Bose-Einstein distribution

Intuition [Beginner]

Particles come in two quantum-mechanical families. Fermions (electrons, protons, neutrons) obey the Pauli exclusion principle: no two can share the same quantum state. Bosons (photons, helium-4 atoms, gluons) have no such restriction. Any number of bosons can pile into the same state at the same time.

The Bose-Einstein distribution tells you the average number of bosons occupying a single-particle state at energy $E$, when the system sits at temperature $T$. The formula is

$$n(E)=\frac{1}{e^{E/k_{B}T}-1},$$

where $k_B$ is Boltzmann's constant. At high temperature ($k_{B}T$ much larger than the level spacing) this reduces to the classical Maxwell-Boltzmann result — quantum indistinguishability stops mattering. At low temperature the denominator in $e^{E/k_{B}T}-1$ is small, and the occupation number $n(E)$ can become very large.

The most important boson in physics is the photon. Applying the Bose-Einstein distribution to photons in a hot cavity gives the Planck blackbody spectrum — the curve that describes the glow of a hot iron, the colour of a star, and the static on an old television. The cosmic microwave background is a photon gas at $T=2.725\text{K}$, and its spectrum matches the Planck curve to extraordinary precision — the afterglow of the Big Bang, described by one line of statistics.

At low enough temperature a gas of massive bosons does something dramatic: a macroscopic fraction of particles collapses into the single-particle ground state. This is Bose-Einstein condensation (BEC), first produced in a dilute gas of rubidium atoms in 1995.

Visual [Beginner]

Three curves on the same axes, each showing the Bose-Einstein occupation number $n(E)$ versus energy $E$ at a different temperature. At low $T$ the curve rises steeply near $E=0$ — the ground state is heavily occupied. At intermediate $T$ the rise is gentler. At high $T$ the three curves converge toward the classical falling exponential.

A second panel shows the Planck spectrum: energy radiated per unit frequency for a blackbody at three temperatures. The peak shifts to higher frequency as $T$ rises (Wien's displacement law), and the total area under the curve grows as $T^4$ (the Stefan-Boltzmann law). Both features follow from the BE distribution applied to photon modes in a cavity.

Worked example [Beginner]

The cosmic microwave background (CMB) is a photon gas filling the universe at temperature $T=2.725\text{K}$. Find the peak frequency of its spectrum.

Step 1. Wien's displacement law gives the peak wavelength of a blackbody: ${\lambda }_{\max }=b/T$, where $b=2.898\times 10^{-3}\text{m}\cdot \text{K}$.

Step 2. Plug in $T=2.725\text{K}$:

$${\lambda }_{\max }=\frac{2.898\times 10^{-3}}{2.725}\approx 1.063\times 10^{-3}\text{m}\approx 1.06\text{mm}.$$

Step 3. Convert to frequency: $f_{\max }=c/{\lambda }_{\max }\approx 3\times 10^8/1.063\times 10^{-3}\approx 282\text{GHz}$.

The CMB peaks in the microwave band at about 282 GHz — which is why it was originally detected as unexplained noise in a radio antenna by Penzias and Wilson in 1965. The photon occupation number at this frequency is roughly $n\approx 1/(e^{hf/k_{B}T}-1)\approx 1$, meaning each electromagnetic mode in the cavity holds about one photon on average: a cool, dilute photon gas.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Consider a system of non-interacting identical bosons in the grand canonical ensemble at temperature $T$ and chemical potential $\mu$. Each single-particle state $k$ has energy ${\epsilon }_k$. The grand partition function for state $k$ is a sum over all possible occupation numbers $n_k=0,1,2,\dots$ (bosons have no exclusion):

$${\mathcal{Z}}_k=\sum _{n_k=0}^{\infty }{e}^{-n_k\beta ({\epsilon }_k-\mu )}=\frac{1}{1-e^{-\beta ({\epsilon }_k-\mu )}},$$

provided $\mu <{\epsilon }_k$ (otherwise the geometric series diverges). The mean occupation number of state $k$ is

$$⟨n_k⟩=-\frac{1}{\beta }\frac{\partial \ln {\mathcal{Z}}_k}{\partial {\epsilon }_k}=\frac{1}{e^{\beta ({\epsilon }_k-\mu )}-1}.$$

This is the Bose-Einstein distribution:

$$\boxed{⟨n(\epsilon )⟩=\frac{1}{e^{(\epsilon -\mu )/k_{B}T}-1}}$$

The chemical potential satisfies $\mu \le {\epsilon }_0$ where ${\epsilon }_0$ is the lowest single-particle energy; for a system with fixed total particle number $N$, $\mu$ is determined implicitly by ${\sum }_k⟨n_k⟩=N$.

