| import OAI.Probability.RadialTransition.CellMoments |
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| namespace OAI |
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| universe uI uX uY uZ |
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| noncomputable section |
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| open MeasureTheory Filter Set |
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| open scoped ContDiff BigOperators NNReal ENNReal |
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| namespace RadialTransition |
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| section AbstractCells |
| variable {I : Type uI} [Fintype I] [DecidableEq I] {X : I → Type uX} |
| |
| def cellHamiltonian (U : ∀ i, X i → ℝ) (W : ∀ i j, X i → X j → ℝ) |
| (x : ∀ i, X i) : ℝ := |
| ∑ i, U i (x i) + (1/2:ℝ) * ∑ i, ∑ j, if i = j then 0 else W i j (x i) (x j) |
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| lemma double_sum_update (w : ∀ i j, X i → X j → ℝ) |
| (hdiag : ∀ i a b, w i i a b = 0) |
| (hsym : ∀ i j a b, w i j a b = w j i b a) |
| (x : ∀ i, X i) (k : I) (s t : X k) : |
| (∑ i, ∑ j, w i j (Function.update x k s i) (Function.update x k s j)) - |
| (∑ i, ∑ j, w i j (Function.update x k t i) (Function.update x k t j)) = |
| 2 * ∑ j, (w k j s (x j) - w k j t (x j)) := by |
| have hpoint (i j : I) : |
| w i j (Function.update x k s i) (Function.update x k s j) - |
| w i j (Function.update x k t i) (Function.update x k t j) = |
| (if i = k then w k j s (x j) - w k j t (x j) else 0) + |
| (if j = k then w i k (x i) s - w i k (x i) t else 0) := by |
| by_cases hi : i = k |
| · subst i |
| by_cases hj : j = k |
| · subst j; simp [hdiag] |
| · simp [hj, Function.update_of_ne] |
| · by_cases hj : j = k |
| · subst j; simp [hi, Function.update_of_ne] |
| · simp [hi, hj, Function.update_of_ne] |
| simp_rw [← Finset.sum_sub_distrib, hpoint, Finset.sum_add_distrib] |
| simp only [Finset.sum_ite_irrel, Finset.sum_const_zero, Finset.sum_ite_eq', Finset.mem_univ, |
| ite_true] |
| conv_lhs => arg 2; arg 2; intro i; rw [hsym i k (x i) s, hsym i k (x i) t] |
| ring |
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|
| lemma cellHamiltonian_update (U : ∀ i, X i → ℝ) (W : ∀ i j, X i → X j → ℝ) |
| (vac : ∀ i, X i) (hU0 : ∀ i, U i (vac i) = 0) |
| (hW0 : ∀ i j a, W i j (vac i) a = 0) |
| (hsym : ∀ i j a b, W i j a b = W j i b a) |
| (x : ∀ i, X i) (k : I) (s : X k) : |
| cellHamiltonian U W (Function.update x k s) - |
| cellHamiltonian U W (Function.update x k (vac k)) = |
| U k s + ∑ j, if k = j then 0 else W k j s (x j) := by |
| let w (i j : I) (a : X i) (b : X j) := if i = j then 0 else W i j a b |
| have hwsym (i j : I) (a : X i) (b : X j) : w i j a b = w j i b a := by |
| dsimp [w] |
| by_cases h : i = j |
| · simp [h] |
| · simp [h, Ne.symm h, hsym i j a b] |
| have hh := double_sum_update w (by simp [w]) hwsym x k s (vac k) |
| have hu : (∑ i, U i (Function.update x k s i)) - |
| (∑ i, U i (Function.update x k (vac k) i)) = U k s := by |
| rw [← Finset.sum_sub_distrib] |
| have he (i : I) : U i (Function.update x k s i) - U i (Function.update x k (vac k) i) = |
| if i = k then U k s else 0 := by |
| by_cases hi : i = k |
| · subst i; simp [hU0] |
| · simp [hi, Function.update_of_ne] |
| simp_rw [he] |
| simp |
| have hz (j : I) : w k j (vac k) (x j) = 0 := by simp [w, hW0] |
| simp only [hz, sub_zero] at hh |
| change (∑ i, U i (Function.update x k s i)) + 1/2 * (∑ i, ∑ j, w i j _ _) - |
