gibbs non gibbs dynamical transitions a large deviation
play

Gibbs-non-Gibbs dynamical transitions. A large-deviation paradigm - PowerPoint PPT Presentation

Mar 11, 2024 •101 likes •1.25k views

Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbs-non-Gibbs dynamical transitions. A large-deviation paradigm R. Fern andez F. den Hollander J. Mart nez Utrecht Leiden Leiden Kyoto, July 2013 Intro Gibbs Non-Gibbs

  1. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition (Lattice) Gibbs measures: formal definition Gibbsian specification: Family Π Φ = { π Φ Λ : Λ ⊂ ⊂ L } with Λ ) = e − βH Λ ( ω | σ ) π Φ Λ ( C ω Norm. [ π Φ Λ ( · | σ ) = equilibrium in Λ given σ ] Gibbs measures: µ is Gibbs for Φ if, equivalently, ◮ µ is left invariant by Π Φ : � π Φ Λ ( C ω Λ ) µ ( dω ) = µ ( C ω Λ ) [ µ = equilibrium in L = every Λ in equilibrium] ◮ µ = w − lim Λ → L π Φ Λ ( · | σ ) + convex combinations [thermodynamic limit]

  2. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbsianness test How to recognize Gibbsianness Kozlov – Sullivan: µ is Gibbs iff it is ◮ Non-null: µ ( C ω Λ ) > 0 for every cylinder C ω Λ ◮ Quasilocal: If Λ ⊂ Γ ⊂ ⊂ L , � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � − γ → L 0 − − → Λ Λ σ,ω,ξ ± Physics in Λ does not depend on state of the Andromeda galaxy

  3. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbsianness test How to recognize Gibbsianness Kozlov – Sullivan: µ is Gibbs iff it is ◮ Non-null: µ ( C ω Λ ) > 0 for every cylinder C ω Λ ◮ Quasilocal: If Λ ⊂ Γ ⊂ ⊂ L , � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � − γ → L 0 − − → Λ Λ σ,ω,ξ ± Physics in Λ does not depend on state of the Andromeda galaxy

  4. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Gibbsianness test How to recognize Gibbsianness Kozlov – Sullivan: µ is Gibbs iff it is ◮ Non-null: µ ( C ω Λ ) > 0 for every cylinder C ω Λ ◮ Quasilocal: If Λ ⊂ Γ ⊂ ⊂ L , � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � − γ → L 0 − − → Λ Λ σ,ω,ξ ± Physics in Λ does not depend on state of the Andromeda galaxy

  5. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality Essential non-quasilocality µ is Gibbs if ∃ Λ and w sp s.t: ∃ ξ ± for which � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � — γ → L 0 / − → Λ Λ σ,ω,ξ ± � � � � · for every realisation of µ C ω Λ ◮ Quasilocality = continuity w.r.t. external conditions ◮ Non-quasilocality = essential discontinuity w.r.t. external conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

  6. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality Essential non-quasilocality µ is Gibbs if ∃ Λ and w sp s.t: ∃ ξ ± for which � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � — γ → L 0 / − → Λ Λ σ,ω,ξ ± � � � � · for every realisation of µ C ω Λ ◮ Quasilocality = continuity w.r.t. external conditions ◮ Non-quasilocality = essential discontinuity w.r.t. external conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

  7. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality Essential non-quasilocality µ is Gibbs if ∃ Λ and w sp s.t: ∃ ξ ± for which � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � — γ → L 0 / − → Λ Λ σ,ω,ξ ± � � � � · for every realisation of µ C ω Λ ◮ Quasilocality = continuity w.r.t. external conditions ◮ Non-quasilocality = essential discontinuity w.r.t. external conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

  8. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Non-quasilocality Essential non-quasilocality µ is Gibbs if ∃ Λ and w sp s.t: ∃ ξ ± for which � Γ ξ − �� � � Γ ξ + � � � � � � σ � σ sup � µ C ω − µ C ω � — γ → L 0 / − → Λ Λ σ,ω,ξ ± � � � � · for every realisation of µ C ω Λ ◮ Quasilocality = continuity w.r.t. external conditions ◮ Non-quasilocality = essential discontinuity w.r.t. external conditions Interpretation: Info from ∞ despite frozen fluctuations Possible explanation: hidden variables

