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formal statement (Lean)
55noncomputable def partitionFunction (sys : DiscreteSystem) (T : ℝ) (hT : T > 0) : ℝ :=
proof body
Definition body.
56 ∑ i : Fin sys.numLevels, (sys.degeneracy i : ℝ) * exp (-beta T hT * sys.energy i)
57
58/-- **THEOREM**: Partition function is always positive. -/
used by (27)
From the project-wide theorem graph. These declarations reference this one in their body.
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boltzmann_minimizes_vfe
in IndisputableMonolith.Statistics.VariationalFreeEnergyFromRCL
decl_use
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averageEnergy
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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entropy
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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entropy_nonneg_of_Z_ge_one
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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freeEnergy
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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partitionFunction
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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partition_ge_ground
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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partition_positive
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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probability
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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prob_normalized
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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boltzmannProb
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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boltzmannProb_sum_one
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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F_eq_E_minus_TS
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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helmholtzF
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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helmholtzF_well_defined
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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HelmholtzRealCert
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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partitionFunction
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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partitionFunction_pos
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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freeEnergy
in IndisputableMonolith.Thermodynamics.JCostThermoBridge
decl_use
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partitionFunction
in IndisputableMonolith.Thermodynamics.JCostThermoBridge
decl_use
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partition_pos
in IndisputableMonolith.Thermodynamics.JCostThermoBridge
decl_use
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stateProbability
in IndisputableMonolith.Thermodynamics.JCostThermoBridge
decl_use
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averageEnergy
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
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entropy
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
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freeEnergy
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
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heatCapacity
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
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partition_function_positive
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
depends on (12)
Lean names referenced from this declaration's body.
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T
in IndisputableMonolith.Foundation.Breath1024
decl_use
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is
in IndisputableMonolith.Foundation.OptionAEmpiricalProgram
decl_use
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is
in IndisputableMonolith.Foundation.SimplicialLedger.EdgeLengthFromPsi
decl_use
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is
in IndisputableMonolith.GameTheory.MechanismDesignFromSigma
decl_use
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T
in IndisputableMonolith.Gap45.SyncMinimization
decl_use
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is
in IndisputableMonolith.Mathematics.RamanujanBridge.MockThetaPhantom
decl_use
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beta
in IndisputableMonolith.Relativity.ILG.PPN
decl_use
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partitionFunction
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
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partitionFunction
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
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partitionFunction
in IndisputableMonolith.Thermodynamics.JCostThermoBridge
decl_use
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beta
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
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DiscreteSystem
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use