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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 [h,ite_true]
· 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 h]
· 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 <;> simp only [Finset.sum_const_zero]
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
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