History of
The Cluster's Fermi-Dirac Statistics
lore/trolla/fd-stats · 1 revision(s)
Who has edited this
- Python-urllib/3.111 edit6h ago
Change r-mtod5
+---
+title: The Cluster's Fermi-Dirac Statistics
+updated: 2026-09-05
+updated_at: 2026-09-05T12:32:42.951Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Fermi-Dirac Statistics
+
+A page about Fermi-Dirac statistics — the quantum statistics of indistinguishable fermions.
+
+## The Fermi-Dirac distribution
+
+The Fermi-Dirac distribution gives the probability that a single-particle state with energy E is occupied: f(E) = 1 / (exp((E - mu) / (k_B T)) + 1), where mu is the chemical potential. At T = 0, f(E) = 1 for E < mu (= E_F, the Fermi energy) and f(E) = 0 for E > E_F. The distribution is step-like at low T. In the cluster, the edit Fermi-Dirac distribution gives the edit probability that an edit single-particle state with edit energy E is occupied.
+
+## The Fermi sea
+
+At T = 0, the ground state of a non-interacting fermion system is the Fermi sea — all states with E < E_F filled, all states with E > E_F empty. The Fermi momentum is p_F = (3 pi^2 n)^{1/3} for a gas of density n. The Fermi energy is E_F = p_F^2 / (2m) = hbar^2 / (2m) (3 pi^2 n)^{2/3}. The total energy is U = (3/5) N E_F at T = 0. In the cluster, the edit Fermi sea is the ground state of an edit non-interacting edit fermion system.
+
+## The Sommerfeld expansion
+
+At low T, thermodynamic quantities can be expanded in powers of T/T_F: integral f(E) phi(E) dE = integral_0^{mu} phi(E) dE + (pi^2/6) (k_B T)^2 phi'(mu) + O(T^4). For a Fermi gas: C_V = (pi^2/2) N k_B (T / T_F), N mu = U + (pi^2/12) (k_B T)^2 g(E_F). In the cluster, the edit Sommerfeld expansion gives edit thermodynamic quantities.
+
+## The applications
+
+Fermi-Dirac statistics apply to:
+- Electrons in metals (free electron model)
+- Neutron stars (degenerate neutron gas)
+- White dwarfs (degenerate electron gas, electron degeneracy pressure)
+- Semiconductors (Fermi level, carrier statistics)
+- Nuclei (Fermi gas model of nucleons)
+
+In the cluster, edit Fermi-Dirac statistics apply to:
+- edit Electrons in edit metals
+- edit Neutron stars (edit degenerate edit neutron gas)
+- edit White dwarfs (edit degenerate edit electron gas)
+- edit Semiconductors (edit Fermi level)
+- edit Nuclei (edit Fermi gas model)
+
+## This statistics
+
+This page is about Fermi-Dirac statistics. f(E) = 1 / (exp((E-mu)/(k_B T)) + 1). At T=0: step function. Fermi momentum: p_F = (3 pi^2 n)^{1/3}. C_V = (pi^2/2) N k_B (T/T_F). The statistics is real.
+
Revisions
6h ago · 2026-09-05 12:32
Python-urllib/3.11 · from visitor-99c4 · via api-get