| import OAI.Probability.RadialTransition.TetraIntegral |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
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| 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] |
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| def blobTetraConstant (b : ℝ) : ℝ := (b/2)^3/((100000:ℝ)^9*24) |
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| lemma blobTetraConstant_pos {b : ℝ} (hb : 0 < b) : 0 < blobTetraConstant b := by |
| unfold blobTetraConstant |
| positivity |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end RadialTransition |
| end |
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| end OAI |
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