  9. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Renormalization transformations General definition: A (stochastic) RT is a map Prob(Ω ′ ) Prob(Ω) − → � µ ′ ( · ) = µ �− → K ( · | ω ) µ ( dω ) where K is a probability kernel The transformation is deterministic if ∃ f : Ω → Ω ′ s.t. K ( · | ω ) = δ f ( ω ) ( · ) A block RT is of the form � K ( dω ′ | ω ) = K ′ x ( dω ′ x ′ | ω Bx ′ ) x ′ ⊂ L is the block associated to x ′ each B x ′ ⊂

  10. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Renormalization transformations General definition: A (stochastic) RT is a map Prob(Ω ′ ) Prob(Ω) − → � µ ′ ( · ) = µ �− → K ( · | ω ) µ ( dω ) where K is a probability kernel The transformation is deterministic if ∃ f : Ω → Ω ′ s.t. K ( · | ω ) = δ f ( ω ) ( · ) A block RT is of the form � K ( dω ′ | ω ) = K ′ x ( dω ′ x ′ | ω Bx ′ ) x ′ ⊂ L is the block associated to x ′ each B x ′ ⊂

  11. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Renormalization transformations General definition: A (stochastic) RT is a map Prob(Ω ′ ) Prob(Ω) − → � µ ′ ( · ) = µ �− → K ( · | ω ) µ ( dω ) where K is a probability kernel The transformation is deterministic if ∃ f : Ω → Ω ′ s.t. K ( · | ω ) = δ f ( ω ) ( · ) A block RT is of the form � K ( dω ′ | ω ) = K ′ x ( dω ′ x ′ | ω Bx ′ ) x ′ ⊂ L is the block associated to x ′ each B x ′ ⊂

  12. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Examples of block transformations Deterministic transformations: ◮ Decimation ◮ Majority (odd block) Stochastic transformations: ◮ Majority (even block) ◮ Kadanoff: � � � p ω ′ exp x ∈ B x ′ ω x x ′ K ′ x ( dω ′ dω ′ x ′ | ω Bx ′ ) = x ′ Norm [weighted majority; → majority as p → ∞ ]

  13. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Hidden variables and non-quasilocality Hidden-variables mechanism: ◮ Each fixed ω ′ Λc determines a constrained Ω system ◮ ω ′ sp is s.t. the constrained system has a phase transition ◮ ξ ′ far away decides the phase → info form ∞ Two-slice point of view: ◮ Ω = original slice = hidden variables ◮ Ω ′ = present slice = observed variables Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

  14. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Hidden variables and non-quasilocality Hidden-variables mechanism: ◮ Each fixed ω ′ Λc determines a constrained Ω system ◮ ω ′ sp is s.t. the constrained system has a phase transition ◮ ξ ′ far away decides the phase → info form ∞ Two-slice point of view: ◮ Ω = original slice = hidden variables ◮ Ω ′ = present slice = observed variables Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

  15. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Hidden variables and non-quasilocality Hidden-variables mechanism: ◮ Each fixed ω ′ Λc determines a constrained Ω system ◮ ω ′ sp is s.t. the constrained system has a phase transition ◮ ξ ′ far away decides the phase → info form ∞ Two-slice point of view: ◮ Ω = original slice = hidden variables ◮ Ω ′ = present slice = observed variables Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

  16. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Hidden variables and non-quasilocality Hidden-variables mechanism: ◮ Each fixed ω ′ Λc determines a constrained Ω system ◮ ω ′ sp is s.t. the constrained system has a phase transition ◮ ξ ′ far away decides the phase → info form ∞ Two-slice point of view: ◮ Ω = original slice = hidden variables ◮ Ω ′ = present slice = observed variables Non-quasilocality: Hidden variables constrained by observed variables exhibit a phase transition

