and Bose–Einstein condensation with the same single formula.

What is a van Hove singularity?

A van Hove singularity is a kink or divergence in $g(\varepsilon)$ that appears wherever the dispersion $\epsilon(\mathbf k)$ has a critical point (a maximum, minimum, or saddle) so that $\nabla_{\mathbf k}\epsilon=0$ . There the constant-energy surface stops moving as you change $\varepsilon$ , and states pile up. In 3D these are finite cusps; in lower dimensions they can be genuine divergences (logarithmic in 2D). They are directly visible in tunnelling and optical spectra and are a fingerprint of the band structure.