For photons the particle number is not conserved (photons are created and destroyed by the walls of the cavity), so $\mu =0$. The distribution reduces to

$$⟨n(\epsilon )⟩=\frac{1}{e^{\epsilon /k_{B}T}-1},(\text{photons: }\mu =0).$$

For a gas of massive bosons with fixed $N$, $\mu$ increases toward the ground-state energy as $T$ drops. When $\mu$ reaches ${\epsilon }_0$, the sum ${\sum }_{k\ne 0}⟨n_k⟩$ saturates at a value less than $N$, and the remaining particles macroscopically occupy the ground state — this is Bose-Einstein condensation.

The photon gas and the Planck spectrum

A photon in a box of volume $V$ has wave vector $\mathbf{k}$ with $|\mathbf{k}|=2\pi /\lambda$ and energy $\epsilon =\hslash \omega =\hslash c|\mathbf{k}|$. The density of photon modes in $k$-space is $V/(2\pi {)}^3$ per polarisation, with two polarisation states per mode. Converting to a frequency density of states:

$$g(\omega )=\frac{V{\omega }^2}{{\pi }^2{c}^3}.$$

The mean energy in the frequency interval $[\omega ,\omega +d\omega ]$ is $g(\omega )⟨n(\omega )⟩\hslash \omega d\omega$, giving the Planck spectral energy density:

$$u(\omega ,T)=\frac{\hslash }{{\pi }^2{c}^3}\frac{{\omega }^3}{e^{\beta \hslash \omega }-1}.$$

This is the Planck blackbody radiation law, derived from the Bose-Einstein distribution. The total energy density is

$$U/V=\frac{{\pi }^2(k_{B}T{)}^4}{15{\hslash }^3{c}^3}=\frac{4\sigma }{c}{T}^4,$$

where $\sigma ={\pi }^2{k}_B^4/(60{\hslash }^3{c}^2)$ is the Stefan-Boltzmann constant — the $T^4$ Stefan-Boltzmann law is a direct consequence.

Debye model of solids

The specific heat of a solid at low temperature is dominated by phonons (quantised lattice vibrations), which are bosons. The Debye model treats the solid as an isotropic elastic continuum with a cutoff frequency ${\omega }_D$ fixed by the number of atoms. The phonon density of states is $g(\omega )=9N{\omega }^2/{\omega }_D^3$ for $\omega \le {\omega }_D$. Applying the BE distribution with $\mu =0$ (phonon number is not conserved) gives the Debye heat capacity

$$C_V=9N{k}_B{\left(\frac{T}{{\Theta }_D}\right)}^3{\int }_0^{{\Theta }_D/T}\frac{x^4{e}^x}{(e^x-1{)}^2}dx,$$

where ${\Theta }_D=\hslash {\omega }_D/k_B$ is the Debye temperature. At low $T$ ($T\ll {\Theta }_D$): $C_V\propto T^3$ — the Debye $T^3$ law, confirmed experimentally for all insulating solids. At high $T$: $C_V\to 3N{k}_B$, the Dulong-Petit value.

Counterexamples to common slips

• The chemical potential $\mu$ in the BE distribution is not the same as the photon energy $\hslash \omega$. For photons, $\mu =0$ because the photon number is not a conserved quantity — photons are freely emitted and absorbed.
• The BE distribution diverges when $\epsilon =\mu$. This signals the onset of Bose-Einstein condensation: the geometric-series derivation breaks down because a macroscopic fraction of particles occupies the ground state and must be treated separately.
• The classical limit $n(\epsilon )\ll 1$ (i.e., $e^{\beta (\epsilon -\mu )}\gg 1$) recovers $n(\epsilon )\approx e^{-\beta (\epsilon -\mu )}$, the Maxwell-Boltzmann distribution. Quantum statistics matter only when occupation numbers are comparable to or exceed unity.
• The BE distribution describes non-interacting bosons. Real systems (liquid helium, ultracold atomic gases) have interactions that modify the condensation temperature and the condensate fraction. The interacting theory (Bogoliubov, Gross-Pitaevskii) is treated in the Master tier.

