lean/OAI/Probability/RadialTransition/GrandPartition.lean

import OAI.Probability.RadialTransition.PartitionReplication

namespace OAI

noncomputable section

open scoped BigOperators NNReal ContDiff ENNReal
open Filter MeasureTheory Set

namespace RadialTransition

def grandOn (φ : Potential) (A : Set Position) (β z : ℝ) : ℝ :=
  ∑' N : ℕ, Real.exp (z * N) * partitionOn φ A N β

theorem partitionOn_stability_upper {φ : Potential} (hφ : Continuous φ)
    (hb : IsBoundedPotential φ) {B β : ℝ} (hstable : StableWith φ B)
    (hβ : 0 ≤ β) {A : Set Position} (hAfin : volume A ≠ ⊤) (N : ℕ) :
    partitionOn φ A N β ≤ (volume.real A) ^ N * Real.exp (β * B * N) / N.factorial := by
  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.mpr hAfin
  have hh : rawPartition φ A (Fin N) β ≤ (volume.real A) ^ N * Real.exp (β * B * N) := by
    calc
      _ ≤ ∫ _ : Fin N → Position, Real.exp (β * B * N)
          ∂Measure.pi (fun _ : Fin N => volume.restrict A) := by
        apply integral_mono (integrable_orderedBoltzmann hφ hb β _) (integrable_const _)
        intro x
        apply Real.exp_le_exp.mpr
        rw [orderedEnergy_fin]
        nlinarith [mul_le_mul_of_nonneg_left (hstable N x) hβ]
      _ = _ := by
        simp [integral_const, measureReal_def, Measure.pi_univ]
  have hh' := mul_le_mul_of_nonneg_left hh
    (show (0 : ℝ) ≤ (N.factorial : ℝ)⁻¹ by positivity)
  simpa only [partitionOn, div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc] using hh'

theorem grandOn_term_upper {φ : Potential} (hφ : Continuous φ)
    (hb : IsBoundedPotential φ) {B β : ℝ} (hstable : StableWith φ B)
    (hβ : 0 ≤ β) {A : Set Position} (hAfin : volume A ≠ ⊤) (z : ℝ) (N : ℕ) :
    Real.exp (z * N) * partitionOn φ A N β ≤
      (volume.real A * Real.exp (β * B + z)) ^ N / N.factorial := by
  have hh := mul_le_mul_of_nonneg_left (partitionOn_stability_upper hφ hb hstable hβ hAfin N)
    (Real.exp_pos (z * N)).le
  apply hh.trans_eq
  rw [Real.exp_add, mul_pow, mul_pow, ← Real.exp_nat_mul (β * B) N,
    ← Real.exp_nat_mul z N]
  have he₁ : β * B * (N : ℝ) = (N : ℝ) * (β * B) := by ring
  have he₂ : z * (N : ℝ) = (N : ℝ) * z := by ring
  rw [he₁, he₂]
  ring

theorem grandOn_summable {φ : Potential} (hφ : Continuous φ)
    (hb : IsBoundedPotential φ) {B β : ℝ} (hstable : StableWith φ B)
    (hβ : 0 ≤ β) {A : Set Position} (hAfin : volume A ≠ ⊤) (z : ℝ) :
    Summable (fun N : ℕ => Real.exp (z * N) * partitionOn φ A N β) :=
  Summable.of_nonneg_of_le
    (fun N => mul_nonneg (Real.exp_pos _).le (partitionOn_nonneg φ A N β))
    (grandOn_term_upper hφ hb hstable hβ hAfin z)
    (Real.summable_pow_div_factorial _)

theorem grandOn_bounds {φ : Potential} (hφ : Continuous φ)
    (hb : IsBoundedPotential φ) {B β : ℝ} (hstable : StableWith φ B)
    (hβ : 0 ≤ β) {A : Set Position} (hAfin : volume A ≠ ⊤) (z : ℝ) :
    1 ≤ grandOn φ A β z ∧ grandOn φ A β z ≤ Real.exp (volume.real A * Real.exp (β * B + z)) := by
  have hs := grandOn_summable hφ hb hstable hβ hAfin z
  constructor
  · have hh := hs.le_tsum 0 (fun N _ => mul_nonneg (Real.exp_pos _).le (partitionOn_nonneg φ A N β))
    convert hh using 1 <;> first | rfl | norm_num [partitionOn_zero]
  · have hh := hs.tsum_le_tsum (grandOn_term_upper hφ hb hstable hβ hAfin z)
      (Real.summable_pow_div_factorial _)
    have he : (∑' N : ℕ, (volume.real A * Real.exp (β * B + z)) ^ N / N.factorial) =
        Real.exp (volume.real A * Real.exp (β * B + z)) := by
      rw [Real.exp_eq_exp_ℝ, NormedSpace.exp_eq_tsum_div]
    exact hh.trans_eq he

