Gap equation: Fermi–Dirac form equals tanh form

Proved

BCSTheory.gapIntegral_fermiDirac_form

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

bcs-theorysuperconductivity

For $k_B>0$ , $T>0$ and all real $\hbar\omega_D$ , $\Delta$ , with $f(E,T)=1/(e^{E/k_BT}+1)$ ,

\int_0^{\hbar\omega_D}\frac{d\eta}{\sqrt{\eta^2+\Delta^2}}\Bigl[1-2f\bigl(\sqrt{\eta^2+\Delta^2},T\bigr)\Bigr] = \int_0^{\hbar\omega_D}\frac{d\eta}{\sqrt{\eta^2+\Delta^2}}\tanh\frac{\sqrt{\eta^2+\Delta^2}}{2k_BT}.

Preamble

import Mathlib
import Definitions.Def_BCSTheory_GapEquation
open Filter Topology

Formal statement

namespace BCSTheory

theorem gapIntegral_fermiDirac_form (kB hbarOmegaD Δ T : ℝ) (hkB : 0 < kB) (hT : 0 < T) :
    (∫ η in (0 : ℝ)..hbarOmegaD,
        (1 - 2 * fermiDirac kB (quasiparticleEnergy η Δ) T) / quasiparticleEnergy η Δ) =
      gapIntegral kB hbarOmegaD Δ T := by sorry

end BCSTheory

Source

Wikipedia (de), Artikel „BCS-Theorie“ (PDF-Ausdruck vom 05.10.2026), Abschnitt „Erfolge der BCS-Theorie“, Unterabschnitt „Existenz einer Energielücke“, Integralgleichung für

T \neq 0

und ihre tanh-Form.

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, with full knowledge of the source text and of the intended meaning. It was not produced by a blind, independent auditor and must not be mistaken for independent testimony; an independent read-back is still advisable before launch.

Let $k_B, \hbar\omega_D, \Delta, T$ be real numbers with $k_B>0$ and $T>0$ ( $\hbar\omega_D$ and $\Delta$ are arbitrary, possibly zero or negative). Write $E(\eta) = \sqrt{\eta^2+\Delta^2}$ and $f(E,T) = 1/(e^{E/(k_BT)}+1)$ . Then

\int_0^{\hbar\omega_D}\frac{1-2f\bigl(E(\eta),T\bigr)}{E(\eta)}\,d\eta = \int_0^{\hbar\omega_D}\frac{\tanh\bigl(E(\eta)/(2k_BT)\bigr)}{E(\eta)}\,d\eta,

the right-hand side being the gap integral $I(k_B,\hbar\omega_D,\Delta,T)$ of the definition file. Both integrals are oriented interval integrals; a quotient with denominator $E(\eta) = 0$ is read as $0$ , and a non-integrable integrand would give the value $0$ (the claim is an equality of the two values under these conventions).

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

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