| import OAI.Probability.ContinuumTransition.PhysicalPerturbation |
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| namespace OAI |
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| section |
| open scoped Topology BigOperators |
| open Filter MeasureTheory Set |
| namespace ContinuumTemperature |
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| theorem finiteCanonical_lower_coercive {t β L : ℝ} (ht : 0 ≤ t) (hβ : 0 ≤ β) (hL : 0 < L) |
| {σ R : ℕ → ℝ} (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hsmall : 2197*t*kernelHeight*(∑' n, σ n)/2 ≤ t^2/8) (N : ℕ) : |
| β*((t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/(4+2/L)^3)*((N : ℝ)/L^3)^2 + |
| ((N : ℝ)/L^3)*Real.log ((N : ℝ)/L^3)-(β*t^2/2+1)*((N : ℝ)/L^3) ≤ |
| finiteCanonical (physicalProfile t σ R) β L N := by |
| by_cases hN : N = 0 |
| · simp [hN] |
| have hφ := measurable_radialPotential (measurable_physicalProfile t hσ hR hs) |
| have hst := physicalProfile_stable ht hσ hR hs hsmall |
| have hh := neg_log_partition_lower hφ hβ hL hst hN |
| (physicalProfile_energy_coercive ht hL.le hσ hR hs hsmall N) |
| have hd := div_le_div_of_nonneg_right hh (pow_pos hL 3).le |
| unfold finiteCanonical |
| convert hd using 1; try rfl |
| field_simp |
| ring |
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| theorem canonicalDensity_lower_coercive {t β d : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) |
| {σ R : ℕ → ℝ} (hσ : ∀ n, 0 ≤ σ n) (hR : ∀ n, 1 ≤ R n) (hs : Summable σ) |
| (hsmall : 2197*t*kernelHeight*(∑' n, σ n)/2 ≤ t^2/8) (hd : 0 ≤ d) : |
| β*((t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/64)*d^2 + |
| d*Real.log d-(β*t^2/2+1)*d ≤ canonicalDensity (physicalProfile t σ R) β d := by |
| let : Fact (0 ≤ d) := ⟨hd⟩ |
| have ht0 : 0 ≤ t := (lt_trans zero_lt_one ht).le |
| have hφ := measurable_radialPotential (measurable_physicalProfile t hσ hR hs) |
| have hst := physicalProfile_stable ht0 hσ hR hs hsmall |
| have hcore (r : ℝ) (_ : 0 < r) := physicalProfile_le_core ht0 hσ hR r |
| have hlim := finiteCanonical_tendsto hφ ht hβ hst hcore hd |
| have hi := tendsto_inv_atTop_zero.comp (densityFilter_length d) |
| have hc : Tendsto (fun p : ℝ × ℕ => |
| (t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/(4+2/p.1)^3) |
| (densityFilter d) (𝓝 ((t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/64)) := by |
| have hh := (tendsto_const_nhds (x := t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)).div |
| ((tendsto_const_nhds.add (hi.const_mul 2)).pow 3) |
| (by norm_num : (4+2*(0 : ℝ))^3 ≠ 0) |
| change Tendsto (fun p : ℝ × ℕ => |
| (t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/(4+2*(p.1)⁻¹)^3) |
| (densityFilter d) (𝓝 ((t^2/8-2197*t*kernelHeight*(∑' n, σ n)/2)/((4+2*(0 : ℝ))^3))) at hh |
| simpa only [mul_zero, add_zero, div_eq_mul_inv, Function.comp_def, Pi.mul_apply, Pi.inv_apply, |
| show (4 : ℝ)^3 = 64 by norm_num] using hh |
| have he := (((hc.const_mul β).mul ((densityFilter_density d).pow 2)).add |
| (Real.continuous_mul_log.continuousAt.tendsto.comp (densityFilter_density d))).sub |
| ((densityFilter_density d).const_mul (β*t^2/2+1)) |
| apply le_of_tendsto_of_tendsto he hlim |
| filter_upwards [(densityFilter_length d).eventually (eventually_gt_atTop 0)] with p hp |
| exact finiteCanonical_lower_coercive ht0 hβ hp hσ hR hs hsmall p.2 |
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| end ContinuumTemperature |
| end |
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| end OAI |
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