theorem grandOn_union_lower {φ : Potential} (hφ : Continuous φ)
    (hb : IsBoundedPotential φ) {B β : ℝ} (hstable : StableWith φ B) (hβ : 0 ≤ β)
    {A D : Set Position} (hA : MeasurableSet A) (hD : MeasurableSet D) (had : Disjoint A D)
    (hAfin : volume A ≠ ⊤) (hDfin : volume D ≠ ⊤)
    (hc : ∀ x ∈ A, ∀ y ∈ D, φ (nndist x y) ≤ 0) (z : ℝ) :
    grandOn φ A β z * grandOn φ D β z ≤ grandOn φ (A ∪ D) β z := by
  have hUfin : volume (A ∪ D) ≠ ⊤ := (measure_union_le A D).trans_lt
    (ENNReal.add_lt_top.mpr ⟨hAfin.lt_top, hDfin.lt_top⟩) |>.ne
  have hsA := grandOn_summable hφ hb hstable hβ hAfin z
  have hsD := grandOn_summable hφ hb hstable hβ hDfin z
  have hsU := grandOn_summable hφ hb hstable hβ hUfin z
  have hcov (N : ℕ) :
      (∑ m ∈ Finset.range (N + 1), (Real.exp (z * m) * partitionOn φ A m β) *
        (Real.exp (z * ((N - m : ℕ) : ℝ)) * partitionOn φ D (N - m) β)) ≤
          Real.exp (z * N) * partitionOn φ (A ∪ D) N β := by
    have he : (∑ m ∈ Finset.range (N + 1), (Real.exp (z * m) * partitionOn φ A m β) *
        (Real.exp (z * ((N - m : ℕ) : ℝ)) * partitionOn φ D (N - m) β)) =
        Real.exp (z * N) * ∑ m ∈ Finset.range (N + 1),
          partitionOn φ A m β * partitionOn φ D (N - m) β := by
      rw [Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro m hm
      have hmN : m ≤ N := Nat.le_of_lt_succ (Finset.mem_range.mp hm)
      have he : z * (m : ℝ) + z * (N - m : ℕ) = z * N := by
        rw [Nat.cast_sub hmN]; ring
      calc
        _ = (Real.exp (z * m) * Real.exp (z * ((N - m : ℕ) : ℝ))) *
            (partitionOn φ A m β * partitionOn φ D (N - m) β) := by ring
        _ = _ := by rw [← Real.exp_add, he]
    rw [he]
    exact mul_le_mul_of_nonneg_left (partitionOn_union_convolution hφ hb hβ hA hD had hAfin hDfin hc N)
      (Real.exp_pos _).le
  have hnn (N : ℕ) : 0 ≤ ∑ m ∈ Finset.range (N + 1),
      (Real.exp (z * m) * partitionOn φ A m β) *
        (Real.exp (z * ((N - m : ℕ) : ℝ)) * partitionOn φ D (N - m) β) := by
    apply Finset.sum_nonneg
    intro m _
    exact mul_nonneg (mul_nonneg (Real.exp_pos _).le (partitionOn_nonneg φ A m β))
      (mul_nonneg (Real.exp_pos _).le (partitionOn_nonneg φ D (N - m) β))
  rw [grandOn, grandOn,
    tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm hsA.norm hsD.norm]
  exact (Summable.of_nonneg_of_le hnn hcov hsU).tsum_le_tsum hcov hsU

theorem grandOn_mono {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ)
    {B β : ℝ} (hstable : StableWith φ B) (hβ : 0 ≤ β) {A D : Set Position}
    (hsub : A ⊆ D) (hDfin : volume D ≠ ⊤) (z : ℝ) : grandOn φ A β z ≤ grandOn φ D β z := by
  have hAfin : volume A ≠ ⊤ := (measure_mono hsub).trans_lt hDfin.lt_top |>.ne
  exact (grandOn_summable hφ hb hstable hβ hAfin z).tsum_le_tsum
    (fun N => mul_le_mul_of_nonneg_left (partitionOn_mono hφ hb β N hsub hDfin) (Real.exp_pos _).le)
    (grandOn_summable hφ hb hstable hβ hDfin z)

theorem grandOn_translate (φ : Potential) (A : Set Position) (a : Position) (β z : ℝ) :
    grandOn φ (translateSet a A) β z = grandOn φ A β z := by
  simp only [grandOn, partitionOn_translate]

theorem grandOn_cube (φ : Potential) (L t β h : ℝ) :
    grandOn φ (cube L) β (t * h) = grandPartition φ t L β h := by
  simp only [grandOn, grandPartition, partitionOn_cube]

end RadialTransition

end

end OAI