| import OAI.Probability.RadialTransition.CroppedCells |
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| namespace OAI |
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| universe uX uY uI |
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| noncomputable section |
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| open Function MeasureTheory Set Filter |
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| open scoped ContDiff NNReal BigOperators ENNReal |
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| namespace RadialTransition |
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| lemma if_zero_sub (p : Prop) [Decidable p] (a b : ℝ) : |
| (if p then 0 else a-b) = (if p then 0 else a) - (if p then 0 else b) := by |
| split_ifs <;> simp |
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| lemma pairWithin_sub {X : Type uX} (v w : X → X → ℝ) (s : ParticleState X) : |
| pairWithin (fun x y => v x y-w x y) s = pairWithin v s - pairWithin w s := by |
| classical |
| unfold pairWithin |
| simp_rw [if_zero_sub] |
| simp_rw [Finset.sum_sub_distrib] |
| ring |
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| lemma pairBetween_sub {X : Type uX} {Y : Type uY} (v w : X → Y → ℝ) (s : ParticleState X) (t : ParticleState Y) : |
| pairBetween (fun x y => v x y-w x y) s t = pairBetween v s t - pairBetween w s t := by |
| unfold pairBetween |
| simp_rw [Finset.sum_sub_distrib] |
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| lemma pairWithin_abs_bound {X : Type uX} (v : X → X → ℝ) {K : ℝ} (hK : 0 ≤ K) |
| (hv : ∀ x y, |v x y| ≤ K) (s : ParticleState X) : |
| |pairWithin v s| ≤ (1/2:ℝ) * (K * particleCount s * particleCount s) := by |
| classical |
| unfold pairWithin |
| rw [abs_mul, abs_of_pos (by norm_num : (0:ℝ) < 1/2)] |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| calc |
| |∑ i : Fin s.1, ∑ j : Fin s.1, if i = j then 0 else v (s.2 i) (s.2 j)| ≤ |
| ∑ i : Fin s.1, ∑ j : Fin s.1, |if i = j then 0 else v (s.2 i) (s.2 j)| := |
| (Finset.abs_sum_le_sum_abs _ _).trans (Finset.sum_le_sum (fun _ _ => Finset.abs_sum_le_sum_abs _ _)) |
| _ ≤ ∑ i : Fin s.1, ∑ j : Fin s.1, K := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| split_ifs |
| · simpa only [abs_zero] using hK |
| · exact hv _ _ |
| _ = _ := by simp [particleCount]; ring |
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| lemma pairBetween_abs_bound {X : Type uX} {Y : Type uY} (v : X → Y → ℝ) {K : ℝ} |
| (hv : ∀ x y, |v x y| ≤ K) (s : ParticleState X) (t : ParticleState Y) : |
| |pairBetween v s t| ≤ K * particleCount s * particleCount t := by |
| unfold pairBetween |
| calc |
| |∑ i : Fin s.1, ∑ j : Fin t.1, v (s.2 i) (t.2 j)| ≤ |
| ∑ i : Fin s.1, ∑ j : Fin t.1, |v (s.2 i) (t.2 j)| := |
| (Finset.abs_sum_le_sum_abs _ _).trans (Finset.sum_le_sum (fun _ _ => Finset.abs_sum_le_sum_abs _ _)) |
| _ ≤ ∑ i : Fin s.1, ∑ j : Fin t.1, K := |
| Finset.sum_le_sum (fun i _ => Finset.sum_le_sum (fun j _ => hv _ _)) |
| _ = _ := by simp [particleCount]; ring |
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| lemma double_sum_diagonal {I : Type uI} [Fintype I] [DecidableEq I] (a : I → I → ℝ) : |
| (∑ i, ∑ j, a i j) = ∑ i, a i i + ∑ i, ∑ j, if i = j then 0 else a i j := by |
| rw [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| conv_lhs => |
| arg 2; ext j |
