Quantum ideal-gas statistics

An ideal quantum gas has mean occupationwith one sign for bosons and the other for fermions.

For $g$ copies of a positive-energy branch $E=\hslash v|\mathbf{k}|$ in area $A$, the density of states is $D(E)=gAE/(2\pi {\hslash }^2{v}^2)$. At zero chemical potential, the thermal energy is $3g\zeta (3)A(k_{B}T{)}^3/(4\pi {\hslash }^2{v}^2)$, giving the displayed fixed-zero-chemical-potential heat capacity. If a symmetric filled negative-energy band also supplies holes, its excitation energy contributes an equal amount. Counting cones, spin species and holes explicitly is necessary before assigning an absolute numerical coefficient.

Maxwell–Boltzmann statistics describes a dilute classical gas whose single-particle occupation factors have the displayed exponential form. It is the low-occupation classical limit of Bose--Einstein and Fermi--Dirac statistics. Integrating this factor over momentum gives the nonrelativistic Maxwell--Boltzmann number density when the particle energy is $m{c}^2+p^2/(2m)$.

Integrating Maxwell–Boltzmann statistics over momentum for nonrelativistic particles of mass $m$, spin-state count $g_s$, and energy $m{c}^2+p^2/(2m)$ gives

The Fermi level is the chemical potential controlling electron occupations. For an ideal Fermi gas at zero temperature it equals the Fermi energy; at nonzero temperature the chemical potential need not equal that zero-temperature energy.

The Maxwell-Boltzmann distribution describes the random velocities and speeds of particles in a classical ideal gas at thermal equilibrium. Individual particles keep changing velocity through collisions, while the equilibrium probability distribution stays fixed. For one species of particle mass $m$ at positive temperature $T$, the characteristic scale iswhere $k_B$ is the Boltzmann constant. A signed velocity component follows a normal distribution; the nonnegative magnitude follows the Maxwell speed distribution. Their peaks differ because a speed measurement groups together all directions of motion.

Use the gas rest frame, three spatial dimensions, and nonrelativistic particles, with $k_{B}T\ll m{c}^2$ where $c$ is the speed of light. Assume a dilute gas whose interactions between encounters can be neglected in the ideal gas approximation and whose translational motion is classical. A sufficient condition for negligible quantum exchange effects is $n{\lambda }^3\ll 1$, where $n$ is the number density and $\lambda$ the thermal de Broglie wavelength. Diluteness alone does not guarantee thermal equilibrium; the gas must have equilibrated. These are the assumptions behind the classical Maxwell-Boltzmann velocity distribution. ^{[ref] [ref]}

The connection to Maxwell–Boltzmann statistics is the Boltzmann factor for translational kinetic energy, $E_{\mathrm{kin}}=m|\mathbf{v}{|}^2/2$. Normalizing $e^{-E_{\mathrm{kin}}/(k_{B}T)}$ over velocity gives a probability density; an occupation factor $e^{(\mu -E)/(k_{B}T)}$ instead specifies a mean occupation, with chemical potential $\mu$. Their normalizations serve different purposes. ^[ref]

This connects microscopic motion to measurements: a gas thermometer uses the ideal gas law, collisions and the mean free path govern transport, and measurements of spectral lines use thermal Doppler broadening to infer temperature. Each probes a different aspect of the same thermal motion.

In 1860, James Clerk Maxwell presented the component and speed laws in Part I, Proposition IV of Illustrations of the Dynamical Theory of Gases, pp. 22–23. His argument combined independence of the three velocity components with isotropy, and he explicitly compared the component law with the normal distribution of observational errors. Independence is an assumption of that argument; perpendicular coordinate axes alone do not establish it. ^[ref]

The speed distribution of atoms striking a wall is weighted by the normal component of velocity, relative to the distribution of atoms present in the bulk. For an isotropic classical monatomic gas, the impact speed density is proportional to $v^3{e}^{-m{v}^2/(2k_{B}T)}$, and the mean impact kinetic energy is $2k_{B}T$, rather than $3k_{B}T/2$.

In the gas rest frame, the Maxwell-Boltzmann velocity distribution is the probability density function per unit volume in velocity space:Thus $f(\mathbf{v})d^3\mathbf{v}$ gives the probability of a small three-dimensional velocity region. For a gas with bulk mean velocity $\mathbf{V}$, replace $\mathbf{v}$ by $\mathbf{v}-\mathbf{V}$; the Maxwell speed distribution then describes the magnitude relative to that bulk motion.

