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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Adiabatic relation log

adiabatic_relation_log

physicsquantum field theoryrelativitybasic

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.

Exact Lean statement

theorem adiabatic_relation_log
    {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) :
    c * log (Ua / Ub) + log (Va / Vb) = 0

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem adiabatic_relation_log    {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) :    c * log (Ua / Ub) + log (Va / Vb) = 0 := by  -- Unfold the entropy and expand every `log (x / y)` into `log x - log y`,  -- so both `hS` and the goal become linear in the individual logarithms.  unfold entropy at hS  rw [Real.log_div hUa.ne' hU0.ne', Real.log_div hUb.ne' hU0.ne',      Real.log_div hVa.ne' hV0.ne', Real.log_div hVb.ne' hV0.ne'] at hS  rw [Real.log_div hUa.ne' hUb.ne', Real.log_div hVa.ne' hVb.ne']  -- The difference of the two entropies is `N * R` times the goal, so the  -- goal is exactly the second factor of a vanishing product.  have key : N * R * (c * (log Ua - log Ub) + (log Va - log Vb)) = 0 := by    linear_combination hS  exact (mul_eq_zero.mp key).resolve_left (mul_ne_zero hN.ne' hR.ne')
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/Thermodynamics/IdealGas/Basic.lean:46-68

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Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

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.

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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