  17. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations ( B x ′ = { x ′ } ) On finite volumes, the two-slice measures are of the form � �� � K Λ ( dω ′ | ω ) µ H Kad ( ω, ω ′ ) + H Λ ( ω ) dω ′ Λ ( dω ) ∝ exp β Λ dω Λ Λ where � p �� � � x ω x − 1 H Kad ( ω, ω ′ ) = β ω ′ β log 2 cosh( p ω x ) Λ x ′ acts on the original spins as an extra magnetic field Constrained internal spins have phase transition if p β ω ′ x compensates h in the average ( ∗ ) and β is large enough

  18. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations ( B x ′ = { x ′ } ) On finite volumes, the two-slice measures are of the form � �� � K Λ ( dω ′ | ω ) µ H Kad ( ω, ω ′ ) + H Λ ( ω ) dω ′ Λ ( dω ) ∝ exp β Λ dω Λ Λ where � p �� � � x ω x − 1 H Kad ( ω, ω ′ ) = β ω ′ β log 2 cosh( p ω x ) Λ x ′ acts on the original spins as an extra magnetic field Constrained internal spins have phase transition if p β ω ′ x compensates h in the average ( ∗ ) and β is large enough

  19. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations ( B x ′ = { x ′ } ) On finite volumes, the two-slice measures are of the form � �� � K Λ ( dω ′ | ω ) µ H Kad ( ω, ω ′ ) + H Λ ( ω ) dω ′ Λ ( dω ) ∝ exp β Λ dω Λ Λ where � p �� � � x ω x − 1 H Kad ( ω, ω ′ ) = β ω ′ β log 2 cosh( p ω x ) Λ x ′ acts on the original spins as an extra magnetic field Constrained internal spins have phase transition if p β ω ′ x compensates h in the average ( ∗ ) and β is large enough

  20. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations (cont.) Let h be the original Ising field and fix β large enough s.t. ◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds If h = 0, alternated ω ′ = ⇒ ( ∗ ) for p/β small enough Hence, ∃ p 1 > p 2 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for 2 > p If h � = 0, ∃ ω ′ s.t. ( ∗ ) only for a range of p/β Hence, ∃ p 1 ≥ p 2 > p 3 ≥ p 4 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for p 2 > p > p 3 ◮ µ ′ is not Gibbs for p 4 > p

  21. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations (cont.) Let h be the original Ising field and fix β large enough s.t. ◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds If h = 0, alternated ω ′ = ⇒ ( ∗ ) for p/β small enough Hence, ∃ p 1 > p 2 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for 2 > p If h � = 0, ∃ ω ′ s.t. ( ∗ ) only for a range of p/β Hence, ∃ p 1 ≥ p 2 > p 3 ≥ p 4 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for p 2 > p > p 3 ◮ µ ′ is not Gibbs for p 4 > p

  22. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations (cont.) Let h be the original Ising field and fix β large enough s.t. ◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds If h = 0, alternated ω ′ = ⇒ ( ∗ ) for p/β small enough Hence, ∃ p 1 > p 2 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for 2 > p If h � = 0, ∃ ω ′ s.t. ( ∗ ) only for a range of p/β Hence, ∃ p 1 ≥ p 2 > p 3 ≥ p 4 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for p 2 > p > p 3 ◮ µ ′ is not Gibbs for p 4 > p

  23. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations (cont.) Let h be the original Ising field and fix β large enough s.t. ◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds If h = 0, alternated ω ′ = ⇒ ( ∗ ) for p/β small enough Hence, ∃ p 1 > p 2 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for 2 > p If h � = 0, ∃ ω ′ s.t. ( ∗ ) only for a range of p/β Hence, ∃ p 1 ≥ p 2 > p 3 ≥ p 4 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for p 2 > p > p 3 ◮ µ ′ is not Gibbs for p 4 > p

  24. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Renormalization and non-Gibbsianness Single-site Kadanoff transformations (cont.) Let h be the original Ising field and fix β large enough s.t. ◮ Original model with h = 0 has phase transition ◮ Pirogov-Sinai theory holds If h = 0, alternated ω ′ = ⇒ ( ∗ ) for p/β small enough Hence, ∃ p 1 > p 2 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for 2 > p If h � = 0, ∃ ω ′ s.t. ( ∗ ) only for a range of p/β Hence, ∃ p 1 ≥ p 2 > p 3 ≥ p 4 s.t. ◮ µ ′ is Gibbs for p > p 1 ◮ µ ′ is not Gibbs for p 2 > p > p 3 ◮ µ ′ is not Gibbs for p 4 > p