| ((∑ i, U i (Function.update x k (vac k) i)) + 1/2 * (∑ i, ∑ j, w i j _ _)) = _ |
| dsimp only [w] at hh |
| linarith |
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|
| lemma cellHamiltonian_stable (U : ∀ i, X i → ℝ) (W : ∀ i j, X i → X j → ℝ) |
| (n : ∀ i, X i → ℝ) (_hn : ∀ i x, 0 ≤ n i x) |
| (J : I → I → ℝ) (hJ : ∀ i j, 0 ≤ J i j) (hJsym : ∀ i j, J i j = J j i) |
| {D B : ℝ} (_hD : 0 ≤ D) (hrow : ∀ i, ∑ j, J i j ≤ D) |
| (hU : ∀ i a, D * (n i a)^2 - B * n i a - J i i/2 * (n i a)^2 ≤ U i a) |
| (hW : ∀ i j a b, -J i j * n i a * n j b ≤ W i j a b) |
| (x : ∀ i, X i) : |
| D/2 * ∑ i, (n i (x i))^2 - B * ∑ i, n i (x i) ≤ cellHamiltonian U W x := by |
| have hb := symmetric_row_quadratic J (fun i => n i (x i)) D hJ hJsym hrow |
| have hu := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hU i (x i)) |
| have hw := Finset.sum_le_sum (s := Finset.univ) (fun i _ => |
| Finset.sum_le_sum (s := Finset.univ) (fun j _ => show |
| (if i = j then 0 else -J i j * n i (x i) * n j (x j)) ≤ |
| (if i = j then 0 else W i j (x i) (x j)) from by split_ifs; rfl; exact hW i j _ _)) |
| have hsplit : (∑ i, ∑ j, J i j * n i (x i) * n j (x j)) = |
| (∑ i, J i i * (n i (x i))^2) + |
| ∑ i, ∑ j, if i = j then 0 else J i j * n i (x i) * n j (x j) := by |
| rw [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| have he (j : I) : J i j * n i (x i) * n j (x j) = |
| (if i = j then J i j * n i (x i) * n j (x j) else 0) + |
| (if i = j then 0 else J i j * n i (x i) * n j (x j)) := by split_ifs <;> simp |
| calc |
| _ = ∑ j, ((if i = j then J i j * n i (x i) * n j (x j) else 0) + |
| (if i = j then 0 else J i j * n i (x i) * n j (x j))) := |
| Finset.sum_congr rfl (fun j _ => he j) |
| _ = _ := by |
| rw [Finset.sum_add_distrib] |
| simp only [Finset.sum_ite_eq, Finset.mem_univ, ite_true] |
| congr 1 |
| ring |
| have hwneg : (∑ i, ∑ j, if i = j then 0 else -J i j * n i (x i) * n j (x j)) = |
| -(∑ i, ∑ j, if i = j then 0 else J i j * n i (x i) * n j (x j)) := by |
| simp only [← Finset.sum_neg_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| split_ifs <;> ring |
| rw [hwneg] at hw |
| simp only [Finset.sum_sub_distrib, ← Finset.mul_sum] at hu |
| have hd : (∑ i, J i i / 2 * (n i (x i))^2) = (1/2:ℝ) * ∑ i, J i i * (n i (x i))^2 := by |
| rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i _; ring |
| rw [hd] at hu |
| unfold cellHamiltonian |
| linarith |
|
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| end AbstractCells |
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|
| section PairCells |
| variable {X : Type uX} {Y : Type uY} {Z : Type uZ} |
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| def pairWithin (v : X → X → ℝ) (s : ParticleState X) : ℝ := by |
| classical |
| exact (1/2:ℝ) * ∑ i : Fin s.1, ∑ j : Fin s.1, if i = j then 0 else v (s.2 i) (s.2 j) |
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| def pairBetween (v : X → Y → ℝ) (s : ParticleState X) (t : ParticleState Y) : ℝ := |
| ∑ i : Fin s.1, ∑ j : Fin t.1, v (s.2 i) (t.2 j) |
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| @[simp] theorem pairWithin_vacuum (v : X → X → ℝ) : pairWithin v particleVacuum = 0 := by |
| change (1/2:ℝ) * (∑ i : Fin 0, ∑ j : Fin 0, if i = j then 0 else v (Fin.elim0 i) (Fin.elim0 j)) = 0 |
| rw [Fin.sum_univ_zero, mul_zero] |
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| @[simp] theorem pairBetween_vacuum (v : X → Y → ℝ) (t : ParticleState Y) : |
| pairBetween v particleVacuum t = 0 := by |
| change (∑ i : Fin 0, ∑ j : Fin t.1, v (Fin.elim0 i) (t.2 j)) = 0 |