Key theorem with proof [Intermediate+]

Theorem (Planck spectrum from Bose-Einstein statistics). The spectral energy density of electromagnetic radiation in thermal equilibrium at temperature $T$ in a cavity of volume $V$ is

$$u(\omega ,T)=\frac{\hslash }{{\pi }^2{c}^3}\frac{{\omega }^3}{e^{\beta \hslash \omega }-1},$$

and the total energy density is $U/V=a{T}^4$ where $a={\pi }^2{k}_B^4/(15{\hslash }^3{c}^3)$.

Proof. A photon in a cubic cavity of side $L$ has wave vector components $k_i=2\pi n_i/L$ with $n_i\in \mathbb{Z}$. Each mode has energy $\epsilon =\hslash \omega =\hslash c|\mathbf{k}|$ and two polarisation states. Photons are bosons with zero chemical potential ($\mu =0$) because they are not conserved — the cavity walls absorb and re-emit them freely.

The number of modes with $|\mathbf{k}|$ between $k$ and $k+dk$ is, by the volume of a spherical shell in $\mathbf{k}$-space divided by the volume per mode $(2\pi /L{)}^3$, times two polarisations:

$$dN=2\cdot \frac{4\pi k^{2}dk}{(2\pi /L{)}^3}=\frac{V{k}^2}{{\pi }^2}dk.$$

Substituting $k=\omega /c$ and $dk=d\omega /c$ gives the mode density in frequency:

$$g(\omega )d\omega =\frac{V{\omega }^2}{{\pi }^2{c}^3}d\omega .$$

The mean energy per mode at frequency $\omega$ is the BE occupation number times the photon energy:

$$⟨E(\omega )⟩=\frac{\hslash \omega }{e^{\beta \hslash \omega }-1}.$$

The spectral energy density (energy per unit volume per unit angular frequency) is

$$u(\omega ,T)=\frac{1}{V}g(\omega )⟨E(\omega )⟩=\frac{{\omega }^2}{{\pi }^2{c}^3}\cdot \frac{\hslash \omega }{e^{\beta \hslash \omega }-1}=\frac{\hslash }{{\pi }^2{c}^3}\frac{{\omega }^3}{e^{\beta \hslash \omega }-1}.$$

For the total energy density, substitute $x=\beta \hslash \omega$:

$$\frac{U}{V}={\int }_0^{\infty }u(\omega ,T)d\omega =\frac{(k_{B}T{)}^4}{{\pi }^2{\hslash }^3{c}^3}{\int }_0^{\infty }\frac{x^3}{e^x-1}dx.$$

The integral ${\int }_0^{\infty }{x}^3/(e^x-1)dx=\Gamma (4)\zeta (4)=6\cdot {\pi }^4/90={\pi }^4/15$. Therefore

$$\frac{U}{V}=\frac{{\pi }^2(k_{B}T{)}^4}{15{\hslash }^3{c}^3}=\frac{4\sigma }{c}{T}^4.\text{\hspace{1em}□}$$

The ${\pi }^4/15$ factor is a pure number arising from the Riemann zeta function at 4 — a fingerprint of the BE statistics. The classical Rayleigh-Jeans result ($u\propto {\omega }^2{k}_{B}T$) is recovered from the low-frequency limit $e^{\beta \hslash \omega }\approx 1+\beta \hslash \omega$; the ultraviolet catastrophe is averted because the high-frequency modes are exponentially suppressed by the BE denominator.

Exercises [Intermediate+]

Bogoliubov theory and the Gross-Pitaevskii equation [Master]

The ideal Bose gas is a pedagogical starting point, but real condensates are interacting. The two foundational theories of interacting bosons at low temperature are due to Bogoliubov (1947) and Gross / Pitaevskii (1961).

