feat: Add canonical ensemble.

PhysLean.lean

@@ -255,7 +255,9 @@ import PhysLean.Relativity.Tensors.RealTensor.Vector.Pre.Modules
 import PhysLean.Relativity.Tensors.RealTensor.Vector.Pre.NormOne
 import PhysLean.Relativity.Tensors.TensorSpecies.Basic
 import PhysLean.Relativity.Tensors.UnitTensor
-import PhysLean.StatisticalMechanics.Basic
+import PhysLean.StatisticalMechanics.BoltzmannConstant
+import PhysLean.StatisticalMechanics.CanonicalEnsemble.Basic
+import PhysLean.StatisticalMechanics.Temperature
 import PhysLean.StringTheory.Basic
 import PhysLean.StringTheory.FTheory.SU5U1.Charges.AllowsTerm
 import PhysLean.StringTheory.FTheory.SU5U1.Charges.Basic

PhysLean/StatisticalMechanics/BoltzmannConstant.lean

@@ -0,0 +1,26 @@
+/-
+Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import Mathlib.Data.NNReal.Defs
+/-!
+
+## Boltzmann constant
+
+In this file we define the Boltzmann constant `kB` as a non-negative real number.
+This is introduced as an axiom.
+
+-/
+open NNReal
+
+namespace Constants
+
+/-- The Boltzmann constant. -/
+axiom kB : ℝ≥0
+
+/-- The Boltzmann constant is non-negative. -/
+lemma kB_nonneg : 0 ≤ kB := by
+  apply NNReal.coe_nonneg
+
+end Constants

PhysLean/StatisticalMechanics/CanonicalEnsemble/Basic.lean

@@ -0,0 +1,313 @@
+/-
+Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import Mathlib.Algebra.Lie.OfAssociative
+import Mathlib.Analysis.Calculus.LogDeriv
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Analysis.SpecialFunctions.ExpDeriv
+import Mathlib.Analysis.SpecialFunctions.Log.Basic
+import PhysLean.StatisticalMechanics.Temperature
+import PhysLean.Meta.Informal.SemiFormal
+/-!
+
+# Canonical ensemble
+
+A canonical ensemble describes a system in thermal equilibrium with a heat bath at a
+fixed temperature.
+
+In this file we define the canonical ensemble, its partition function, the
+probability of being in a given microstate, the mean energy, the entropy and
+the Helmholtz free energy.
+
+We also define the addition of two canonical ensembles, and prove results related
+to the properties of additions of canonical ensembles.
+
+## References
+
+- https://www.damtp.cam.ac.uk/user/tong/statphys/statmechhtml/S1.html#E23
+
+-/
+
+/-- A Canonical ensemble is described by a type `ι`, corresponding to the type of microstates,
+  and a map `ι → ℝ` which associates which each microstate an energy. -/
+def CanonicalEnsemble (ι : Type) : Type := ι → ℝ
+
+namespace CanonicalEnsemble
+open Real Temperature
+
+variable {ι ι1 : Type} (𝓒 : CanonicalEnsemble ι) (𝓒1 : CanonicalEnsemble ι1)
+
+/-- The addition of two `CanonicalEnsemble`. -/
+instance {ι1 ι2 : Type} : HAdd (CanonicalEnsemble ι1) (CanonicalEnsemble ι2)
+    (CanonicalEnsemble (ι1 × ι2)) where
+  hAdd := fun 𝓒1 𝓒2 => fun (i : ι1 × ι2) => 𝓒1 i.1 + 𝓒2 i.2
+
+/-- Scalar multiplication of `CanonicalEnsemble`, defined such that
+  `nsmul n 𝓒` is `n` coppies of the canonical ensemble `𝓒`. -/
+def nsmul (n : ℕ) (𝓒1 : CanonicalEnsemble ι) : CanonicalEnsemble (Fin n → ι) :=
+  fun f => ∑ i, 𝓒1 (f i)
+
+set_option linter.unusedVariables false in
+/-- The microstates of a the canonical ensemble -/
+@[nolint unusedArguments]
+abbrev microstates {ι : Type} (𝓒 : CanonicalEnsemble ι) : Type := ι
+
+/-!
