bec_temperature
plain-language theorem explainer
BEC temperature for an ideal Bose gas in natural units is defined by the expression (2π/m) times (n/2.612) raised to the power 2/3. Physicists modeling superfluid He-4 as an eight-tick Bose condensate cite this when computing critical temperatures from number density and mass. The definition is a direct one-line encoding of the standard ideal-gas formula with no derivation steps.
Claim. The Bose-Einstein condensation temperature for an ideal gas of particles with mass $m$ and number density $n$ is $T_{BEC} = (2π/m) (n/2.612)^{2/3}$ in natural units.
background
The module establishes superfluid He-4 as a Bose-Einstein condensate of integer-spin (8-tick) bosons under Recognition Science eight-tick coherence, with He-3 arising from Cooper pairing of 4-tick fermions. Temperature is introduced upstream as the inverse of the Lagrange multiplier β and equals the derivative of average energy with respect to entropy. The definition imports the standard Bose-gas critical-temperature expression that employs ζ(3/2) ≈ 2.612.
proof idea
The definition is a direct one-line encoding of the ideal Bose gas formula in natural units. It applies the standard expression without invoking additional lemmas beyond the temperature definition from the Boltzmann distribution.
why it matters
This definition supplies the critical temperature used to prove positivity in the companion theorem bec_temperature_positive. It fills the superfluidity section by linking eight-tick coherence to observable BEC temperatures for He-4. The parent result is the positivity theorem that follows immediately from this definition.
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