Bose–Einstein statistics describes the equilibrium occupation of single-particle energy levels by indistinguishable bosons with integer spin. Because bosons are not constrained by the Pauli exclusion principle, many particles can occupy the same quantum state. Photons, some atomic species in ultracold gases, and various quasiparticles such as phonons all obey Bose–Einstein statistics. In high-occupancy regimes, this leads to strong enhancement of low-energy states compared with classical expectations.

In thermal and diffusive equilibrium at temperature \( T \) and chemical potential \( \mu \), the probability of finding a given number of particles in a state is determined by the grand canonical ensemble. Maximizing the entropy under constraints of fixed average particle number and energy leads to a characteristic form for the mean occupation number of each energy level.

Consider an ideal gas of non-interacting bosons with single-particle energies \( \epsilon_i \). The mean occupation number \( \bar{n}_i \) of the level \( \epsilon_i \) in thermal and chemical equilibrium is

$$ \bar{n}_i = \frac{1}{\exp\!\left(\frac{\epsilon_i - \mu}{kT}\right) - 1}, $$

where \( k \) is Boltzmann’s constant and \( \mu \) is the chemical potential. The total particle number and energy are obtained by summing over all single-particle states,

$$ N = \sum_i \bar{n}_i, \qquad E = \sum_i \epsilon_i \bar{n}_i. $$

In the continuum limit, one replaces the sum by an integral over energy with density of states \( g(\epsilon) \),

$$ n = \frac{N}{V} = \int_0^\infty \frac{g(\epsilon)}{\exp\!\left(\frac{\epsilon - \mu}{kT}\right) - 1} \, d\epsilon. $$

For nonrelativistic particles of mass \( m \) in three dimensions, \( g(\epsilon) \propto V\,\epsilon^{1/2} \), so low-energy states contribute strongly to the total number and energy.

Because the denominator can become small as \( \epsilon_i \rightarrow \mu \), the mean occupation \( \bar{n}_i \) can become very large. For an ideal Bose gas in equilibrium one always has \( \mu \leq \epsilon_0 \), where \( \epsilon_0 \) is the ground-state energy. Approaching this limit leads to qualitatively new behavior.

In the dilute, high-temperature regime where

$$ \exp\!\left(\frac{\epsilon_i - \mu}{kT}\right) \gg 1, $$

the “\(-1\)” in the denominator of the Bose–Einstein distribution can be neglected. The mean occupation number then reduces to the classical Maxwell–Boltzmann form,

$$ \bar{n}_i \approx \exp\!\left(-\frac{\epsilon_i - \mu}{kT}\right). $$

In this limit, the average occupation of each quantum state satisfies \( \bar{n}_i \ll 1 \), and the discrete nature of the energy levels becomes unimportant. Quantum statistics becomes necessary when the phase-space density approaches unity. This condition can be expressed using the thermal de Broglie wavelength,

$$ \lambda_{\mathrm{th}} = \sqrt{\frac{2\pi \hbar^2}{m k T}}, $$

so that quantum degeneracy sets in when

$$ n \lambda_{\mathrm{th}}^3 \gtrsim 1. $$

At this point, either Bose–Einstein or Fermi–Dirac statistics must be used depending on the particle spin.

As a Bose gas is cooled at fixed particle density, the chemical potential \( \mu \) rises toward the ground-state energy \( \epsilon_0 \). For a uniform ideal gas, \( \mu \) cannot exceed \( \epsilon_0 \), and at a critical temperature \( T_{\mathrm{c}} \) the excited states can no longer accommodate all particles. The excess particles accumulate in the ground state, producing a macroscopic occupation \( \bar{n}_0 \).

This phenomenon is known as Bose–Einstein condensation. Below \( T_{\mathrm{c}} \), a finite fraction of the total particle number resides in the lowest-energy state even in the thermodynamic limit. The condensed component behaves as a coherent matter wave and exhibits properties such as long-range phase coherence and superfluidity in interacting systems.

In astrophysics and cosmology, Bose–Einstein statistics is essential for describing photon fields such as the cosmic microwave background, where the chemical potential \( \mu = 0 \), and for modeling bosonic dark-matter candidates in certain theoretical scenarios.

It is often convenient to introduce a dimensionless energy variable

$$ x = \frac{\epsilon}{kT}, $$

and a dimensionless chemical potential parameter

$$ a = -\frac{\mu}{kT} \geq 0. $$

In these variables the Bose–Einstein occupation number becomes

$$ \bar{n}(x; a) = \frac{1}{\exp(x + a) - 1}. $$

For fixed \( a \), the curve \( \bar{n}(x; a) \) rises sharply at small \( x \), indicating strong occupation of low-energy states, and then falls off approximately as \( \exp(-x) \) at large \( x \). Smaller values of \( a \) (corresponding to larger \( \mu \)) enhance the low-energy occupation even further. In the special case of photons, such as in Blackbody radiation, one has \( \mu = 0 \) and therefore \( a = 0 \).

The interactive Plotly figure below shows \( \bar{n}(x; a) \) as a function of \( x = \epsilon/(kT) \) for different values of \( a = -\mu/(kT) \). Moving the slider changes the value of \( a \), and the curve deforms accordingly.

• The Bose–Einstein occupation number \( \bar{n}_i = [\exp((\epsilon_i - \mu)/(kT)) - 1]^{-1} \) allows arbitrarily large occupation of a single quantum state, reflecting bosonic bunching.
• In the dilute limit where \( \exp((\epsilon_i - \mu)/(kT)) \gg 1 \), the Bose–Einstein distribution reduces to the Maxwell–Boltzmann form \( \bar{n}_i \approx \exp[-(\epsilon_i - \mu)/(kT)] \).
• Quantum degeneracy becomes important when the phase-space density satisfies \( n \lambda_{\mathrm{th}}^3 \gtrsim 1 \) with \( \lambda_{\mathrm{th}} = \sqrt{2\pi \hbar^2/(m k T)} \), requiring Bose–Einstein or Fermi–Dirac statistics.
• Bose–Einstein condensation occurs when \( \mu \rightarrow \epsilon_0 \) as the gas is cooled, leading to a macroscopic ground-state occupation below the critical temperature \( T_{\mathrm{c}} \).
• For photons in blackbody radiation, the chemical potential vanishes (\( \mu = 0 \)), so the distribution reduces to \( \bar{n}(x) = [\exp(x) - 1]^{-1} \) and underlies the Planck spectrum.

• Starting from the grand canonical ensemble, derive the Bose–Einstein occupation number \( \bar{n}_i = [\exp((\epsilon_i - \mu)/(kT)) - 1]^{-1} \) by maximizing the entropy under constraints on \( N \) and \( E \).
• Show explicitly how the Maxwell–Boltzmann distribution \( \bar{n}_i \approx \exp[-(\epsilon_i - \mu)/(kT)] \) arises as the low-occupation limit of the Bose–Einstein distribution.
• Using the condition \( n \lambda_{\mathrm{th}}^3 \sim 1 \), estimate the temperature at which a given Bose gas becomes quantum degenerate for specified \( n \) and \( m \).
• Explain qualitatively how varying \( a = -\mu/(kT) \) changes the shape of \( \bar{n}(x; a) \) in the Plotly figure, especially at small \( x \).
• Discuss why photons in thermal equilibrium must have \( \mu = 0 \) and how this leads to the blackbody spectrum described in the Blackbody radiation article.