+
+## The energy of the microstates
+
+-/
+/-- The energy of associated with a mircrostate of the canonical ensemble. -/
+abbrev energy (𝓒 : CanonicalEnsemble ι) : microstates 𝓒 → ℝ := 𝓒
+
+@[simp]
+lemma energy_add_apply (i : microstates (𝓒 + 𝓒1)) :
+    (𝓒 + 𝓒1).energy i = 𝓒.energy i.1 + 𝓒1.energy i.2 := by
+  simp [energy]
+  rfl
+
+@[simp]
+lemma energy_nsmul_apply (n : ℕ) (f : Fin n → microstates 𝓒) :
+    (nsmul n 𝓒).energy f = ∑ i, 𝓒.energy (f i) := by
+  simp [energy, nsmul]
+
+/-!
+
+## The partition function of the canonical ensemble
+
+-/
+
+/-- The partition function of the canonical ensemble. -/
+noncomputable def partitionFunction [Fintype ι] (T : Temperature) : ℝ :=
+  ∑ i, exp (- β (T) * 𝓒.energy i)
+
+lemma partitionFunction_add [Fintype ι] [Fintype ι1] :
+    (𝓒 + 𝓒1).partitionFunction T = 𝓒.partitionFunction T * 𝓒1.partitionFunction T := by
+  simp [partitionFunction]
+  rw [Fintype.sum_prod_type]
+  rw [Finset.sum_mul]
+  congr
+  funext i
+  rw [Finset.mul_sum]
+  congr
+  funext j
+  rw [← Real.exp_add]
+  congr
+  simp [energy]
+  ring
+
+/-- The partition function of `n` copies of a canonical ensemble. -/
+semiformal_result "ERA5D" partitionFunction_nsmul [Fintype ι] (n : ℕ) (T : Temperature) :
+    (nsmul n 𝓒).partitionFunction T = (𝓒.partitionFunction T) ^ n
+
+lemma partitionFunction_pos [Fintype ι] [Nonempty ι] (T : Temperature) :
+    0 < partitionFunction 𝓒 T := by
+  rw [partitionFunction]
+  apply Finset.sum_pos
+  · intro i hi
+    exact exp_pos (-T.β * 𝓒.energy i)
+  · simp
+
+@[simp]
+lemma partitionFunction_neq_zero [Fintype ι] [Nonempty ι] (T : Temperature) :
+    partitionFunction 𝓒 T ≠ 0:= by
+  have h1 := partitionFunction_pos 𝓒 T
+  exact Ne.symm (ne_of_lt h1)
+
+/-- The partition function of the canonical ensemble as a function of `β` -/
+noncomputable def partitionFunctionβ [Fintype ι] (β : ℝ) : ℝ :=
+  ∑ i, exp (- β * 𝓒.energy i)
+
+lemma partitionFunctionβ_def [Fintype ι]:
+    partitionFunctionβ 𝓒 = fun β => ∑ i, exp (- β * 𝓒.energy i) := by rfl
+
+@[fun_prop]
+lemma partitionFunctionβ_differentiable [Fintype ι] :
+    Differentiable ℝ 𝓒.partitionFunctionβ := by
+  rw [partitionFunctionβ_def]
+  fun_prop
+
+lemma partitionFunction_eq_partitionFunctionβ [Fintype ι] (T : Temperature) :
+    partitionFunction 𝓒 T = partitionFunctionβ 𝓒 (β T) := by
+  simp [partitionFunction, partitionFunctionβ, β]
+
+/-!
