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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

physicsquantum field theoryrelativitybasic

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

Exact Lean statement

theorem adiabatic_relation_UaUbVaVb
    {s0 U0 V0 N0 c R : ℝ}
    {Ua Ub Va Vb N : ℝ}
    (hUa : 0 < Ua) (hUb : 0 < Ub)
    (hVa : 0 < Va) (hVb : 0 < Vb)
    (hN : 0 < N)
    (hU0 : 0 < U0) (hV0 : 0 < V0)
    (hR : 0 < R)
    (hS :
      entropy c R s0 U0 V0 N0 Ua Va N =
      entropy c R s0 U0 V0 N0 Ub Vb N) :
    (Real.rpow (Ua / Ub) c) * (Va / Vb) = 1

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem adiabatic_relation_UaUbVaVb    {s0 U0 V0 N0 c R : ℝ}    {Ua Ub Va Vb N : ℝ}    (hUa : 0 < Ua) (hUb : 0 < Ub)    (hVa : 0 < Va) (hVb : 0 < Vb)    (hN : 0 < N)    (hU0 : 0 < U0) (hV0 : 0 < V0)    (hR : 0 < R)    (hS :      entropy c R s0 U0 V0 N0 Ua Va N =      entropy c R s0 U0 V0 N0 Ub Vb N) :    (Real.rpow (Ua / Ub) c) * (Va / Vb) = 1 := by    have hlog := adiabatic_relation_log hUa hUb hVa hVb hN hU0 hV0 hR hS    -- The product is `exp` of the left-hand side of `hlog`, i.e. `exp 0 = 1`.    show (Ua / Ub) ^ c * (Va / Vb) = 1    rw [Real.rpow_def_of_pos (div_pos hUa hUb), ← Real.exp_log (div_pos hVa hVb),        ← Real.exp_add, mul_comm (log (Ua / Ub)) c, hlog, Real.exp_zero]
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/Thermodynamics/IdealGas/Basic.lean:75-91

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Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Project-declaredLean 4.32.0

Differential Entropy eq k B beta mean Energy add k B log math Z

CanonicalEnsemble.differentialEntropy_eq_kB_beta_meanEnergy_add_kB_log_mathZ

Plain-language statement

General identity: S_diff = kB β ⟨E⟩ + kB log Z_math. This connects the differential entropy to the mean energy and the mathematical partition function. Integrability of log (probability …) follows from the pointwise formula.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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