For any one Cartesian velocity component $v_x$, integrating out the other two givesThe factorization $f(\mathbf{v})=q(v_x)q(v_y)q(v_z)$ establishes independence: all three components are centered normal random variables, with variance $s^2$. Both the component density $q$ and the vector density $f$ have their largest values at zero. This concerns densities for small equal-sized regions; any single exact velocity still has probability zero. ^[ref]

The standard deviation $s$ measures thermal motion even though the mean velocity is zero. Positive and negative components cancel in the mean, but their squares add. This is why thermal Doppler broadening measures a nonzero line width in a gas with no bulk motion.

The Maxwell speed distribution is the probability density function for the magnitude $v=|\mathbf{v}|$. Unlike a signed velocity component, speed cannot be negative:For $0\le a<b$, the probability of a speed in $[a,b]$ is ${\int }_a^{b}g(v)dv$, and ${\int }_0^{\infty }g(v)dv=1$. The density has units of inverse speed, so its height alone is not a probability. ^[ref]

All velocities with magnitudes from $v$ to $v+dv$ occupy a thin spherical shell of volume $4\pi v^{2}dv$ in velocity space. The isotropy of the Maxwell-Boltzmann velocity distribution makes its density constant over each shell, giving $g(v)=4\pi v^{2}f(v{\mathbf{e}}_x)$ for any fixed unit direction ${\mathbf{e}}_x$. A larger radius offers more directions and more volume. The increasing shell factor initially wins over the decreasing Boltzmann factor, so $g(0)=0$ and the most probable speed is positive. This is the geometric reason that the peaks of the vector and speed densities disagree.

With the dimensionless quantity $u=v/s$, every such gas has the same rescaled speed density:For comparison, $w=v_x/s$ has the standard normal distribution density $\varphi (w)=e^{-w^2/2}/\sqrt{2\pi }$ on the whole real line. The Maxwell speed distribution is neither the positive half of this curve nor the distribution of $|v_x|$: it combines three components.

The first two moments of the Maxwell speed distribution give the mean and root mean square:Consequently the mean translational kinetic energy is $m⟨v^2⟩/2=3k_{B}T/2$, as required by the equipartition theorem. ^[ref] The mean speed weights fast particles more than the density peak does, and the root mean square weights them more strongly still.

Changing temperature or particle mass only changes $s\propto \sqrt{T/m}$. At fixed mass, four times the temperature doubles every characteristic speed; at fixed temperature, four times the mass halves them. A stretched density becomes lower so that its area stays one. For two species at the same temperature, equal mean translational kinetic energy therefore means different typical speeds.

The most probable speed is the mode, or density maximum, of the Maxwell speed distribution:For small equal-width speed bins, the bin near $v_{\mathrm{mp}}$ contains the largest expected fraction of particles. This does not mean that any exact speed has positive probability, or that every particle has this speed. The three common summaries obeyso $v_{\mathrm{mp}}<⟨v⟩<v_{\mathrm{rms}}$. The first describes a density peak, the second the mean speed, and the third the root mean square associated with kinetic energy. ^[ref]

For a concrete scale $s=300\mathrm{m}{\mathrm{s}}^{-1}$, these are approximately $424$, $479$, and $520\mathrm{m}{\mathrm{s}}^{-1}$. The most probable velocity vector and each most probable component remain zero in the gas rest frame. A zero vector labels one small region of velocity space, whereas a positive speed labels an entire spherical shell.

A short check uses the derivative of the logarithm of the Maxwell speed distribution for $v>0$:It is positive below $\sqrt{2}s$ and negative above it, locating the unique peak. The factor $v^2$ from the spherical shell supplies the term $2/v$; omitting it would incorrectly move the speed peak to zero.

Doppler broadening is the spread of an observed spectral line caused by the distribution of emitter velocities along the line of sight.

