Bose–Einstein (BE) statistics describes the equilibrium occupation of single–particle energy levels by indistinguishable bosons (integer spin), which are not constrained by the Pauli exclusion principle. A given energy level with energy <math>\epsilon_i</math> can therefore host an arbitrarily large number of particles.

For an ideal Bose gas in thermal and diffusive equilibrium at temperature <math>T</math> and chemical potential <math>\mu</math>, the mean occupation number of the level <math>\epsilon_i</math> is <math> \bar{n}_i \;=\; \frac{1}{\exp\!\left(\dfrac{\epsilon_i-\mu}{k_{\mathrm B}T}\right) - 1}, </math> where <math>k_{\mathrm B}</math> is Boltzmann’s constant. The total particle number and total energy are <math> N \;=\; \sum_i \bar{n}_i, \qquad E \;=\; \sum_i \epsilon_i \bar{n}_i. </math>

In the continuum limit with density of states <math>g(\epsilon)</math>, the number density is <math> n \;=\; \frac{N}{V} \;=\; \int_0^\infty \frac{g(\epsilon)}{\exp\!\left(\dfrac{\epsilon-\mu}{k_{\mathrm B}T}\right) - 1}\,d\epsilon. </math> For non–relativistic particles in three dimensions, <math>g(\epsilon) \propto V\,\epsilon^{1/2}</math>, so low–energy states have a strong influence on the thermodynamics.

Because the denominator can become small as <math>\epsilon_i \to \mu</math>, the mean occupation <math>\bar{n}_i</math> can become very large. In an ideal gas, the chemical potential satisfies <math>\mu \le \epsilon_0</math>, the ground–state energy. As <math>T</math> decreases at fixed density, <math>\mu</math> approaches <math>\epsilon_0</math> from below and a macroscopic population of the ground state appears: Bose–Einstein condensation.

In the dilute, high–temperature limit, <math> \exp\!\left(\dfrac{\epsilon_i-\mu}{k_{\mathrm B}T}\right) \gg 1 </math> and the “–1” in the denominator can be neglected, giving the classical Maxwell–Boltzmann distribution <math> \bar{n}_i \;\approx\; \exp\!\left(-\dfrac{\epsilon_i-\mu}{k_{\mathrm B}T}\right). </math> Quantum degeneracy is important when the phase–space density <math> n \lambda_{\mathrm{th}}^3 \gtrsim 1, \qquad \lambda_{\mathrm{th}} = \sqrt{\frac{2\pi \hbar^2}{m k_{\mathrm B}T}} </math> (thermal de Broglie wavelength) is of order unity, signalling that BE statistics is required.

In astrophysics and cosmology, BE statistics underlies the distribution of photons (with <math>\mu = 0</math>) and is important for bosonic quasiparticles (e.g. phonons) and certain dark–matter candidates in high–occupation–number regimes.

For visualization it is convenient to introduce the dimensionless variables <math> x \;=\; \frac{\epsilon}{k_{\mathrm B}T}, \qquad a \;=\; -\frac{\mu}{k_{\mathrm B}T} \;\ge\; 0, </math> so that <math> \bar{n}(x) \;=\; \frac{1}{\exp(x + a) - 1}. </math>

The following embeddable Plotly block (HTML + JavaScript) can be pasted directly into DokuWiki (with HTML embedding enabled) to plot <math>\bar{n}(x)</math> for several values of <math>a</math>:

This reproduces the characteristic strong enhancement of low–energy occupation for bosons, especially when <math>a</math> is small (i.e. when <math>\mu</math> is close to the ground–state energy).

* The Bose–Einstein occupation number <math>\bar{n}_i = [\exp((\epsilon_i-\mu)/k_{\mathrm B}T)-1]^{-1}</math> allows arbitrarily large <math>\bar{n}_i</math>, reflecting bosonic bunching at low energy. * In the limit <math>\exp((\epsilon_i-\mu)/k_{\mathrm B}T) \gg 1</math>, BE statistics reduces to the Maxwell–Boltzmann form <math>\bar{n}_i \approx \exp[-(\epsilon_i-\mu)/k_{\mathrm B}T]</math>. * Quantum degeneracy appears when <math>n \lambda_{\mathrm{th}}^3 \gtrsim 1</math>, with <math>\lambda_{\mathrm{th}} = \sqrt{2\pi\hbar^2/(m k_{\mathrm B}T)}</math>, signalling the need for BE (or FD) statistics. * As <math>T</math> is lowered at fixed <math>n</math>, the approach of <math>\mu</math> to <math>\epsilon_0</math> leads to Bose–Einstein condensation with macroscopic ground–state occupation. * For photons with <math>\mu = 0</math>, BE statistics combined with the electromagnetic density of states yields the Planck blackbody spectrum and the shape of the cosmic microwave background.

* Derive <math>\bar{n}_i = [\exp((\epsilon_i-\mu)/k_{\mathrm B}T)-1]^{-1}</math> from the grand–canonical ensemble by maximizing entropy with constraints on <math>N</math> and <math>E</math>. * Show analytically how the BE distribution reduces to <math>\bar{n}_i \approx \exp[-(\epsilon_i-\mu)/k_{\mathrm B}T]</math> when <math>\bar{n}_i \ll 1</math>, and interpret this limit physically. * For a 3D ideal Bose gas with <math>g(\epsilon) \propto \epsilon^{1/2}</math>, derive the critical temperature <math>T_{\mathrm c}(n,m)</math> of Bose–Einstein condensation. * Using <math>\bar{n}(x) = [\exp(x + a)-1]^{-1}</math>, explain how increasing <math>a = -\mu/(k_{\mathrm B}T)</math> modifies the low–energy part of the curve in the Plotly plot. * Discuss why a photon gas in thermal equilibrium must have <math>\mu = 0</math>, and how this condition, together with BE statistics, leads to the observed Planck spectrum.