| rw [show a i j = (if i = j then a i j else 0) + (if i = j then 0 else a i j) by split_ifs <;> simp] |
| simp only [Finset.sum_add_distrib, Finset.sum_ite_eq, Finset.mem_univ, ite_true] |
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| section CellDifferences |
| variable {I : Type uI} [Fintype I] [DecidableEq I] {X : I → Type uX} |
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| lemma pairCellEnergy_sub (v w : ∀ i j, X i → X j → ℝ) (x : ∀ i, ParticleState (X i)) : |
| pairCellEnergy (fun i j a b => v i j a b-w i j a b) x = pairCellEnergy v x-pairCellEnergy w x := by |
| unfold pairCellEnergy cellHamiltonian |
| simp_rw [pairWithin_sub, pairBetween_sub, if_zero_sub, Finset.sum_sub_distrib] |
| ring |
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| lemma pairCellEnergy_abs_bound (v : ∀ i j, X i → X j → ℝ) (K : I → I → ℝ) |
| (hK : ∀ i j, 0 ≤ K i j) (hv : ∀ i j a b, |v i j a b| ≤ K i j) |
| (x : ∀ i, ParticleState (X i)) : |
| |pairCellEnergy v x| ≤ (1/2:ℝ) * ∑ i, ∑ j, K i j * cellN i x * cellN j x := by |
| let a i j := K i j * cellN i x * cellN j x |
| have hi : (∑ i, |pairWithin (v i i) (x i)|) ≤ (1/2:ℝ) * ∑ i, a i i := by |
| rw [Finset.mul_sum] |
| exact Finset.sum_le_sum (fun i _ => pairWithin_abs_bound _ (hK i i) (hv i i) (x i)) |
| have hc : (∑ i, ∑ j, |if i = j then 0 else pairBetween (v i j) (x i) (x j)|) ≤ |
| ∑ i, ∑ j, if i = j then 0 else a i j := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| split_ifs |
| · simp |
| · exact pairBetween_abs_bound _ (hv i j) _ _ |
| unfold pairCellEnergy cellHamiltonian |
| calc |
| _ ≤ |∑ i, pairWithin (v i i) (x i)| + |
| |(1/2:ℝ) * ∑ i, ∑ j, if i = j then 0 else pairBetween (v i j) (x i) (x j)| := abs_add_le _ _ |
| _ ≤ (∑ i, |pairWithin (v i i) (x i)|) + (1/2:ℝ) * |
| ∑ i, ∑ j, |if i = j then 0 else pairBetween (v i j) (x i) (x j)| := by |
| rw [abs_mul, abs_of_pos (by norm_num : (0:ℝ) < 1/2)] |
| apply add_le_add (Finset.abs_sum_le_sum_abs _ _) |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| exact (Finset.abs_sum_le_sum_abs _ _).trans |
| (Finset.sum_le_sum (fun _ _ => Finset.abs_sum_le_sum_abs _ _)) |
| _ ≤ (1/2:ℝ) * ∑ i, a i i + (1/2:ℝ) * ∑ i, ∑ j, if i = j then 0 else a i j := |
| add_le_add hi (mul_le_mul_of_nonneg_left hc (by norm_num)) |
| _ = _ := by rw [← mul_add, ← double_sum_diagonal] |
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| lemma pairCellEnergy_abs_difference (v w : ∀ i j, X i → X j → ℝ) (K : I → I → ℝ) |
| (hK : ∀ i j, 0 ≤ K i j) (hv : ∀ i j a b, |v i j a b-w i j a b| ≤ K i j) |
| (x : ∀ i, ParticleState (X i)) : |
| |pairCellEnergy v x-pairCellEnergy w x| ≤ (1/2:ℝ) * ∑ i, ∑ j, K i j * cellN i x * cellN j x := by |
| rw [← pairCellEnergy_sub] |
| exact pairCellEnergy_abs_bound _ K hK hv x |
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| end CellDifferences |
| section PhysicalControl |
| variable {I : Type uI} [Fintype I] [DecidableEq I] {X : I → Type uX} |
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| lemma pairCell_quadratic_bound |
| (pos : ∀ i, X i → Position) (hdiam : ∀ i x y, dist (pos i x) (pos i y) ≤ 1) |
| {t : ℝ} (ht : 0 ≤ t) (v : ∀ i j, X i → X j → ℝ) |
| (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) |
| (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) (x : ∀ i, ParticleState (X i)) : |
| (t^2/8)/2 * ∑ i, (cellN i x)^2 - (t^2/2) * ∑ i, cellN i x ≤ pairCellEnergy v x := by |
