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9b6e014 | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Probability.ContinuumTransition.GrandPartition
namespace OAI
section
open scoped Topology BigOperators ENNReal
open Filter MeasureTheory Set
namespace ContinuumTemperature
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
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
end OAI
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