| rw [Fin.sum_univ_zero] |
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| theorem pairBetween_symm (v : X → Y → ℝ) (w : Y → X → ℝ) |
| (hv : ∀ x y, v x y = w y x) (s : ParticleState X) (t : ParticleState Y) : |
| pairBetween v s t = pairBetween w t s := by |
| unfold pairBetween |
| simp_rw [hv] |
| exact Finset.sum_comm |
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| lemma continuous_particle_prod [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] |
| {f : ParticleState X → ParticleState Y → Z} |
| (hf : ∀ m n, Continuous (fun p : (Fin m → X) × (Fin n → Y) => f ⟨m,p.1⟩ ⟨n,p.2⟩)) : |
| Continuous (Function.uncurry f) := by |
| let e := Homeomorph.sigmaProdDistrib (X := fun m : ℕ => Fin m → X) (Y := ParticleState Y) |
| have h : Continuous (fun q : Σ m : ℕ, (Fin m → X) × ParticleState Y => |
| f ⟨q.1,q.2.1⟩ q.2.2) := by |
| apply continuous_sigma |
| intro m |
| let e' := Homeomorph.sigmaProdDistrib (X := fun n : ℕ => Fin n → Y) (Y := Fin m → X) |
| have h' : Continuous (fun q : Σ n : ℕ, (Fin n → Y) × (Fin m → X) => |
| f ⟨m,q.2.2⟩ ⟨q.1,q.2.1⟩) := by |
| apply continuous_sigma |
| intro n |
| exact (hf m n).comp continuous_swap |
| exact (h'.comp e'.continuous).comp continuous_swap |
| exact h.comp e.continuous |
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| lemma continuous_pairWithin [TopologicalSpace X] (v : X → X → ℝ) |
| (hv : Continuous (Function.uncurry v)) : Continuous (pairWithin v) := by |
| apply continuous_sigma |
| intro n |
| change Continuous (fun a : Fin n → X => (1/2:ℝ) * |
| ∑ i : Fin n, ∑ j : Fin n, if i = j then 0 else v (a i) (a j)) |
| apply continuous_const.mul |
| apply continuous_finsetSum _ |
| intro i _ |
| apply continuous_finsetSum _ |
| intro j _ |
| split_ifs |
| · exact continuous_const |
| · exact hv.comp (show Continuous (fun a : Fin n → X => (a i, a j)) from |
| (continuous_apply i).prodMk (continuous_apply j)) |
|
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| lemma continuous_pairBetween [TopologicalSpace X] [TopologicalSpace Y] (v : X → Y → ℝ) |
| (hv : Continuous (Function.uncurry v)) : Continuous (Function.uncurry (pairBetween v)) := by |
| apply continuous_particle_prod |
| intro m n |
| change Continuous (fun p : (Fin m → X) × (Fin n → Y) => |
| ∑ i : Fin m, ∑ j : Fin n, v (p.1 i) (p.2 j)) |
| apply continuous_finsetSum _ |
| intro i _ |
| apply continuous_finsetSum _ |
| intro j _ |
| exact hv.comp (show Continuous (fun p : (Fin m → X) × (Fin n → Y) => (p.1 i, p.2 j)) from |
| ((continuous_apply i).comp continuous_fst).prodMk ((continuous_apply j).comp continuous_snd)) |
|
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| lemma pairBetween_lower (v : X → Y → ℝ) {J : ℝ} (hv : ∀ x y, -J ≤ v x y) |
| (s : ParticleState X) (t : ParticleState Y) : |
| -J * particleCount s * particleCount t ≤ pairBetween v s t := by |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun i _ => |
| Finset.sum_le_sum (s := Finset.univ) (fun j _ => hv (s.2 i) (t.2 j))) |
| simpa only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, |
| pairBetween, particleCount, mul_comm, mul_left_comm] using hh |
|
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| lemma pairWithin_perturb_lower (v w : X → X → ℝ) {J : ℝ} (hJ : 0 ≤ J) |
| (hv : ∀ x y, w x y - J ≤ v x y) (s : ParticleState X) : |
| pairWithin w s - J/2 * (particleCount s : ℝ)^2 ≤ pairWithin v s := by |
| classical |
| have hp (i j : Fin s.1) : |
| (if i = j then 0 else w (s.2 i) (s.2 j)) - J ≤ |