A Maxwellian emitter has Gaussian Doppler frequency FWHM $\Delta {\nu }_D/\nu =\sqrt{8k_{B}T\ln 2/(m{c}^2)}$. Equating this width to the minimum adjacent weak-field sodium component separation $2{\mu }_{B}B/(3h)$ gives the title resolution scale. A Gaussian blend has no unique resolution threshold: disappearance of individual components and disappearance of the whole broadened multiplet are different criteria. Instrumental resolution, pressure broadening, selected polarization and line strengths must be specified for a sharper prediction.

If nonrelativistic emitters of mass $m$ have rest-frame angular frequency ${\omega }_0$, then the one-dimensional Maxwell-Boltzmann velocity distribution produces a Gaussian line with standard deviationMeasuring the line width therefore determines the gas temperature when nonthermal broadening is negligible or separately modeled.

A purely thermal hydrogen absorption line has velocity-space optical depth proportional to $\exp [-(v-v_0{)}^2/b^2]$ with the displayed width. Additional independent Gaussian broadening contributes positive terms to $b^2$. Fitting the lower envelope of width against neutral-hydrogen column density, with numerical calibration for density and pressure smoothing, estimates the temperature-density relation of photoionized intergalactic gas. A single broad line only bounds the temperature from above when its nonthermal width is unknown.

The density of states $g(E)$ is defined so that $g(E)dE$ counts one-particle states with energies in an interval of width $dE$. A sum over closely spaced states then becomes an integral weighted by $g(E)$.

Two spin states per wavevector and parabolic dispersion give $n=2v_d{k}_F^d/(2\pi {)}^d$, where $v_d$ is the unit-ball volume. Thus $E_F={\hslash }^2[n{a}_d{]}^{2/d}/(2m)$ with $a_1=\pi /2$, $a_2=2\pi$, $a_3=3{\pi }^2$. Differentiating the cumulative count yields $g(E_F)=dn/(2E_F)$, normalized per length, area or volume as appropriate.

For the continuum dispersion ${\epsilon }_{\mathbf{k}}={\hslash }^2(k_x^2+k_y^2+k_z^2/\gamma )/(2m)$ with $\gamma >0$, substituting $q_z=k_z/\sqrt{\gamma }$ makes the energy isotropic and gives Jacobian $\sqrt{\gamma }$. Every momentum-state counting integral therefore acquires that factor. Per internal state and unit volume, $N=\sqrt{\gamma }(2m/{\hslash }^2{)}^{3/2}\sqrt{\epsilon }/(4{\pi }^2)$. The result assumes an unrestricted three-dimensional continuum; increasing this mass anisotropy is not the same operation as imposing a two-dimensional Brillouin zone and a finite interlayer band width.

For an area $A$ with periodic in-plane boundary conditions, one polarization contributes $A2\pi kdk/(2\pi {)}^2$ normal modes of the electromagnetic field. For linear dispersion relation $\omega =ck$ and effective physical volume $Ad$, this gives the density of states in the title. No extra factor two for polarization is allowed. Opposite in-plane directions are already counted in the full circle of wavevectors.

For a spinless nonrelativistic particle in a central potential, $g(ϵ)=(2m{)}^{3/2}(4{\pi }^2{\hslash }^3{)}^{-1}{\int }_{U<ϵ}\sqrt{ϵ-U}{d}^{3}r$. Spherical symmetry reduces this to a radial integral over every allowed shell. It is a leading phase-space approximation.

Periodic boundaries on a cube give one spinless state per $(2\pi /L{)}^3$ in wave-vector space. Counting the full spherical shell gives $d\mathcal{N}/V=k^{2}dk/(2{\pi }^2)$. Hard-wall counting in the positive octant gives the same leading volume term.

The number of lattice-vibration modes per unit angular frequency. For a three-dimensional isotropic monatomic crystal with three linear acoustic branches of a common speed $v_s$, wavevector-shell counting gives $g(\omega )=3V{\omega }^2/(2{\pi }^2{v}_s^3)$ below a suitable cutoff. Different branch speeds replace $3v_s^{-3}$ by $v_L^{-3}+2v_T^{-3}$.

For the nearest-neighbour monatomic-chain dispersion relation, ${\omega }_m=2\sqrt{\alpha /M}$ and $d\omega /dq=(a/2)\sqrt{{\omega }_m^2-{\omega }^2}$ on the positive half-zone. The folded wave-number count gives the displayed density of states, including both propagation directions. Its upper-edge Van Hove singularity is integrable: ${\int }_0^{{\omega }_m}g(\omega )d\omega =N$. A finite chain has discrete modes; the continuous density describes its large-size limit.

