| import OAI.Probability.RadialTransition.PackingBlob |
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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 core_nonpos_outer {t : ℝ} (ht : 1 ≤ t) : |
| ∀ r : ℝ≥0, 2 ≤ (r:ℝ) → corePotential t r ≤ 0 := by |
| intro r hr |
| have ht0 : 0 < t := by linarith |
| have hinv : 1/t ≤ 1 := (div_le_one ht0).mpr ht |
| rw [corePotential, coreReal_zero_outer ht0 (by linarith)] |
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| lemma core_finitePressure_upper {t α L β h : ℝ} (hL : 1 ≤ L) |
| (ht0 : 0 < t) (ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1) |
| (hβ : 0 ≤ β) (hα : 0 ≤ α) (h1 : h ≤ α) (h4 : 4*h-9 ≤ α) : |
| finiteGrandPressure (corePotential t) t L β h ≤ |
| α*((packingNumber L:ℝ)/L^3)+(216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t := by |
| have hL0 : 0 < L := by linarith |
| have hh := core_grand_upper hL0.le ht0.le ht hβ (mul_nonneg ht0.le hα) h1 h4 |
| have hp := (grandPartition_bounds (continuous_corePotential t) (bounded_corePotential t) |
| (stable_corePotential t) hL0.le hβ t h).1 |
| have hreal := ENNReal.toReal_mono (by finiteness : |
| ENNReal.ofReal (Real.exp (t*α*packingNumber L)) * |
| ((15001:ℝ≥0∞)^(cubeCells L).card * ENNReal.ofReal (Real.exp (L^3*Real.exp (t*h-β*t^2/2)))) ≠ ⊤) hh |
| simp only [ENNReal.toReal_ofReal (show 0 ≤ grandPartition (corePotential t) t L β h by linarith), |
| ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_ofReal (Real.exp_pos _).le, |
| ENNReal.toReal_ofNat] at hreal |
| have hlog := Real.log_le_log (show 0 < grandPartition (corePotential t) t L β h by linarith) hreal |
| rw [Real.log_mul (Real.exp_ne_zero _) (by positivity), Real.log_exp, |
| Real.log_mul (by positivity) (Real.exp_ne_zero _), Real.log_pow, Real.log_exp] at hlog |
| have hc := mul_le_mul_of_nonneg_right (card_cubeCells hL) (show 0 ≤ Real.log (15001:ℝ) from Real.log_nonneg (by norm_num)) |
| have hden : 0 < t*L^3 := by positivity |
| dsimp only [finiteGrandPressure] |
| apply (div_le_iff₀ hden).mpr |
| have heq : (α*((packingNumber L:ℝ)/L^3)+(216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t)*(t*L^3) = |
| t*α*packingNumber L+216*L^3*Real.log 15001+L^3*Real.exp (t*h-β*t^2/2) := by |
| field_simp |
| ring |
| rw [heq] |
| linarith |
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| lemma corePressure_upper {t α β h : ℝ} (ht0 : 1 ≤ t) |
| (ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1) |
| (hβ : 0 ≤ β) (hα : 0 ≤ α) (h1 : h ≤ α) (h4 : 4*h-9 ≤ α) : |
| grandPressure (corePotential t) t β h ≤ |
| α*packingDensity+(216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t := by |
| have htpos : 0 < t := by linarith |
| have hl := (grandPressure_spec (continuous_corePotential t) (bounded_corePotential t) |
| (stable_corePotential t) htpos hβ (core_nonpos_outer ht0) h).1 |
| have hr := packingDensity_limit.const_mul α |>.add_const ((216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t) |
| apply le_of_tendsto_of_tendsto hl hr |
| filter_upwards [eventually_ge_atTop (1:ℝ)] with L hL |
| exact core_finitePressure_upper hL htpos ht hβ hα h1 h4 |
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| end RadialTransition |
| end |
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| end OAI |
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