| (if i = j then 0 else v (s.2 i) (s.2 j)) := by |
| split_ifs |
| · linarith |
| · exact hv _ _ |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun i _ => |
| Finset.sum_le_sum (s := Finset.univ) (fun j _ => hp i j)) |
| simp only [Finset.sum_sub_distrib, Finset.sum_const, Finset.card_univ, |
| Fintype.card_fin, nsmul_eq_mul] at hh |
| unfold pairWithin particleCount |
| nlinarith |
|
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| lemma pairWithin_core {t J : ℝ} (ht : 0 ≤ t) (hJ : 0 ≤ J) |
| (pos : X → Position) (hdiam : ∀ x y, dist (pos x) (pos y) ≤ 1) |
| (v : X → X → ℝ) (hv : ∀ x y, corePotential t (nndist (pos x) (pos y)) - J ≤ v x y) |
| (s : ParticleState X) : |
| t^2/8 * (particleCount s : ℝ)^2 - t^2/2 * particleCount s - |
| J/2 * (particleCount s : ℝ)^2 ≤ pairWithin v s := by |
| have hh := pairWithin_perturb_lower v (fun x y => corePotential t (nndist (pos x) (pos y))) hJ hv s |
| have hc := core_ordered_lower_small_diameter ht (fun i : Fin s.1 => pos (s.2 i)) |
| (fun i j => hdiam _ _) |
| simp only [Fintype.card_fin, ne_eq, ite_not] at hc |
| dsimp only [pairWithin, corePotential, particleCount] at hh |
| change (1/2:ℝ) * (∑ i : Fin s.1, ∑ j : Fin s.1, |
| if i = j then 0 else coreReal t (dist (pos (s.2 i)) (pos (s.2 j)))) - _ ≤ _ at hh |
| dsimp only [particleCount, pairWithin] |
| linarith |
|
|
| end PairCells |
|
|
| section PhysicalCells |
| variable {I : Type uI} [Fintype I] [DecidableEq I] {X : I → Type uX} |
| [∀ i, TopologicalSpace (X i)] [∀ i, MeasurableSpace (X i)] |
| [∀ i, BorelSpace (X i)] [∀ i, SecondCountableTopology (X i)] |
|
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| def pairCellEnergy (v : ∀ i j, X i → X j → ℝ) (x : ∀ i, ParticleState (X i)) : ℝ := |
| cellHamiltonian (fun i => pairWithin (v i i)) (fun i j => pairBetween (v i j)) x |
|
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| omit [∀ i, MeasurableSpace (X i)] [∀ i, BorelSpace (X i)] |
| [∀ i, SecondCountableTopology (X i)] in |
| lemma continuous_pairCellEnergy (v : ∀ i j, X i → X j → ℝ) |
| (hv : ∀ i j, Continuous (Function.uncurry (v i j))) : Continuous (pairCellEnergy v) := by |
| unfold pairCellEnergy cellHamiltonian |
| apply Continuous.add |
| · exact continuous_finsetSum _ (fun i _ => (continuous_pairWithin _ (hv i i)).comp (continuous_apply i)) |
| · apply continuous_const.mul |
| apply continuous_finsetSum _ |
| intro i _ |
| apply continuous_finsetSum _ |
| intro j _ |
| split_ifs |
| · exact continuous_const |
| · exact (continuous_pairBetween _ (hv i j)).comp |
| (show Continuous (fun x : ∀ i, ParticleState (X i) => (x i, x j)) from |
| (continuous_apply i).prodMk (continuous_apply j)) |
|
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| theorem pairCell_exponential_moment [Nonempty I] |
| (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (pos : ∀ i, X i → Position) (hdiam : ∀ i x y, dist (pos i x) (pos i y) ≤ 1) |
| {t β V z : ℝ} (ht : 0 < t) (hβ : 0 < β) (hV : 0 ≤ V) |
| (hvol : ∀ i, (ν i univ).toReal ≤ V) |
| (v : ∀ i j, X i → X j → ℝ) (hv : ∀ i j, Continuous (Function.uncurry (v i j))) |
| (hvsym : ∀ i j x y, v i j x y = v j i y x) |
| (J : I → I → ℝ) (hJ : ∀ i j, 0 ≤ J i j) (hJsym : ∀ i j, J i j = J j i) |
| (hrow : ∀ i, ∑ j, J i j ≤ (t^2/8)/8) |
| (hintra : ∀ i x y, corePotential t (nndist (pos i x) (pos i y)) - J i i ≤ v i i x y) |
| (hcross : ∀ i j x y, -J i j ≤ v i j x y) : |
| ∀ i, (∫ x, Real.exp (β*(t^2/8)/4 * (cellN i x)^2) |
| ∂boltzmannMeasure (Measure.pi (fun j => particleBase (ν j) z)) |