A single acoustic branch $\omega =v|k|$ has thermal energy per length $U_T/\ell =\int dk\hslash v|k|/[2\pi (e^{\beta \hslash v|k|}-1)]=\pi (k_{B}T{)}^2/(6\hslash v)$. Differentiating gives the displayed linear heat capacity. The regime requires temperatures below the microscopic phonon cutoff but above the finite-size level spacing, with the translational zero mode treated separately.

A nonanalyticity of the density of states associated with a stationary point of a band dispersion relation. Its form depends on dimension and on whether the stationary point is an extremum or a saddle.

A single two-dimensional quadratic saddle with $A={\hslash }^2/(2m^{\ast })$ contributes this leading joint density of states when both contour branches are included and $g_s$ counts independent spin channels. The contour integral is proportional to ${\int }_{-\Lambda }^{\Lambda }dy/\sqrt{y^2+\delta /A}=2\mathrm{arsinh}(\Lambda \sqrt{A/\delta })$ above the saddle. Its asymptote gives the logarithm with $E_{\Lambda }$ of order $A{\Lambda }^2$. The cutoff-dependent constant is not universal. This is a density per area per energy; changing to a density per frequency requires a Jacobian. ^[ref]

A spin factor and disconnected contour branches count different independent contributions. For the transition energy $E_s+{\hslash }^2(k_x^2-k_y^2)/(2m^{\ast })$, one branch contributes $g_s{m}^{\ast }I/(4{\pi }^2{\hslash }^2)$, with $I=\int d{k}_y/\sqrt{k_y^2+\delta /A}$. The full two-branch saddle doubles this. A formula for one branch with spin degeneracy two can therefore equal a full-contour formula with spin degeneracy one, but those are distinct counting conventions. An extra valley or inequivalent saddle requires its own multiplicity.

For a one-dimensional subband $E=E_0+\alpha k^2$ with $\alpha >0$, the density of states varies as $(E-E_0{)}^{-1/2}$ just above its edge. Optical peaks also depend on the joint density of states, selection rules and excitons.

The density of available one-particle states multiplied by their mean occupation. For equilibrium electrons, $Z(E)=D(E)f(E)$ with the Fermi-Dirac distribution $f$. The integral of $Z$ counts electrons; the integral of $D$ counts available states.

A many-to-one change of energy variable requires summing every inverse branch. For $E(\xi )=\sqrt{{\xi }^2+{\Delta }^2}$ and constant normal density of states $g_F$ on both sides of zero, there are two roots for $E>\Delta$, giving $D_{+}(E)=2g_{F}E/\sqrt{E^2-{\Delta }^2}$. A single branch instead gives $g_{F}E/\sqrt{E^2-{\Delta }^2}$. The distinction persists when $\Delta \to 0$: folding $\xi$ into $|\xi |$ doubles a two-sided constant density on the positive axis. Spin degeneracy is a separate normalization choice; it must be applied consistently to both normal and transformed densities. The BCS electronic density of states has the single-factor expression because it is spectrally weighted rather than an unweighted branch count.

For filled valence and empty conduction energy bands, the joint density of states counts pairs of states at a prescribed interband energy difference. In a direct optical transition the photon momentum is usually negligible compared with a Brillouin zone dimension, so initial and final states have nearly the same crystal momentum. The absorption spectrum weights this count by transition matrix elements; its onset and structure can constrain the band gap and electronic band structure. Indirect transitions additionally involve phonons to supply crystal momentum.

For isotropic direct-gap parabolic bands, $m_r^{-1}=m_e^{\ast -1}+m_h^{\ast -1}$ and transition energy is $E_g+{\hslash }^2{k}^2/(2m_r)$. Differentiating the two-spin state count $k^3/(3{\pi }^2)$ per volume gives the displayed density for $E>E_g$, and zero below. Density per angular frequency is $\hslash g_j(\hslash \omega )$, not the same dimensioned quantity.

