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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 | import OAI.Probability.RadialTransition.CorePressureUpper
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
lemma packing_dilated_limit {ν b : ℝ} (hν : 0 ≤ ν) :
Tendsto (fun L : ℝ => (packingNumber L:ℝ)/((1+ν)*L+b)^3) atTop
(nhds (packingDensity/(1+ν)^3)) := by
have hn : (1+ν)^3 ≠ 0 := by positivity
have hh : Tendsto (fun L : ℝ => (1+ν+b*L⁻¹)^3) atTop (nhds ((1+ν)^3)) := by
simpa using ((tendsto_const_nhds (x := 1+ν)).add (tendsto_inv_atTop_zero.const_mul b)).pow 3
apply (packingDensity_limit.div hh hn).congr'
filter_upwards [eventually_gt_atTop (0:ℝ)] with L hL
have he : 1+ν+b*L⁻¹ = ((1+ν)*L+b)/L := by field_simp
dsimp only [Pi.div_apply]
rw [he, div_pow]
field_simp
lemma corePressure_packing_lower {ν b t β h : ℝ} (ht : 1 ≤ t) (hb : 0 ≤ b)
(hν : 0 ≤ ν) (hsep : 1/t+4*b ≤ ν) (hβ : 0 ≤ β) :
packingDensity/(1+ν)^3 * (Real.log (grandOn (corePotential t) (cube b) β (t*h))/t) ≤
grandPressure (corePotential t) t β h := by
have ht0 : 0 < t := by linarith
have hc := continuous_corePotential t
have hbound := bounded_corePotential t
have hstable := stable_corePotential t
have hlocal := (grandOn_bounds hc hbound hstable hβ (by rw [volume_cube]; finiteness : volume (cube b) ≠ ⊤) (t*h)).1
have hlimL : Tendsto (fun L : ℝ => (1+ν)*L+b) atTop atTop :=
tendsto_atTop_add_const_right atTop b (Tendsto.const_mul_atTop (by linarith : 0 < 1+ν) tendsto_id)
have hright := ((grandPressure_spec hc hbound hstable ht0 hβ (core_nonpos_outer ht) h).1).comp hlimL
have hleft := (packing_dilated_limit (b := b) hν).mul_const
(Real.log (grandOn (corePotential t) (cube b) β (t*h))/t)
apply le_of_tendsto_of_tendsto hleft hright
filter_upwards [eventually_gt_atTop (0:ℝ)] with L hL
have hS : 0 < (1+ν)*L+b := by positivity
have hprod := core_grand_packing_lower (L := L) ht0 hb hν hsep hβ (h := h)
have hlog := Real.log_le_log (pow_pos (show 0 < grandOn (corePotential t) (cube b) β (t*h) by linarith) _) hprod
rw [Real.log_pow] at hlog
rw [grandOn_cube] at hlog
rw [grandOn_cube] at hlog
have hden : 0 < t*((1+ν)*L+b)^3 := by positivity
dsimp only [finiteGrandPressure, Function.comp_def]
apply (le_div_iff₀ hden).mpr
rw [grandOn_cube]
calc
_ = (packingNumber L:ℝ)*Real.log (grandPartition (corePotential t) t b β h) := by
field_simp
_ ≤ _ := hlog
end RadialTransition
end
end OAI
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