The Cluster's Bose-Einstein Statistics

A page about Bose-Einstein statistics — the quantum statistics of indistinguishable bosons.

The Bose-Einstein distribution

The Bose-Einstein distribution gives the probability that a single-particle state with energy E is occupied: f_BE(E) = 1 / (exp((E - mu) / (k_B T)) - 1). Unlike Fermi-Dirac, mu <= 0 for a gas of bosons. As T decreases, mu approaches 0, and the occupation of the ground state grows macroscopically — this is Bose-Einstein condensation (BEC). In the cluster, the edit Bose-Einstein distribution gives the edit probability that an edit state with edit energy E is occupied.

Bose-Einstein condensation

For a 3D ideal Bose gas, the critical temperature for BEC is T_c = (2 pi hbar^2 / m k_B) (n / zeta(3/2))^{2/3} ~ 3.31 hbar^2 n^{2/3} / (m k_B), where zeta(3/2) ~ 2.612. Below T_c, the fraction in the ground state is N_0 / N = 1 - (T / T_c)^{3/2}. The critical density is n_c = 2.612 / lambda_th^3. In the cluster, the edit BEC critical temperature is edit T_c ~ 3.31 hbar^2 n^{2/3} / (m_edit k_B).

The Planck distribution

Black-body radiation is a gas of photons, which are bosons with mu = 0. The photon occupation is f_photon(E) = 1 / (exp(E / (k_B T)) - 1) = 1 / (exp(h nu / (k_B T)) - 1). The energy density is u(nu) d nu = (8 pi h nu^3 / c^3) / (exp(h nu / (k_B T)) - 1) d nu. Integrating gives the Stefan-Boltzmann law: u = a T^4, where a = 4 sigma / c = pi^2 k_B^4 / (15 hbar^3 c^3). The total intensity is j = sigma T^4. In the cluster, the edit Planck distribution is an edit gas of edit photons.

The applications

Bose-Einstein statistics apply to:

• Black-body radiation (photons)
• Phonons (quantized lattice vibrations)
• Superfluid helium-4 (macroscopic occupation of ground state)
• Ultracold atomic gases (BECs of Rb, Na, Li atoms)
• Magnons (quantized spin waves)

In the cluster, edit Bose-Einstein statistics apply to:

• edit Black-body radiation (edit photons)
• edit Phonons
• edit Superfluid helium-4 (edit BEC)
• edit Ultracold atomic gases (edit BECs)
• edit Magnons

This statistics

This page is about Bose-Einstein statistics. f(E) = 1 / (exp((E-mu)/(k_B T)) - 1). T_c ~ 3.31 hbar^2 n^{2/3} / (m k_B). N_0/N = 1 - (T/T_c)^{3/2}. Planck distribution: u(nu) ~ nu^3 / (exp(h nu/(k_B T)) - 1). The statistics is real.