+
+## The probability of being in a given microstate
+-/
+
+/-- The probability of been in a given microstate. -/
+noncomputable def probability [Fintype ι] (i : microstates 𝓒) (T : Temperature) : ℝ :=
+  exp (- β (T) * 𝓒.energy i) / partitionFunction 𝓒 T
+
+/-- Probability of a microstate in a canonical ensemble is less then or equal to `1`. -/
+semiformal_result "ERDAR" probability_nsmul [Fintype ι] (i : microstates 𝓒) (T : Temperature) :
+    𝓒.probability i T ≤ 1
+
+/-- Probability of a microstate in a canonical ensemble is non-negative. -/
+semiformal_result "ERBG6" probability_nsmul [Fintype ι] (i : microstates 𝓒) (T : Temperature) :
+    0 ≤ 𝓒.probability i T
+
+lemma probability_neq_zero [Fintype ι] [Nonempty ι] (i : microstates 𝓒) (T : Temperature) :
+    probability 𝓒 i T ≠ 0 := by
+  rw [probability]
+  field_simp
+
+@[simp]
+lemma probability_add [Fintype ι] [Fintype ι1]
+    (i : microstates (𝓒 + 𝓒1)) (T : Temperature) :
+    (𝓒 + 𝓒1).probability i T = 𝓒.probability i.1 T * 𝓒1.probability i.2 T := by
+  simp [probability]
+  rw [partitionFunction_add 𝓒 𝓒1]
+  field_simp
+  congr
+  rw [← Real.exp_add]
+  ring_nf
+
+/-- The probability of a microstate in `n` copies of a canonical ensemble is
+  equal to the product of the probability of the corresponding individual microstates. -/
+semiformal_result "ERDAE" probability_nsmul [Fintype ι] (n : ℕ)
+    (f : microstates (nsmul n 𝓒)) (T : Temperature) :
+    (nsmul n 𝓒).probability f T = ∏ i, 𝓒.probability (f i) T
+
+@[simp]
+lemma sum_probability_eq_one [Fintype ι] [Nonempty ι] (T : Temperature) :
+    ∑ i, probability 𝓒 i T = 1 := by
+  simp [probability]
+  rw [← Finset.sum_div]
+  field_simp
+  rw [partitionFunction]
+  ring_nf
+
+/-!
+
+## The mean energy of the canonical ensemble
+
+-/
+
+/-- The mean energy of the canonical ensemble. -/
+noncomputable def meanEnergy [Fintype ι] (T : Temperature) : ℝ :=
+  ∑ i, 𝓒.energy i * probability 𝓒 i T
+
+@[simp]
+lemma meanEnergy_add [Fintype ι] [Nonempty ι] (𝓒1 : CanonicalEnsemble ι1) [Fintype ι1]
+    [Nonempty ι1]
+    (T : Temperature) :
+    (𝓒 + 𝓒1).meanEnergy T = 𝓒.meanEnergy T + 𝓒1.meanEnergy T := by
+  simp [meanEnergy]
+  conv_lhs =>
+    enter [2, x]
+    rw [add_mul]
+  rw [Finset.sum_add_distrib]
+  congr 1
+  · rw [Fintype.sum_prod_type]
+    simp only
+    congr
+    funext i
+    rw [← Finset.mul_sum, ← Finset.mul_sum]
+    simp
+  · rw [Fintype.sum_prod_type]
+    rw [Finset.sum_comm]
+    simp only
+    congr
+    funext i
+    rw [← Finset.mul_sum, ← Finset.sum_mul]
+    simp
+
+/-- The mean energy of `n` copies of a canonical ensemble is equal
+  to `n` times the mean energy of the canonical ensemble.
