| import OAI.Probability.FreeIsing.Posterior.FinitePosterior | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace Problem367.FinitePosterior | |
| variable {S V : Type*} [Fintype S] [Fintype V] | |
| def fieldEnergy (energy : S → ℝ) (spin : S → V → ℝ) (x : V → ℝ) (s : S) : ℝ := | |
| energy s + ∑ v, x v * spin s v | |
| def fieldMean (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ) | |
| (x : V → ℝ) : ℝ := | |
| (∑ s, observable s * Real.exp (fieldEnergy energy spin x s)) / | |
| (∑ s, Real.exp (fieldEnergy energy spin x s)) | |
| def fieldMass (energy : S → ℝ) (spin : S → V → ℝ) (x : V → ℝ) (s : S) : ℝ := | |
| Real.exp (fieldEnergy energy spin x s) / (∑ r, Real.exp (fieldEnergy energy spin x r)) | |
| lemma fieldMean_eq_sum_fieldMass (energy : S → ℝ) (spin : S → V → ℝ) | |
| (observable : S → ℝ) (x : V → ℝ) : | |
| fieldMean energy spin observable x = ∑ s, observable s * fieldMass energy spin x s := by | |
| simp only [fieldMean, fieldMass, ← mul_div_assoc, ← Finset.sum_div] | |
| lemma fieldMean_along_line (energy : S → ℝ) (spin : S → V → ℝ) | |
| (observable : S → ℝ) (x direction : V → ℝ) (t : ℝ) : | |
| fieldMean energy spin observable (fun v => x v + t * direction v) = | |
| mean (fieldEnergy energy spin x) (fun s => ∑ v, direction v * spin s v) | |
| observable t := by | |
| have heq (s : S) : | |
| fieldEnergy energy spin (fun v => x v + t * direction v) s = | |
| fieldEnergy energy spin x s + t * ∑ v, direction v * spin s v := by | |
| simp only [fieldEnergy, add_mul, Finset.sum_add_distrib, | |
| Finset.mul_sum, mul_assoc] | |
| ring | |
| simp only [fieldMean, heq, mean, partition, weight] | |
| lemma mean_fieldEnergy_zero (energy : S → ℝ) (spin : S → V → ℝ) | |
| (signal observable : S → ℝ) (x : V → ℝ) : | |
| mean (fieldEnergy energy spin x) signal observable 0 = | |
| fieldMean energy spin observable x := by | |
| simp only [mean, partition, weight, zero_mul, add_zero, fieldMean] | |
| lemma fieldPartition_pos [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ) | |
| (x : V → ℝ) : 0 < ∑ s, Real.exp (fieldEnergy energy spin x s) := by | |
| exact Finset.sum_pos (fun s _ => Real.exp_pos _) Finset.univ_nonempty | |
| lemma fieldMass_pos [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ) | |
| (x : V → ℝ) (s : S) : 0 < fieldMass energy spin x s := | |
| div_pos (Real.exp_pos _) (fieldPartition_pos energy spin x) | |
| theorem continuous_fieldMean [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ) | |
| (observable : S → ℝ) : Continuous (fieldMean energy spin observable) := by | |
| unfold fieldMean | |
| apply Continuous.div | |
| · unfold fieldEnergy | |
| fun_prop | |
| · unfold fieldEnergy | |
| fun_prop | |
| · intro x | |
| exact ne_of_gt (fieldPartition_pos energy spin x) | |
| lemma fieldMean_const [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ) | |
| (x : V → ℝ) (c : ℝ) : fieldMean energy spin (fun _ => c) x = c := by | |
| rw [← mean_fieldEnergy_zero energy spin (fun _ => 0), mean_const] | |
| lemma fieldMean_spin_mem_Ioo [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ) | |
| (observable : S → ℝ) (x : V → ℝ) | |
| (hs : ∀ s, observable s = -1 ∨ observable s = 1) | |
| (hn : ∃ s, observable s = -1) (hp : ∃ s, observable s = 1) : | |
| fieldMean energy spin observable x ∈ Set.Ioo (-1 : ℝ) 1 := by | |
| rw [← mean_fieldEnergy_zero energy spin (fun _ => 0)] | |
| exact spin_mean_mem_Ioo _ _ _ _ hs hn hp | |
| theorem hasDerivAt_fieldMean [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ) | |
| (observable : S → ℝ) (x direction : V → ℝ) : | |
| HasDerivAt (fun t => fieldMean energy spin observable (fun v => x v + t * direction v)) | |
| (fieldMean energy spin (fun s => observable s * ∑ v, direction v * spin s v) x - | |
| fieldMean energy spin observable x * | |
| fieldMean energy spin (fun s => ∑ v, direction v * spin s v) x) 0 := by | |
| simp_rw [fieldMean_along_line] | |
| simpa only [mean_fieldEnergy_zero] using | |
| hasDerivAt_mean (fieldEnergy energy spin x) | |
| (fun s => ∑ v, direction v * spin s v) observable 0 | |
| theorem hasDerivAt_fieldMean_coordinate [Nonempty S] [DecidableEq V] | |
| (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ) | |
| (x : V → ℝ) (u : V) : | |
| HasDerivAt | |
| (fun t => fieldMean energy spin observable (fun v => x v + if v = u then t else 0)) | |
| (fieldMean energy spin (fun s => observable s * spin s u) x - | |
| fieldMean energy spin observable x * fieldMean energy spin (fun s => spin s u) x) 0 := by | |
| have h := hasDerivAt_fieldMean energy spin observable x (fun v => if v = u then 1 else 0) | |
| simpa only [mul_ite, mul_one, mul_zero, ite_mul, one_mul, zero_mul, | |
| Finset.sum_ite_eq', Finset.mem_univ, ite_true] using h | |
| end Problem367.FinitePosterior | |
| end | |
| end OAI | |