| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.HeatBathStationary |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def kernelEvent {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) : ℝ := ∑ y : Config n, if A y then discreteKernel β J k x y else 0 |
|
|
| theorem kernelEvent_nonneg {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) : 0 ≤ kernelEvent β J A k x := by |
| apply Finset.sum_nonneg |
| intro y _ |
| split_ifs |
| · exact discreteKernel_nonneg β J k x y |
| · exact le_rfl |
|
|
| theorem kernelEvent_le_one {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| (A : Config n → Prop) [DecidablePred A] (k : ℕ) (x : Config n) : kernelEvent β J A k x ≤ 1 := by |
| rw [← discreteKernel_sum hn β J k x] |
| apply Finset.sum_le_sum |
| intro y _ |
| split_ifs |
| · exact le_rfl |
| · exact discreteKernel_nonneg β J k x y |
|
|
| theorem kernelEvent_zero {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (x : Config n) : kernelEvent β J A 0 x = if A x then 1 else 0 := by |
| classical |
| unfold kernelEvent |
| simp only [discreteKernel] |
| by_cases h : A x |
| · rw [ite_eq_left h] |
| have he (y : Config n) : (if A y then if x=y then (1:ℝ) else 0 else 0) = if x=y then 1 else 0 := by |
| by_cases hxy : x=y |
| · subst y |
| · simp only [ite_eq_right hxy,ite_self] |
| simp_rw [he] |
| simp only [Finset.sum_ite_eq,Finset.mem_univ,ite_true] |
| · rw [ite_eq_right h] |
| apply Finset.sum_eq_zero |
| intro y _ |
| by_cases hxy : x=y |
| · subst y |
| · simp only [ite_eq_right hxy,ite_self] |
|
|
| theorem kernelEvent_succ {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) : kernelEvent β J A (k+1) x = |
| ∑ z : Config n, heatBath β J x z*kernelEvent β J A k z := by |
| unfold kernelEvent |
| simp only [discreteKernel,Finset.mul_sum,mul_ite,mul_zero] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro y _ |
| split_ifs < |
|
|
| theorem kernelEvent_stationary {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| (A : Config n → Prop) [DecidablePred A] (k : ℕ) : |
| (∑ x : Config n, gibbs β J x*kernelEvent β J A k x) = |
| ∑ y : Config n, if A y then gibbs β J y else 0 := by |
| unfold kernelEvent |
| simp only [Finset.mul_sum,mul_ite,mul_zero] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro y _ |
| split_ifs |
| · exact discreteKernel_stationary hn β J k y |
| · exact Finset.sum_const_zero |
|
|
| def occupation {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) : ℝ := ∑ i ∈ Finset.range (k+1), kernelEvent β J A i x |
|
|
| theorem occupation_nonneg {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) : 0 ≤ occupation β J A k x := |
| Finset.sum_nonneg (fun i _ => kernelEvent_nonneg β J A i x) |
|
|
| theorem one_le_occupation {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) (hx : A x) : 1 ≤ occupation β J A k x := by |
| have H := Finset.single_le_sum (f := fun i => kernelEvent β J A i x) |
| (fun i (_ : i ∈ Finset.range (k+1)) => kernelEvent_nonneg β J A i x) |
| (show 0 ∈ Finset.range (k+1) by simp) |
| simpa only [kernelEvent_zero,ite_eq_left hx,occupation] using H |
|
|
| theorem occupation_recursion {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A] |
| (k : ℕ) (x : Config n) : occupation β J A (k+1) x = |
| kernelEvent β J A 0 x+∑ z : Config n, heatBath β J x z*occupation β J A k z := by |
| unfold occupation |
| rw [Finset.sum_range_succ'] |
| simp_rw [kernelEvent_succ,Finset.mul_sum] |
| rw [Finset.sum_comm] |
| ring |
| |
| theorem occupation_stationary {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| (A : Config n → Prop) [DecidablePred A] (k : ℕ) : |
| (∑ x : Config n, gibbs β J x*occupation β J A k x) = |
| ((k+1:ℕ):ℝ)*(∑ y : Config n, if A y then gibbs β J y else 0) := by |
| unfold occupation |
| simp only [Finset.mul_sum] |
| rw [Finset.sum_comm] |
| simp_rw [kernelEvent_stationary hn] |
| simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] |
| rw [Finset.mul_sum] |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