For a local transition-energy model $E_c-E_v=E_s+A(k_x^2-k_y^2)$ with $A>0$, positive detuning has two hyperbola branches opening along x. Negative detuning opens along y; the zero-detuning limiting contour is a pair of intersecting lines. The coarea integrand on one positive-detuning branch reduces to $d{k}_y/[2A\sqrt{k_y^2+\delta /A}]$. The gradient vanishes at the saddle, so this is a limiting description as nonzero detuning approaches zero.

For positive detuning inside a strict square region $|k_x|,|k_y|\le \Lambda$, substituting $k_x^2=k_y^2+\delta /A$ bounds $k_y$ by the displayed value, provided $0<\delta /A<{\Lambda }^2$. Integrating to $\Lambda$ instead extends the contour slightly outside that square. Either prescription has the same leading logarithmic singularity near zero detuning; it does not give the same exact finite-detuning result.

Parabolic electron-hole continua with $d$ unconfined directions have cumulative state count proportional to $(E-E_{\mathrm{th}}{)}^{d/2}$. Differentiating gives the threshold joint density proportional to $(E-E_{\mathrm{th}}{)}^{d/2-1}$: a square root for bulk, a step for a well, and an inverse square root for a wire. A fully confined dot instead has discrete lines. Optical matrix elements, occupations, excitons and broadening determine the actual absorption beyond this counting argument.

For a smooth two-dimensional transition energy $D(\mathbf{k})$ and a regular level $D=E$, the coarea identity gives $\int d^{2}k\delta (D(\mathbf{k})-E)={\int }_{D=E}dl/|\nabla D|$. Critical points require a limiting or distributional treatment; the apparent small contour at a parabolic edge is compensated by its small energy gradient.

For two isotropic parabolic subbands, the joint density of states per area and per independent channel is $\mu /(2\pi {\hslash }^2)\Theta (\hslash \omega -E_{\mathrm{edge}})$. The constant is a two-dimensional phase-space effect. Degeneracy and optical matrix elements determine the actual step weights.

For a length $L$ and parabolic free-electron dispersion relation $E={\hslash }^2{k}^2/(2m)$, the displayed total density of states includes both signs of $k$ and both spin states. Per spin it is half as large. Its $E^{-1/2}$ dependence follows by transforming the constant signed-$k$ density $L/(2\pi )$ to energy and including both momenta. The continuum singularity at the band bottom is integrable; a finite specimen still has discrete levels.

The local density of states resolves the density of states by position as well as energy. With a consistent counting of spin states, its integral over the specimen's volume gives the total density of states, $g(E)=\int \rho (\mathbf{r},E)d^3\mathbf{r}$. It describes the spatial distribution of available states, rather than their occupation.

For a spin-degenerate band, the total density of states includes both spin species and is twice the single-spin density of states. Per lattice site, $\nu (ϵ)=2N^{-1}{\sum }_k\delta (ϵ-{ϵ}_k)$. Using a separate explicit spin sum then requires a compensating factor one half when converting momentum sums. The normalization matters in the Stoner criterion.

The single-spin density of states counts momentum states for one spin species. For degenerate up and down species, the spin-summed density of states is twice this value. Specify separately whether the density is per unit volume or extensive. In the BCS gap equation, the momentum sum labels opposite-spin pairs once and therefore uses the single-spin density per volume when the interaction has normalization $V/\mathcal{V}$.

The Fermi-Dirac distribution follows from Fermi-Dirac statistics and is $n(E)=[e^{\beta (E-\mu )}+1{]}^{-1}$.

The mean occupation of a fermionic one-particle state of energy $E$ isAt zero temperature this becomes the step function $1_{E<E_F}$.

The Fermi window measures sensitivity of occupation to energy near the chemical potential. It is positive, even about $\mu$, has width of order $k_{B}T$, and integrates to one over the full energy line. Its centred variance is ${\pi }^2(k_{B}T{)}^2/3$. A band restricted to nonnegative energy contains weight $n_F(0,\mu )$ instead of exactly one; extending to the full line is exponentially accurate when $\mu \gg k_{B}T$. At zero temperature the window tends to a Dirac delta function at the chemical potential.