Bogoliubov theory. Consider $N$ bosons in a box with a weak repulsive contact interaction $V(\mathbf{x}-\mathbf{y})=g\delta (\mathbf{x}-\mathbf{y})$, $g>0$. Below $T_c$, the ground state is macroscopically occupied. Bogoliubov's insight: replace the creation/annihilation operators for the zero-momentum mode by c-numbers, $a_0\approx a_0^{†}\approx \sqrt{N_0}$, and diagonalise the resulting quadratic Hamiltonian via a canonical (Bogoliubov) transformation

$${\alpha }_k=u_k{a}_k+v_k{a}_{-k}^{†},$$

with coefficients $u_k^2-v_k^2=1$ fixed by the diagonalisation condition. The resulting excitation spectrum is

$$E_k=\sqrt{{\epsilon }_k({\epsilon }_k+2g{n}_0)},{\epsilon }_k=\frac{{\hslash }^2{k}^2}{2m},$$

where $n_0=N_0/V$ is the condensate density. For small $k$ this gives $E_k\approx \hslash c_{s}k$ — a phonon spectrum linear in $k$, with sound speed $c_s=\sqrt{g{n}_0/m}$. The linear dispersion at low $k$ is the hallmark of a superfluid: the critical velocity for dissipation is $v_c=c_s$, and below this velocity the fluid flows without resistance. The crossover to free-particle behaviour at large $k$ ($E_k\to {\hslash }^2{k}^2/2m$) is the Bogoliubov spectrum observed in neutron-scattering experiments on superfluid helium.

The Gross-Pitaevskii equation (GPE) is the mean-field equation for the condensate wave function $\psi (\mathbf{x},t)$:

$$i\hslash \frac{\partial \psi }{\partial t}=\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{\mathrm{ext}}(\mathbf{x})+g|\psi {|}^2\right]\psi ,$$

where $V_{\mathrm{ext}}$ is an external potential (the magnetic trap in BEC experiments) and $g|\psi {|}^2$ is the mean-field interaction energy. The GPE is a nonlinear Schrodinger equation; the nonlinearity $g|\psi {|}^2$ encodes the two-body contact interaction at mean-field level. Stationary solutions $\psi (\mathbf{x},t)=\varphi (\mathbf{x})e^{-i\mu t/\hslash }$ satisfy

$$\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{\mathrm{ext}}(\mathbf{x})+g|\varphi {|}^2\right]\varphi =\mu \varphi ,$$

which determines the condensate profile $\varphi (\mathbf{x})$ and the chemical potential $\mu$ self-consistently. The Thomas-Fermi approximation ($g|\varphi {|}^2\gg$ kinetic energy) gives $|\varphi {|}^2\approx (\mu -V_{\mathrm{ext}})/g$ inside the trap — an inverted parabola — and reproduces the observed condensate density profiles.

Superfluidity and the lambda transition [Master]

The Bogoliubov spectrum $E_k=\sqrt{{\epsilon }_k({\epsilon }_k+2g{n}_0)}$ has the property $E_k/k\to c_s>0$ as $k\to 0$. Landau's argument (1941) shows that this linear dispersion implies a finite critical velocity: a body moving through the fluid at speed $v<c_s$ cannot create excitations while conserving energy and momentum, so the fluid flows without dissipation. This is superfluidity.

Liquid helium-4. Below the lambda temperature $T_{\lambda }=2.17\text{K}$, liquid ${}^4\text{He}$ undergoes a phase transition to a superfluid state (He II). The transition is so named because the specific heat diverges logarithmically at $T_{\lambda }$, producing a shape resembling the Greek letter $\lambda$. The superfluid fraction rises from zero at $T_{\lambda }$ to nearly 100% at $T\to 0$. The order parameter is the macroscopic wave function of the condensate, and the transition is in the 3D XY universality class — distinct from the ideal-gas BEC transition (mean-field with logarithmic corrections in 3D).