  25. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Dynamic Gibbs – non-Gibbs transitions Simulations, particle systems, PCA: µ t = S t µ with S t = semigroup of operators. Dynamic G–non-G : µ Gibbs but µ t non-Gibbs at some t Example: ◮ µ =low- T Ising model ◮ S t = S n infinite- T discrete-time Glauber � S x ( ω x | ω x ) = 1 − ǫ S = S { x } with S x ( − ω x | ω x ) = ǫ x [invariant measure = product measure = infinite- T Gibbs] Unquenching: heating up a low- T Ising model

  26. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Dynamic Gibbs – non-Gibbs transitions Simulations, particle systems, PCA: µ t = S t µ with S t = semigroup of operators. Dynamic G–non-G : µ Gibbs but µ t non-Gibbs at some t Example: ◮ µ =low- T Ising model ◮ S t = S n infinite- T discrete-time Glauber � S x ( ω x | ω x ) = 1 − ǫ S = S { x } with S x ( − ω x | ω x ) = ǫ x [invariant measure = product measure = infinite- T Gibbs] Unquenching: heating up a low- T Ising model

  27. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Dynamic Gibbs – non-Gibbs transitions Simulations, particle systems, PCA: µ t = S t µ with S t = semigroup of operators. Dynamic G–non-G : µ Gibbs but µ t non-Gibbs at some t Example: ◮ µ =low- T Ising model ◮ S t = S n infinite- T discrete-time Glauber � S x ( ω x | ω x ) = 1 − ǫ S = S { x } with S x ( − ω x | ω x ) = ǫ x [invariant measure = product measure = infinite- T Gibbs] Unquenching: heating up a low- T Ising model

  28. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Dynamic Gibbs – non-Gibbs transitions Simulations, particle systems, PCA: µ t = S t µ with S t = semigroup of operators. Dynamic G–non-G : µ Gibbs but µ t non-Gibbs at some t Example: ◮ µ =low- T Ising model ◮ S t = S n infinite- T discrete-time Glauber � S x ( ω x | ω x ) = 1 − ǫ S = S { x } with S x ( − ω x | ω x ) = ǫ x [invariant measure = product measure = infinite- T Gibbs] Unquenching: heating up a low- T Ising model

  29. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Un-quenching G–non-G transitions � � ω ω ′ ≡ S x ( ω ′ In matrix form S x x | ω x ) � 1 − ǫ � � 1 + a n � x = 1 ǫ 1 − a n S n S x = , ǫ 1 − ǫ 1 − a n 1 + a n 2 with a n = (1 − 2 ǫ ) n . Hence � 1 + a n � x | ω x ) = A n e p n ω ′ x ( ω ′ S n x ω x , p n = log 1 − a n Kadanoff with p n − − n → 0 ∞ and p n − − → n →∞ 0 − − →

  30. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Un-quenching G–non-G transitions � � ω ω ′ ≡ S x ( ω ′ In matrix form S x x | ω x ) � 1 − ǫ � � 1 + a n � x = 1 ǫ 1 − a n S n S x = , ǫ 1 − ǫ 1 − a n 1 + a n 2 with a n = (1 − 2 ǫ ) n . Hence � 1 + a n � x | ω x ) = A n e p n ω ′ x ( ω ′ S n x ω x , p n = log 1 − a n Kadanoff with p n − − n → 0 ∞ and p n − − → n →∞ 0 − − →

  31. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Un-quenching G–non-G transitions (cont.) Using previous results on Kadanoff-renormalized measures: Gibbs Non-Gibbs . . . . . . ( h = 0) ✲ 0 n 1 n 2 Gibbs Non-Gibbs Gibbs . . . . . . ( h > 0) ✲ n 3 n 4 0 n 1 n 2 Mathematical mechanism: hidden variables (two-slice view) Physical mechanism?

