recognition /
Thermodynamics /
Thermodynamics.PartitionFunction /
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formal statement (Lean)
79 noncomputable def averageEnergy (sys : DiscreteSystem) (T : ℝ) (hT : T > 0) : ℝ :=
proof body
Definition body.
80 (∑ i : Fin sys.numLevels,
81 sys.energy i * (sys.degeneracy i : ℝ) * exp (-beta T hT * sys.energy i)) /
82 partitionFunction sys T hT
83
84 /-- Entropy S = k_B(ln Z + β⟨E⟩). -/
used by (6)
From the project-wide theorem graph. These declarations reference this one in their body.
averageEnergy
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
entropy
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
entropy_nonneg_of_Z_ge_one
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
free_energy_identity
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
entropy
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
heatCapacity
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
depends on (18)
Lean names referenced from this declaration's body.
T
in IndisputableMonolith.Foundation.Breath1024
decl_use
E
in IndisputableMonolith.Foundation.SpectralEmergence
decl_use
T
in IndisputableMonolith.Gap45.SyncMinimization
decl_use
k_B
in IndisputableMonolith.Information.ComputationLimitsStructure
decl_use
Z
in IndisputableMonolith.Masses.Anchor
decl_use
Z
in IndisputableMonolith.Physics.AnchorPolicyCertified
decl_use
k_B
in IndisputableMonolith.Quantum.BekensteinHawking
decl_use
k_B
in IndisputableMonolith.Quantum.PageCurve
decl_use
S
in IndisputableMonolith.Relativity.ILG.Action
decl_use
beta
in IndisputableMonolith.Relativity.ILG.PPN
decl_use
averageEnergy
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
partitionFunction
in IndisputableMonolith.Thermodynamics.BoltzmannDistribution
decl_use
partitionFunction
in IndisputableMonolith.Thermodynamics.HelmholtzReal
decl_use
partitionFunction
in IndisputableMonolith.Thermodynamics.JCostThermoBridge
decl_use
beta
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
DiscreteSystem
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
k_B
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use
partitionFunction
in IndisputableMonolith.Thermodynamics.PartitionFunction
decl_use