The ideal-gas BEC transition and the He-II lambda transition differ fundamentally: the ideal-gas condensation is driven purely by quantum statistics (no interactions), while the lambda transition involves strong interactions that renormalise the transition temperature, the condensate fraction (only about 10% of ${}^4\text{He}$ atoms are in the condensate even at $T=0$, despite 100% superfluid fraction), and the critical exponents. The Bogoliubov and Gross-Pitaevskii theories bridge these extremes by treating interactions perturbatively.

The ideal Bose gas in the thermodynamic limit [Master]

The thermodynamic limit of the ideal Bose gas exhibits a genuine phase transition at $T_c$, despite the non-interacting Hamiltonian. The key mathematical objects are the polylogarithms ${\mathrm{Li}}_s(z)={\sum }_{l=1}^{\infty }{z}^l/l^s$. Above $T_c$, the fugacity $z=e^{\beta \mu }$ is determined by $N=(V/{\lambda }_T^3){\mathrm{Li}}_{3/2}(z)$ and the pressure is

$$P=\frac{k_{B}T}{{\lambda }_T^3}{\mathrm{Li}}_{5/2}(z).$$

Below $T_c$, $z=1$ ($\mu =0$), and the pressure saturates at $P=(k_{B}T/{\lambda }_T^3)\zeta (5/2)$, independent of $N$ — the isotherm is flat. The condensate acts as a reservoir that absorbs or releases particles at constant $\mu =0$ and constant $P$.

The equation of state $P(V,T,N)$ has a kink at $T_c$: the isothermal compressibility ${\kappa }_T=-V^{-1}(\partial V/\partial P{)}_T$ is finite above $T_c$ but diverges below $T_c$ because $\partial P/\partial V=0$ in the mixed phase (condensate + excited states). This is the mechanical signature of the phase transition. The specific heat $C_V$ is continuous at $T_c$ but has a discontinuous derivative — a third-order transition in the Ehrenfest classification.

The BEC transition is possible in 3D but not in 1D or 2D for the non-interacting gas. The integral $\int {\epsilon }^{d/2-1}/(e^{\beta \epsilon }-1)d\epsilon$ diverges at the lower limit for $d\le 2$, so ${\mathrm{Li}}_{d/2}(1)$ is finite only for $d>2$ — the Mermin-Wagner theorem applied to the BEC. Weakly-interacting 2D Bose gases can undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition to a superfluid state without long-range order, driven by vortex-antivortex unbinding rather than condensation.

Connections to other areas [Master]

Quantum field theory. The Bose-Einstein distribution is the finite-temperature occupation number for bosonic fields. In quantum field theory at temperature $T$, the Feynman propagator is replaced by the thermal propagator, and loop diagrams acquire factors of $⟨n(\omega )⟩$ from thermal occupation of intermediate states. The Matsubara formalism replaces continuous imaginary time by a discrete set of frequencies ${\omega }_n=2\pi n/\beta$ (for bosons) — the residue of the BE statistics in the Euclidean path integral.

Condensed matter. Phonons (quantised lattice vibrations), magnons (spin waves), and excitons are all bosonic quasiparticles governed by BE statistics. The Debye $T^3$ law, the Bloch $T^3$ law for magnon specific heat, and the Planck radiation law are all instances of the same $u\propto T^{d+1}$ scaling for massless bosons in $d$ spatial dimensions.

Cosmology. The cosmic microwave background (CMB) is a photon gas at $T=2.725\text{K}$, and the cosmic neutrino background is a Fermi-Dirac gas at $T\approx 1.95\text{K}$. The photon-to-baryon ratio $\eta \approx 6\times 10^{-10}$ sets the baryon density of the universe and is determined from the relative heights of the acoustic peaks in the CMB power spectrum — a direct application of the photon-gas equation of state to observational cosmology.

Quantum information. Bose-Einstein condensation is a macroscopic quantum coherence phenomenon: the condensate wave function $\psi$ has a well-defined phase, and the number-phase uncertainty relation $\Delta N\Delta \varphi \ge 1/2$ constrains the precision of atom interferometry and atomic clocks. Squeezed states of BECs are being explored for quantum-enhanced sensing.

Historical and philosophical context [Master]

Bose 1924. Satyendra Nath Bose, then an unknown lecturer at Dacca University, derived Planck's radiation law by counting photon states in a new way — treating photons as indistinguishable particles whose microstates are characterised by occupation numbers rather than particle labels. He sent his paper to Einstein, who recognised its significance, translated it into German, and had it published in Zeitschrift fur Physik with a translator's note.