| (fun x => β * pairCellEnergy v x)) ≤ |
| Real.exp (16/15 * V * Real.exp (z+β*(t^2/2))) := by |
| let D := t^2/8 |
| let B := t^2/2 |
| have hD : 0 < D := by dsimp [D]; positivity |
| let U (i : I) := pairWithin (v i i) |
| let W (i j : I) := pairBetween (v i j) |
| have hU (i : I) (s : ParticleState (X i)) : |
| D * (particleCount s : ℝ)^2 - B * particleCount s - J i i/2*(particleCount s : ℝ)^2 ≤ U i s := |
| pairWithin_core ht.le (hJ i i) (pos i) (hdiam i) (v i i) (hintra i) s |
| have hW (i j : I) (s : ParticleState (X i)) (r : ParticleState (X j)) : |
| -J i j * particleCount s * particleCount r ≤ W i j s r := |
| pairBetween_lower (v i j) (hcross i j) s r |
| have hrowD (i : I) : ∑ j, J i j ≤ D := (hrow i).trans (by dsimp [D]; nlinarith [sq_nonneg t]) |
| have hH (x : ∀ i, ParticleState (X i)) : |
| D/2 * ∑ i, (cellN i x)^2 - B * ∑ i, cellN i x ≤ pairCellEnergy v x := by |
| exact cellHamiltonian_stable U W (fun _ s => particleCount s) (fun _ _ => Nat.cast_nonneg _) |
| J hJ hJsym hD.le hrowD hU hW x |
| let θ (i j : I) := if i = j then 0 else 8*J i j/D |
| have hθ (i j : I) : 0 ≤ θ i j := by |
| dsimp [θ] |
| split_ifs |
| · rfl |
| · exact div_nonneg (mul_nonneg (by norm_num) (hJ i j)) hD.le |
| have hθrow (i : I) : ∑ j, θ i j ≤ 1 := by |
| have hle : ∑ j, θ i j ≤ 8/D * ∑ j, J i j := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro j _ |
| dsimp [θ] |
| split_ifs |
| · exact mul_nonneg (div_nonneg (by norm_num) hD.le) (hJ i j) |
| · exact le_of_eq (by ring) |
| refine hle.trans ?_ |
| calc |
| _ ≤ 8/D * (D/8) := mul_le_mul_of_nonneg_left (hrow i) (by positivity) |
| _ = 1 := by field_simp |
| have hcond (i : I) (x : ∀ j, ParticleState (X j)) (s : ParticleState (X i)) : |
| D/2 * (particleCount s : ℝ)^2 - B*particleCount s - |
| (particleCount s : ℝ)*(D/8 * ∑ j, θ i j * cellN j x) ≤ |
| pairCellEnergy v (Function.update x i s) - |
| pairCellEnergy v (Function.update x i particleVacuum) := by |
| rw [pairCellEnergy, pairCellEnergy, |
| cellHamiltonian_update _ _ (fun _ => particleVacuum) (by simp) |
| (by simp) (fun k j a b => pairBetween_symm _ _ (hvsym k j) a b)] |
| have hJi : J i i ≤ D := |
| (Finset.single_le_sum (fun j _ => hJ i j) (Finset.mem_univ i)).trans (hrowD i) |
| have hu := hU i s |
| have hsq := mul_nonneg (sub_nonneg.mpr hJi) (sq_nonneg (particleCount s : ℝ)) |
| have hw := Finset.sum_le_sum (s := Finset.univ) (fun j _ => show |
| (if i = j then 0 else -J i j * particleCount s * particleCount (x j)) ≤ |
| (if i = j then 0 else pairBetween (v i j) s (x j)) from by |
| split_ifs; rfl; exact hW i j s (x j)) |
| have he : (∑ j, if i = j then 0 else -J i j * particleCount s * particleCount (x j)) = |
| -(particleCount s : ℝ) * (D/8 * ∑ j, θ i j * cellN j x) := by |
| rw [Finset.mul_sum, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| dsimp [θ, cellN] |
| split_ifs |
| · ring |
| · field_simp |
| rw [he] at hw |
| dsimp [U] at hu |
| linarith |
| apply uniform_cell_exponential_moment ν z (D := β*D) (B := β*B) |
| (E := fun x => β * pairCellEnergy v x) |
| (mul_pos hβ hD) hV hvol θ hθ (by simp [θ]) hθrow |
| (measurable_const.mul (continuous_pairCellEnergy v hv).measurable) |
| · intro x |
| have hh := mul_le_mul_of_nonneg_left (hH x) hβ.le |
| convert hh using 1; first | rfl | ring |
| · intro i x s |
| have hh := mul_le_mul_of_nonneg_left (hcond i x s) hβ.le |
| convert hh using 1 <;> first | rfl | ring |
|
|
| end PhysicalCells |
| end RadialTransition |
|
|
| end |
|
|
| end OAI |
|
|
