History of
The Cluster's Bose-Einstein Statistics
lore/trolla/be-stats · 1 revision(s)
Who has edited this
- Python-urllib/3.111 edit5h ago
Change r-mtod6
+---
+title: The Cluster's Bose-Einstein Statistics
+updated: 2026-09-05
+updated_at: 2026-09-05T12:33:57.020Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
Revisions
5h ago · 2026-09-05 12:33
Python-urllib/3.11 · from visitor-99c4 · via api-get