Einstein 1924-25. Einstein extended Bose's counting method to a gas of massive particles and discovered that the resulting distribution function predicted a phase transition: below a critical temperature, a macroscopic fraction of particles would accumulate in the ground state. This was the first prediction of Bose-Einstein condensation — a purely theoretical prediction with no experimental realisation for 70 years.

London 1938. Fritz London proposed that the superfluid transition in liquid helium-4 (discovered by Kapitza and by Allen and Misener in 1937) was a Bose-Einstein condensation of ${}^4\text{He}$ atoms. The transition temperature predicted by the ideal-gas formula ($T_c\approx 3.1\text{K}$ for liquid-helium density) is close to the observed $T_{\lambda }=2.17\text{K}$ — the discrepancy reflects the strong interactions in liquid helium, which the ideal-gas theory neglects.

Bogoliubov 1947. Nikolai Bogoliubov developed the microscopic theory of the weakly-interacting Bose gas, showing that interactions change the excitation spectrum from quadratic (${\epsilon }_k={\hslash }^2{k}^2/2m$) to linear ($E_k\approx \hslash c_{s}k$) at long wavelengths — the phonon spectrum of a superfluid.

Cornell, Wieman, and Ketterle 1995. Eric Cornell and Carl Wieman produced the first Bose-Einstein condensate in a dilute vapour of rubidium-87 atoms at $T\approx 170\text{nK}$, using laser cooling followed by evaporative cooling in a magnetic trap. Wolfgang Ketterle independently produced a sodium-23 condensate with a much larger atom number, enabling the observation of interference between two condensates and the demonstration of an atom laser. The 2001 Nobel Prize was awarded to Cornell, Wieman, and Ketterle for this achievement.

Bibliography [Master]

• Bose, S. N. "Plancks Gesetz und Lichtquantenhypothese." Zeitschrift fur Physik 26, 178-181 (1924). Original derivation of Planck's law from indistinguishable-particle counting.

• Einstein, A. "Quantentheorie des einatomigen idealen Gases." Sitzungsberichte der Preussischen Akademie der Wissenschaften 1924, 261-267; 1925, 3-14. Extension to massive particles and prediction of BEC.

• London, F. "The $\lambda$-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy." Nature 141, 643-644 (1938). Identification of superfluidity with BEC.

• Bogoliubov, N. N. "On the theory of superfluidity." Journal of Physics (Moscow) 11, 23-32 (1947). Bogoliubov transformation and the weakly-interacting Bose gas.

• Gross, E. P. "Structure of a quantized vortex in boson systems." Nuovo Cimento 20, 454-477 (1961). Pitaevskii, L. P. "Vortex lines in an imperfect Bose gas." Soviet Physics JETP 13, 451-454 (1961). The Gross-Pitaevskii equation.

• Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Cornell, E. A. "Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor." Science 269, 198-201 (1995). First BEC in dilute gas.

• Davis, K. B., Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. "Bose-Einstein Condensation in a Gas of Sodium Atoms." Physical Review Letters 75, 3969-3973 (1995). Independent BEC observation.

• Pethick, C. J. and Smith, H. Bose-Einstein Condensation in Dilute Gases, 2nd ed. (Cambridge University Press, 2008). Standard textbook on BEC physics.

• Pitaevskii, L. and Stringari, S. Bose-Einstein Condensation and Superfluidity (Oxford University Press, 2016). Theoretical treatment of BEC, superfluidity, and the GPE.

• Landau, L. D. and Lifshitz, E. M. Statistical Physics, Part 1, 3rd ed. (Pergamon, 1980), §54-55, §71. The Bose distribution, blackbody radiation, and the ideal Bose gas.

• Huang, K. Statistical Mechanics, 2nd ed. (Wiley, 1987), Ch. 12. Detailed treatment of BEC, the interacting Bose gas, and superfluidity.

• Tong, D. "Statistical Physics." University of Cambridge Part II Lectures. §4. Derivation of quantum statistics from the grand canonical ensemble.