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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Log phys eq log math sub const on Ioi

CanonicalEnsemble.log_phys_eq_log_math_sub_const_on_Ioi

physicsquantum field theoryrelativitylemmas

Plain-language statement

Helper: equality (on Set.Ioi 0) between the β–parametrized logarithm of the physical partition function and the β–parametrized logarithm of the mathematical partition function up to the (β–independent) semiclassical correction. This is used only to identify derivatives (the correction drops). We add the hypothesis h_fin giving finiteness of the Bolt...

Exact Lean statement

lemma log_phys_eq_log_math_sub_const_on_Ioi
    (𝓒 : CanonicalEnsemble ι) [NeZero 𝓒.μ]
    (h_fin :
      ∀ β > 0,
        IsFiniteMeasure (𝓒.μBolt (Temperature.ofβ (Real.toNNReal β)))) :
    Set.EqOn
      (fun β : ℝ =>
        Real.log (𝓒.partitionFunction (Temperature.ofβ (Real.toNNReal β))))
      (fun β : ℝ =>
        Real.log (∫ i, Real.exp (-β * 𝓒.energy i) ∂ 𝓒.μ)
          - (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit)
      (Set.Ioi (0 : ℝ))

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma log_phys_eq_log_math_sub_const_on_Ioi    (𝓒 : CanonicalEnsemble ι) [NeZero 𝓒.μ]    (h_fin :      ∀ β > 0,        IsFiniteMeasure (𝓒.μBolt (Temperature.ofβ (Real.toNNReal β)))) :    Set.EqOn      (fun β : ℝ =>        Real.log (𝓒.partitionFunction (Temperature.ofβ (Real.toNNReal β))))      (fun β : ℝ =>        Real.log (∫ i, Real.exp (-β * 𝓒.energy i) ∂ 𝓒.μ)          - (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit)      (Set.Ioi (0 : ℝ)) := by  intro β hβ  have hβpos : 0 < β := hβ  have _inst : IsFiniteMeasure (𝓒.μBolt (Temperature.ofβ (Real.toNNReal β))) :=    h_fin β hβpos  simp only [log_partitionFunction, log_mathematicalPartitionFunction_eq, β_ofβ,    Real.coe_toNNReal β hβpos.le]
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean:320-337

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Project-declaredLean 4.32.0

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Plain-language statement

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physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Project-declaredLean 4.32.0

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Plain-language statement

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Source project: Physlib

Person-level attribution pending.

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