| import OAI.Probability.ContinuumTransition.GrandPartition |
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| namespace OAI |
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| section |
| open scoped Topology BigOperators ENNReal |
| open Filter MeasureTheory Set |
| namespace ContinuumTemperature |
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|
| theorem log_factorial_lower (N : ℕ) : |
| (N : ℝ) * Real.log N - N ≤ Real.log (N.factorial : ℝ) := by |
| by_cases hN : N = 0 |
| · simp [hN] |
| · have hpos : (0 : ℝ) < N := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hN) |
| have hf : (0 : ℝ) < N.factorial := Nat.cast_pos.mpr (Nat.factorial_pos N) |
| have hh := Real.pow_div_factorial_le_exp (N : ℝ) hpos.le N |
| have hl := Real.log_le_log (div_pos (pow_pos hpos _) hf) hh |
| rw [Real.log_div (pow_ne_zero _ hpos.ne') hf.ne', Real.log_pow, Real.log_exp] at hl |
| linarith |
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|
| theorem boltzmann_le_of_energy_lower {φ : ℝ → ℝ} {β E : ℝ} (hβ : 0 ≤ β) |
| {N : ℕ} {x : Fin N → Space} (hE : (E : EReal) ≤ energy φ x) : |
| boltzmann φ β x ≤ Real.exp (-β * E) := by |
| unfold boltzmann |
| split_ifs with htop |
| · exact (Real.exp_pos _).le |
| · apply Real.exp_le_exp.mpr |
| have he := EReal.toReal_le_toReal hE (EReal.coe_ne_bot _) htop |
| simp only [EReal.toReal_coe] at he |
| exact mul_le_mul_of_nonpos_left he (neg_nonpos.mpr hβ) |
|
|
| theorem partition_le_factorial_of_energy_bound {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {β B L E : ℝ} (hβ : 0 ≤ β) (hL : 0 ≤ L) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) {N : ℕ} (hN : N ≠ 0) |
| (hE : ∀ x : Fin N → Space, (∀ i, x i ∈ cube L) → (E : EReal) ≤ energy φ x) : |
| partition φ β L N ≤ (L ^ 3) ^ N / (N.factorial : ℝ) * Real.exp (-β * E) := by |
| let S := {x : Fin N → Space | ∀ i, x i ∈ cube L} |
| have hS : MeasurableSet S := (isCompact_configurations L N).measurableSet |
| have hfin : IsFiniteMeasure (volume.restrict S) := |
| ⟨by simpa only [Measure.restrict_apply_univ] using |
| (isCompact_configurations L N).measure_lt_top (μ := volume)⟩ |
| have hh : ∀ᵐ x ∂volume.restrict S, boltzmann φ β x ≤ Real.exp (-β * E) := by |
| apply (ae_restrict_iff' hS).mpr |
| exact ae_of_all _ (fun x hx => boltzmann_le_of_energy_lower hβ (hE x hx)) |
| have hi := integral_mono_ae (integrableOn_boltzmann hφ hβ hstable L N) |
| (integrable_const (Real.exp (-β * E))) hh |
| have hc : (∫ x in S, Real.exp (-β * E)) = L ^ (3 * N) * Real.exp (-β * E) := by |
| rw [setIntegral_const, Measure.real, volume_configurations, ENNReal.toReal_pow, |
| ENNReal.toReal_ofReal hL, smul_eq_mul] |
| rw [hc] at hi |
| unfold partition |
| rw [ite_eq_right hN] |
| calc |
| _ ≤ (1 / (N.factorial : ℝ)) * (L ^ (3 * N) * Real.exp (-β * E)) := |
| mul_le_mul_of_nonneg_left hi (by positivity) |
| _ = _ := by rw [pow_mul]; ring |
|
|
| theorem neg_log_partition_lower {φ : ℝ → ℝ} |
| (hφ : Measurable (radialPotential φ)) {β B L E : ℝ} (hβ : 0 ≤ β) (hL : 0 < L) |
| (hstable : ∀ (N : ℕ) (x : Fin N → Space), |
| ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x) {N : ℕ} (hN : N ≠ 0) |
| (hE : ∀ x : Fin N → Space, (∀ i, x i ∈ cube L) → (E : EReal) ≤ energy φ x) : |
| β * E + N * Real.log ((N : ℝ) / L ^ 3) - N ≤ -Real.log (partition φ β L N) := by |
| have hnpos : (0 : ℝ) < N := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hN) |
| have hfp : (0 : ℝ) < N.factorial := Nat.cast_pos.mpr (Nat.factorial_pos N) |
| have hLp : 0 < L ^ 3 := pow_pos hL _ |
| have hZ := partition_pos hφ hβ hL hstable N |
| have hu := partition_le_factorial_of_energy_bound hφ hβ hL.le hstable hN hE |
| have hl := Real.log_le_log hZ hu |
| rw [Real.log_mul (div_pos (pow_pos hLp _) hfp).ne' (Real.exp_pos _).ne', |
| Real.log_div (pow_pos hLp _).ne' hfp.ne', Real.log_pow, Real.log_exp] at hl |
| rw [Real.log_div hnpos.ne' hLp.ne'] |
| have hf := log_factorial_lower N |
| nlinarith |
|
|
| end ContinuumTemperature |
| end |
|
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| end OAI |
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