Bose–Einstein statistics is the framework in statistical mechanics for counting the states of identical bosons and determining their equilibrium populations. Unlike classical particle counting, it treats particles as indistinguishable and permits any nonnegative integer occupation of a single-particle state. Its characteristic distribution describes ideal bosonic systems in thermodynamic equilibrium and underlies both thermal radiation and Bose–Einstein condensation. The statistical rules apply generally to bosons, although the familiar occupation formula assumes noninteracting particles or an appropriate independent-mode description. (damtp.cam.ac.uk)
Historical development
In 1924, Satyendra Nath Bose introduced a new counting method for light quanta that reproduced the spectrum of blackbody radiation. Albert Einstein extended this approach to material particles in papers published in 1924 and 1925. The extension predicted that, below a sufficiently low temperature, a macroscopic number of particles could occupy the lowest-energy state. This became the theoretical basis of Bose–Einstein condensation. (nobelprize.org)
Condensation in dilute atomic gases was experimentally achieved in 1995. The 2001 Nobel Prize in Physics recognized Eric Cornell, Carl Wieman, and Wolfgang Ketterle for achieving condensation in dilute alkali-atom gases and investigating condensate properties. These experiments demonstrated a particularly direct realization of the quantum-statistical behavior predicted decades earlier. (nist.gov)
Indistinguishability and state counting
In quantum mechanics, exchanging two identical bosons leaves their many-particle wave function unchanged. Bosonic states therefore have exchange symmetry. Bosons have integer spin, and their statistical behavior contrasts with that of fermions, whose states are antisymmetric under exchange and obey the Pauli exclusion principle. (ocw.mit.edu)
A bosonic many-particle configuration is specified by occupation numbers (n_i=0,1,2,\ldots), rather than by assigning distinguishable particle labels. If an energy level contains (g_i) distinct single-particle states and (N_i) bosons, the number of possible occupation configurations is
[ W_i=\frac{(N_i+g_i-1)!}{N_i!,(g_i-1)!}. ]
The total multiplicity for specified level populations is the product of these factors. This counting differs from classical counting: exchanging particle labels does not create a new physical configuration. Multiple occupation is allowed, but is not compulsory, and does not itself imply an attractive force between particles. (ocw.mit.edu)
Equilibrium occupation distribution
For independent bosonic states, the mean occupation of a state with energy (\epsilon_i) is
[ \bar n_i= \frac{1}{\exp[(\epsilon_i-\mu)/(k_{\mathrm B}T)]-1}. ]
Here (T) is absolute temperature, (k_{\mathrm B}) is the Boltzmann constant, and (\mu) is the chemical potential. The formula gives a mean particle number, not a normalized probability over energy levels. If a level has degeneracy (g_i), its mean total population is (g_i\bar n_i). Summing occupations gives the mean total particle number; summing (\epsilon_i\bar n_i) gives the mean total energy. (damtp.cam.ac.uk)
The distribution follows naturally from the grand canonical ensemble. With (\beta=1/(k_{\mathrm B}T)), the single-state grand partition function is the geometric series
[ \Xi_i=\sum_{n=0}^{\infty}e^{-\beta n(\epsilon_i-\mu)} =\frac{1}{1-e^{-\beta(\epsilon_i-\mu)}}. ]
For convergence, (\mu) must lie below the lowest single-particle energy in a finite ideal system. A large ground-state occupation develops as (\mu) approaches that energy from below. (homepages.ucl.ac.uk)
Occupation fluctuations and the classical limit
For an independent mode, writing (q=e^{-\beta(\epsilon-\mu)}), the occupation probability distribution is
[ P(n)=(1-q)q^n,\qquad n=0,1,2,\ldots. ]
Its variance satisfies
[ \operatorname{Var}(n)=\bar n(1+\bar n). ]
The quadratic term produces enhanced fluctuations compared with a Poisson distribution having the same mean. This result concerns a thermal independent mode; additional constraints or correlations can change fluctuations in the complete system. (ocw.mit.edu)
When (\exp[(\epsilon-\mu)/(k_{\mathrm B}T)]\gg1), the denominator’s minus one becomes negligible, yielding the classical exponential occupation law. Thus Bose–Einstein statistics approaches Maxwell–Boltzmann statistics in the dilute, low-occupation limit. By comparison, Fermi–Dirac statistics has a plus one in the denominator and restricts the mean occupation of a single state to at most one. (damtp.cam.ac.uk)
Bose–Einstein condensation
For a uniform, three-dimensional, nonrelativistic ideal gas of bosons with one internal state, the excited states can accommodate only a finite particle density at a given temperature as the chemical potential approaches the ground-state energy. Additional particles then accumulate macroscopically in that state, forming a Bose–Einstein condensate. (ocw.mit.edu)
For number density (\rho=N/V) and particle mass (m),
[ T_c=\frac{2\pi\hbar^2}{mk_{\mathrm B}} \left[\frac{\rho}{\zeta(3/2)}\right]^{2/3}, \qquad \frac{N_0}{N}=1-\left(\frac{T}{T_c}\right)^{3/2} \quad(T<T_c). ]
Here (\hbar=h/(2\pi)), with (h) the Planck constant, and (\zeta) is the Riemann zeta function. These relations describe the ideal homogeneous gas in the thermodynamic limit, not every trapped or interacting system. (web.mit.edu)
Radiation, vibrations, and applicability
For ordinary equilibrium thermal radiation, photons can be created and absorbed, so their number is not conserved and their chemical potential is zero. Their occupation becomes
[ \bar n(\omega)=\frac{1}{e^{\hbar\omega/(k_{\mathrm B}T)}-1}, ]
the Planck distribution. It determines thermal mode populations rather than imposing a fixed total photon number. (farside.ph.utexas.edu)
The same occupation factor describes harmonic phonon modes in solids. (oden.utexas.edu) Interactions require further treatment: bosonic exchange symmetry remains valid, but an interacting system need not have ideal-gas occupations. Likewise, condensation and superfluidity are related but distinct; the noninteracting condensate model alone does not provide a complete description of superfluid behavior. (damtp.cam.ac.uk)