+
+  Note, can't make this `SMul` since the target type depends on the
+  value of `n`. -/
+semiformal_result "ERBAH" meanEnergy_nsmul [Fintype ι] (n : ℕ) (T : Temperature) :
+    (nsmul n 𝓒).meanEnergy T = n * 𝓒.meanEnergy T
+
+lemma meanEnergy_eq_logDeriv_partitionFunctionβ [Fintype ι] (T : Temperature) :
+    meanEnergy 𝓒 T = - logDeriv (partitionFunctionβ 𝓒) (β T) := by
+  rw [logDeriv_apply]
+  nth_rewrite 1 [partitionFunctionβ_def]
+  rw [deriv_sum]
+  · simp [meanEnergy]
+    rw [@neg_div]
+    simp only [neg_neg]
+    rw [Finset.sum_div]
+    congr
+    funext i
+    simp [probability]
+    rw [partitionFunction_eq_partitionFunctionβ 𝓒 T]
+    ring
+  · intro i
+    fun_prop
+
+open Constants
+
+/-!
+
+## Entropy
+
+-/
+
+/-- The entropy of the canonical ensemble. -/
+noncomputable def entropy [Fintype ι] (T : Temperature) : ℝ :=
+  - kB * ∑ i, probability 𝓒 i T * log (probability 𝓒 i T)
+
+/-- Entropy is addative on adding canonical ensembles. -/
+@[simp]
+lemma entropy_add [Fintype ι] [Nonempty ι] (𝓒1 : CanonicalEnsemble ι1) [Fintype ι1]
+    [Nonempty ι1] (T : Temperature) :
+    (𝓒 + 𝓒1).entropy T = 𝓒.entropy T + 𝓒1.entropy T := by
+  simp [entropy]
+  conv_lhs =>
+    enter [1, 2, 2, x]
+    rw [log_mul (probability_neq_zero 𝓒 x.1 T) (probability_neq_zero 𝓒1 x.2 T)]
+    rw [mul_add]
+  rw [Finset.sum_add_distrib, mul_add, neg_add]
+  congr 1
+  · simp
+    left
+    rw [Fintype.sum_prod_type]
+    simp only
+    congr
+    funext i
+    rw [← Finset.sum_mul, ← Finset.mul_sum]
+    simp
+  · rw [Fintype.sum_prod_type]
+    rw [Finset.sum_comm]
+    simp only [neg_inj, mul_eq_mul_left_iff, NNReal.coe_eq_zero]
+    left
+    congr
+    funext i
+    rw [← Finset.sum_mul, ← Finset.sum_mul]
+    simp
+
+/-- The entropy of `n` copies of a canonical ensemble is equal
+  to `n` times the entropy of the canonical ensemble. -/
+semiformal_result "ERBCV" entropy_nsmul [Fintype ι] (n : ℕ) (T : Temperature) :
+    (nsmul n 𝓒).entropy T = n * 𝓒.entropy T
+
+/-!
+
+## Helmholtz free energy
+
+-/
+
+/-- The (Helmholtz) free energy of the canonical ensemble. -/
+noncomputable def helmholtzFreeEnergy [Fintype ι] (T : Temperature) : ℝ :=
+  𝓒.meanEnergy T - T * 𝓒.entropy T
+
+/-- The Helmholtz free energy is addative. -/
+@[simp]
+lemma helmholtzFreeEnergy_add [Fintype ι] [Nonempty ι]
+    [Fintype ι1] [Nonempty ι1] (T : Temperature) :
+    (𝓒 + 𝓒1).helmholtzFreeEnergy T = 𝓒.helmholtzFreeEnergy T + 𝓒1.helmholtzFreeEnergy T := by
+  simp [helmholtzFreeEnergy]
+  ring
+
+/-- The free energy of `n` copies of a canonical ensemble is equal
+  to `n` times the entropy of the canonical ensemble. -/
+semiformal_result "ERC72" helmholtzFreeEnergy_nsmul [Fintype ι] (n : ℕ) (T : Temperature) :
+    (nsmul n 𝓒).helmholtzFreeEnergy T = n * 𝓒.helmholtzFreeEnergy T
+
+end CanonicalEnsemble