File size: 7,146 Bytes
a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | 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
|