| apply cellHamiltonian_stable (fun i => pairWithin (v i i)) (fun i j => pairBetween (v i j)) |
| (fun _ s => particleCount s) (fun _ _ => Nat.cast_nonneg _) J hJ hJsym (by positivity) hrow |
| · intro i s |
| exact pairWithin_core ht (hJ i i) (pos i) (hdiam i) (v i i) (hintra i) s |
| · intro i j s r |
| exact pairBetween_lower (v i j) (hcross i j) s r |
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| variable [∀ i, TopologicalSpace (X i)] [∀ i, MeasurableSpace (X i)] |
| [∀ i, BorelSpace (X i)] [∀ i, SecondCountableTopology (X i)] |
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| omit [DecidableEq I] in |
| lemma cell_boltzmann_integrable (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (z : ℝ) {D B : ℝ} (hD : 0 ≤ D) {E : (∀ i, ParticleState (X i)) → ℝ} |
| (hE : Measurable E) |
| (hstable : ∀ x, D/2 * ∑ i, (cellN i x)^2 - B * ∑ i, cellN i x ≤ E x) : |
| Integrable (fun x => Real.exp (-E x)) (Measure.pi (fun i => particleBase (ν i) z)) := by |
| simpa only [zero_sub] using integrable_cell_numerator ν z hE measurable_const hstable |
| (f := fun _ => 0) (fun x => mul_nonneg (by positivity) (Finset.sum_nonneg (fun _ _ => sq_nonneg _))) |
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| omit [DecidableEq I] in |
| lemma cell_exponential_integrable (ν : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (ν i)] |
| (z : ℝ) {D B : ℝ} (hD : 0 ≤ D) {E : (∀ i, ParticleState (X i)) → ℝ} |
| (hE : Measurable E) |
| (hstable : ∀ x, D/2 * ∑ i, (cellN i x)^2 - B * ∑ i, cellN i x ≤ E x) (i : I) : |
| Integrable (fun x => Real.exp (D/4*(cellN i x)^2)) |
| (boltzmannMeasure (Measure.pi (fun j => particleBase (ν j) z)) E) := by |
| have hp := cell_partition_pos ν z (cell_boltzmann_integrable ν z hD hE hstable) |
| apply integrable_boltzmann hE _ hp |
| have hbound (x : ∀ j, ParticleState (X j)) : D/4*(cellN i x)^2 ≤ D/2*∑ j, (cellN j x)^2 := by |
| have hh := Finset.single_le_sum (s := Finset.univ) (f := fun j => (cellN j x)^2) |
| (fun _ _ => sq_nonneg _) (Finset.mem_univ i) |
| nlinarith [mul_nonneg hD (sq_nonneg (cellN i x)), mul_le_mul_of_nonneg_left hh hD] |
| have hi := integrable_cell_numerator ν z hE (measurable_const.mul ((measurable_cellN i).pow_const 2)) |
| hstable hbound |
| convert hi using 1 |
| ext x |
| rw [← Real.exp_add] |
| congr 1 |
| dsimp only [Pi.mul_apply] |
| ring |
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| lemma pairCell_second_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) : |
| let μ := Measure.pi (fun j => particleBase (ν j) z) |
| let E := fun x => β * pairCellEnergy v x |
| Integrable (fun x => Real.exp (-E x)) μ ∧ 0 < boltzmannPartition μ E ∧ |
| ∀ i, Integrable (fun x => (cellN i x)^2) (boltzmannMeasure μ E) ∧ |
| (∫ x, (cellN i x)^2 ∂boltzmannMeasure μ E) ≤ |
| Real.exp (16/15 * V * Real.exp (z+β*(t^2/2))) / (β*(t^2/8)/4) := by |
| dsimp only |
| have hD : 0 < β*(t^2/8) := by positivity |
| have hr (i : I) : ∑ j, J i j ≤ t^2/8 := (hrow i).trans (by nlinarith [sq_nonneg t]) |
| have hE : Measurable (fun x => β * pairCellEnergy v x) := |
| measurable_const.mul (continuous_pairCellEnergy v hv).measurable |
| have hstable (x : ∀ i, ParticleState (X i)) : |
| (β*(t^2/8))/2 * ∑ i, (cellN i x)^2 - (β*(t^2/2))*∑ i, cellN i x ≤ β*pairCellEnergy v x := by |
| have hh := mul_le_mul_of_nonneg_left |
