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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 | import OAI.Probability.RadialTransition.TetraIntegral
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
lemma partitionOn_one (φ : Potential) {b : ℝ} (hb : 0 ≤ b) (β : ℝ) :
partitionOn φ (cube b) 1 β = b^3 := by
simp [partitionOn, rawPartition, orderedEnergy, Measure.real, Measure.pi_univ,
volume_cube, ENNReal.toReal_ofReal hb]
def blobTetraConstant (b : ℝ) : ℝ := (b/2)^3/((100000:ℝ)^9*24)
lemma blobTetraConstant_pos {b : ℝ} (hb : 0 < b) : 0 < blobTetraConstant b := by
unfold blobTetraConstant
positivity
lemma partitionOn_blob_tetra {t b β : ℝ} (hb : 0 ≤ b) (ht : 3*microLength t ≤ b/2) :
blobTetraConstant b*Real.exp (-9*t) ≤ partitionOn (corePotential t) (cube b) 4 β := by
have hsub : cube (b/2+3*microLength t) ⊆ cube b := by
intro x hx k
have hh := hx k
exact ⟨hh.1, by linarith [hh.2]⟩
have hp := partitionOn_mono (continuous_corePotential t) (bounded_corePotential t) β 4 hsub
(by rw [volume_cube]; finiteness)
have h1 := partitionOn_tetra_lower t (b/2) β (by positivity)
have he : (b/2)^3*tetraEpsilon t^9/24 = blobTetraConstant b*Real.exp (-9*t) := by
dsimp [tetraEpsilon, microLength, blobTetraConstant]
rw [div_pow, div_pow, ← Real.exp_nat_mul]
norm_num only [Nat.cast_ofNat]
rw [show (9:ℝ)*(-t) = -9*t by ring]
ring
rw [he] at h1
exact h1.trans hp
lemma core_local_grand_monomer {t b β h : ℝ} (hb : 0 ≤ b) (hβ : 0 ≤ β) :
Real.exp (t*h)*b^3 ≤ grandOn (corePotential t) (cube b) β (t*h) := by
have hs := grandOn_summable (continuous_corePotential t) (bounded_corePotential t)
(stable_corePotential t) hβ (by rw [volume_cube]; finiteness : volume (cube b) ≠ ⊤) (t*h)
have hh := hs.le_tsum 1 (fun n _ => mul_nonneg (Real.exp_pos _).le (partitionOn_nonneg _ _ _ _))
simpa only [Nat.cast_one, mul_one, partitionOn_one _ hb, grandOn] using hh
lemma core_local_grand_tetra {t b β h : ℝ} (hb : 0 ≤ b) (hβ : 0 ≤ β)
(ht : 3*microLength t ≤ b/2) :
blobTetraConstant b*Real.exp (t*(4*h-9)) ≤ grandOn (corePotential t) (cube b) β (t*h) := by
have hs := grandOn_summable (continuous_corePotential t) (bounded_corePotential t)
(stable_corePotential t) hβ (by rw [volume_cube]; finiteness : volume (cube b) ≠ ⊤) (t*h)
have hh := hs.le_tsum 4 (fun n _ => mul_nonneg (Real.exp_pos _).le (partitionOn_nonneg _ _ _ _))
have hl := mul_le_mul_of_nonneg_left (partitionOn_blob_tetra hb ht (β := β)) (Real.exp_pos (t*h*4)).le
have he : Real.exp (t*h*4)*(blobTetraConstant b*Real.exp (-9*t)) =
blobTetraConstant b*Real.exp (t*(4*h-9)) := by
rw [mul_left_comm, ← Real.exp_add]
congr 2
ring
rw [he] at hl
exact hl.trans hh
lemma core_local_log_monomer {t b β h : ℝ} (ht : 0 < t) (hb : 0 < b) (hβ : 0 ≤ β) :
h+Real.log (b^3)/t ≤ Real.log (grandOn (corePotential t) (cube b) β (t*h))/t := by
have hh := Real.log_le_log (mul_pos (Real.exp_pos _) (pow_pos hb 3)) (core_local_grand_monomer hb.le hβ (t := t) (h := h))
rw [Real.log_mul (Real.exp_ne_zero _) (pow_ne_zero _ hb.ne'), Real.log_exp] at hh
apply (le_div_iff₀ ht).mpr
calc
_ = t*h+Real.log (b^3) := by field_simp
_ ≤ _ := hh
lemma core_local_log_tetra {t b β h : ℝ} (ht : 0 < t) (hb : 0 < b) (hβ : 0 ≤ β)
(htb : 3*microLength t ≤ b/2) :
4*h-9+Real.log (blobTetraConstant b)/t ≤ Real.log (grandOn (corePotential t) (cube b) β (t*h))/t := by
have hh := Real.log_le_log (mul_pos (blobTetraConstant_pos hb) (Real.exp_pos _)) (core_local_grand_tetra hb.le hβ htb (h := h))
rw [Real.log_mul (blobTetraConstant_pos hb).ne' (Real.exp_ne_zero _), Real.log_exp] at hh
apply (le_div_iff₀ ht).mpr
calc
_ = Real.log (blobTetraConstant b)+t*(4*h-9) := by field_simp; ring
_ ≤ _ := hh
end RadialTransition
end
end OAI
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