A density of states $g(ϵ)=g_0[1+\alpha \cos (2\pi ϵ/\Delta )]$ has zero-temperature oscillatory grand potential $g_0\alpha {\Delta }^2\cos (2\pi \mu /\Delta )/(4{\pi }^2)$. Its Fermi-window convolution multiplies that oscillation by $R_T$ when the lower band edge is far below the chemical potential. The Fourier transform of the even window removes its sine component and gives the title factor. At high $X$, $R_T\sim 2X{e}^{-X}$; at small $X$, $R_T=1-X^2/6+O(X^4)$. The corresponding thermal change relative to zero temperature has amplitude proportional to $1-R_T$, rather than $R_T$.

For the sinusoidal density-of-states model, divide the oscillatory amplitude of $\Phi (T,\mu )-\Phi (0,\mu )$ by its smooth magnitude $g_0{\pi }^2(k_{B}T{)}^2/6$. This gives the title ratio. Its limit as $T\to 0$ is $|\alpha |$, although both changes vanish at exactly zero temperature. The expansion is $|\alpha |[1-7X^2/60+O(X^4)]$, and its large-$X$ envelope is $6|\alpha |/X^2$. A fixed chemical potential adds the actual cosine phase, which can make the oscillatory contribution zero even when the envelope is nonzero. Exponential suppression of the full oscillatory potential therefore must not be confused with this algebraic thermal-change ratio.

When $E-\mu \gg kT$, the unit term in the denominator of the Fermi-Dirac distribution is negligible. The resulting exponentially small occupation is the Boltzmann approximation. It justifies the usual nondegenerate semiconductor carrier formulas; it fails near strongly occupied band states.

The Sommerfeld expansion approximates integrals weighted by the Fermi-Dirac distribution when $k_{B}T\ll \mu$ and $f$ is smooth near $\mu$. Applying it to the number integral at fixed particle number gives $\mu -E_F=-({\pi }^2/6)(k_{B}T{)}^2{g}^{\prime }(E_F)/g(E_F)+\cdots$. The sign depends on the slope of the density of states. For an unbounded parabolic band, $g(E)\propto E^{d/2-1}$, yielding opposite quadratic shifts in one and three dimensions. In two dimensions the leading algebraic shift vanishes, but the finite band-bottom cutoff still gives an exponentially small negative shift.

For a three-dimensional parabolic band and fixed positive $\mu -E_c$, the Sommerfeld expansion gives $n(T)=n(0)[1+{\pi }^2(k_{B}T/(\mu -E_c){)}^2/8+O(T^4)]$. This increase differs from the usual fixed-density calculation, in which the chemical potential decreases with temperature. The exact density is proportional to $T^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$.

The temperature derivative of the Fermi occupation has opposite signs below and above $\mu$. Pairing energies $\mu \pm x$ leaves a positive contribution because the density of states increases with energy. The unpaired higher-energy tail is also positive. Hence the density increases at fixed $\mu$ for an unbounded three-dimensional parabolic band.

Thermal electron-positron pairs in chemical equilibrium with photons have opposite chemical potentials when the one-particle energy includes rest energy. For negligible degeneracy and $k_{B}T\ll m_e{c}^2$, the zero-chemical-potential density per charge species is $n_0=2[m_e{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}{e}^{-m_e{c}^2/(k_{B}T)}$. In that nondegenerate limit, charge neutrality and the classical equilibrium product $n_{-}{n}_{+}=n_0^2$ give $n_{+}=(\sqrt{n_b^2+4n_0^2}-n_b)/2$, with net electron density $n_b$. Pairs are important when $n_0\gtrsim n_b$. At arbitrary temperature the zero-chemical-potential marker uses the integral $n_0=({\pi }^2{\hslash }^3{)}^{-1}{\int }_0^{\infty }{p}^2[\exp (\sqrt{m_e^2{c}^4+p^2{c}^2}/k_{B}T)+1{]}^{-1}dp$. Its ultrarelativistic limit is $3\zeta (3)(k_{B}T/\hslash c{)}^3/(2{\pi }^2)$. A nonzero net chemical potential shifts the exact abundance boundary; the zero-potential curve is an approximate crossover marker.

For $s>0$,This follows from $(e^x+1{)}^{-1}=(e^x-1{)}^{-1}-2(e^{2x}-1{)}^{-1}$. In three spatial dimensions it gives the fermion-to-boson ratios $3/4$ for ultrarelativistic particle number and $7/8$ for energy density.

A Fermi gas consists of noninteracting fermions whose one-particle states are occupied according to the Fermi-Dirac distribution.