| (pairCell_quadratic_bound pos hdiam ht.le v J hJ hJsym hr hintra hcross x) hβ.le |
| convert hh using 1; first | rfl | ring |
| have hi := cell_boltzmann_integrable ν z hD.le hE hstable |
| refine ⟨hi, cell_partition_pos ν z hi, fun i => ?_⟩ |
| exact secondMoment_of_exponential (measurable_cellN i) (by positivity) |
| (cell_exponential_integrable ν z hD.le hE hstable i) |
| (pairCell_exponential_moment ν pos hdiam ht hβ hV hvol v hv hvsym J hJ hJsym hrow hintra hcross i) |
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| theorem pairCell_log_perturbation [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 w : ∀ i j, X i → X j → ℝ) |
| (hv : ∀ i j, Continuous (Function.uncurry (v i j))) |
| (hw : ∀ i j, Continuous (Function.uncurry (w i j))) |
| (hvsym : ∀ i j x y, v i j x y = v j i y x) |
| (hwsym : ∀ i j x y, w i j x y = w 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) |
| (hintrav : ∀ i x y, corePotential t (nndist (pos i x) (pos i y)) - J i i ≤ v i i x y) |
| (hcrossv : ∀ i j x y, -J i j ≤ v i j x y) |
| (hintraw : ∀ i x y, corePotential t (nndist (pos i x) (pos i y)) - J i i ≤ w i i x y) |
| (hcrossw : ∀ i j x y, -J i j ≤ w i j x y) |
| (K : I → I → ℝ) (hK : ∀ i j, 0 ≤ K i j) |
| (hd : ∀ i j a b, |w i j a b-v i j a b| ≤ K i j) : |
| let μ := Measure.pi (fun i => particleBase (ν i) z) |
| |Real.log (boltzmannPartition μ (fun x => β*pairCellEnergy v x)) - |
| Real.log (boltzmannPartition μ (fun x => β*pairCellEnergy w x))| ≤ |
| β/2 * (Real.exp (16/15 * V * Real.exp (z+β*(t^2/2))) / (β*(t^2/8)/4)) * ∑ i, ∑ j, K i j := by |
| dsimp only |
| have hvdata := pairCell_second_moment (z := z) ν pos hdiam ht hβ hV hvol v hv hvsym J hJ hJsym hrow hintrav hcrossv |
| have hwdata := pairCell_second_moment (z := z) ν pos hdiam ht hβ hV hvol w hw hwsym J hJ hJsym hrow hintraw hcrossw |
| let K' i j := β/2*K i j |
| have hd' (x : ∀ i, ParticleState (X i)) : |
| |β*pairCellEnergy w x-β*pairCellEnergy v x| ≤ ∑ i, ∑ j, K' i j * cellN i x * cellN j x := by |
| rw [← mul_sub, abs_mul, abs_of_pos hβ] |
| calc |
| _ ≤ β*((1/2:ℝ)*∑ i, ∑ j, K i j*cellN i x*cellN j x) := |
| mul_le_mul_of_nonneg_left (pairCellEnergy_abs_difference w v K hK hd x) hβ.le |
| _ = _ := by |
| simp only [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| dsimp [K'] |
| ring |
| have hh := log_partition_perturbation (E := fun x => β*pairCellEnergy v x) (F := fun x => β*pairCellEnergy w x) |
| (measurable_const.mul (continuous_pairCellEnergy v hv).measurable) |
| (measurable_const.mul (continuous_pairCellEnergy w hw).measurable) |
| hvdata.1 hwdata.1 hvdata.2.1 hwdata.2.1 (cellN (X := X)) measurable_cellN |
| K' (fun i j => mul_nonneg (by positivity) (hK i j)) |
| (fun i => (hvdata.2.2 i).1) (fun i => (hwdata.2.2 i).1) |
| (fun i => (hvdata.2.2 i).2) (fun i => (hwdata.2.2 i).2) hd' |
| have hs : (∑ i, ∑ j, K' i j) = β/2*(∑ i, ∑ j, K i j) := by |
| simp only [K', Finset.mul_sum] |
| rw [hs] at hh |
| simpa only [Pi.mul_apply, mul_assoc, mul_left_comm, mul_comm] using hh |
|
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| end PhysicalControl |
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| end RadialTransition |
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| end |
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| end OAI |
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