  32. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Un-quenching G–non-G transitions (cont.) Using previous results on Kadanoff-renormalized measures: Gibbs Non-Gibbs . . . . . . ( h = 0) ✲ 0 n 1 n 2 Gibbs Non-Gibbs Gibbs . . . . . . ( h > 0) ✲ n 3 n 4 0 n 1 n 2 Mathematical mechanism: hidden variables (two-slice view) Physical mechanism?

  33. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Definition and example Un-quenching G–non-G transitions (cont.) Using previous results on Kadanoff-renormalized measures: Gibbs Non-Gibbs . . . . . . ( h = 0) ✲ 0 n 1 n 2 Gibbs Non-Gibbs Gibbs . . . . . . ( h > 0) ✲ n 3 n 4 0 n 1 n 2 Mathematical mechanism: hidden variables (two-slice view) Physical mechanism?

  34. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: Heuristic version A. van Enter: most probable history of an improbable state Given a large improbable droplet. How did it get there? ◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived To compete: typical of the other phase (cost exp-perimeter) Heuristic version: ◮ Short t : Only nature, no time to change much ◮ Mid t : ◮ ω sp nurtured, but ξ ± nature ◮ Hence ξ ± determines original phase → discontinuity ◮ Long t : If h � = 0 only one phase → no tilting mechanism

  35. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: Heuristic version A. van Enter: most probable history of an improbable state Given a large improbable droplet. How did it get there? ◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived To compete: typical of the other phase (cost exp-perimeter) Heuristic version: ◮ Short t : Only nature, no time to change much ◮ Mid t : ◮ ω sp nurtured, but ξ ± nature ◮ Hence ξ ± determines original phase → discontinuity ◮ Long t : If h � = 0 only one phase → no tilting mechanism

  36. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: Heuristic version A. van Enter: most probable history of an improbable state Given a large improbable droplet. How did it get there? ◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived To compete: typical of the other phase (cost exp-perimeter) Heuristic version: ◮ Short t : Only nature, no time to change much ◮ Mid t : ◮ ω sp nurtured, but ξ ± nature ◮ Hence ξ ± determines original phase → discontinuity ◮ Long t : If h � = 0 only one phase → no tilting mechanism

  37. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: Heuristic version A. van Enter: most probable history of an improbable state Given a large improbable droplet. How did it get there? ◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived To compete: typical of the other phase (cost exp-perimeter) Heuristic version: ◮ Short t : Only nature, no time to change much ◮ Mid t : ◮ ω sp nurtured, but ξ ± nature ◮ Hence ξ ± determines original phase → discontinuity ◮ Long t : If h � = 0 only one phase → no tilting mechanism

  38. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: Heuristic version A. van Enter: most probable history of an improbable state Given a large improbable droplet. How did it get there? ◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived To compete: typical of the other phase (cost exp-perimeter) Heuristic version: ◮ Short t : Only nature, no time to change much ◮ Mid t : ◮ ω sp nurtured, but ξ ± nature ◮ Hence ξ ± determines original phase → discontinuity ◮ Long t : If h � = 0 only one phase → no tilting mechanism

  39. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: Heuristic version A. van Enter: most probable history of an improbable state Given a large improbable droplet. How did it get there? ◮ Nurture: Created by the dynamics (cost exp-volume) ◮ Nature: Present at t = 0 and survived To compete: typical of the other phase (cost exp-perimeter) Heuristic version: ◮ Short t : Only nature, no time to change much ◮ Mid t : ◮ ω sp nurtured, but ξ ± nature ◮ Hence ξ ± determines original phase → discontinuity ◮ Long t : If h � = 0 only one phase → no tilting mechanism

  40. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: rigorous version most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories of measures conditioned to a given final empirical measure ◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness Perturbation of conditioning → discontinuous choice of trajectory

  41. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: rigorous version most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories of measures conditioned to a given final empirical measure ◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness Perturbation of conditioning → discontinuous choice of trajectory

  42. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: rigorous version most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories of measures conditioned to a given final empirical measure ◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness Perturbation of conditioning → discontinuous choice of trajectory

  43. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: rigorous version most probable history = most probable trajectory most probable = minimizer of the large-deviation rate Paradigm: Establish a large-deviation principle for trajectories of measures conditioned to a given final empirical measure ◮ Single minimizer = Gibbsianness ◮ Multiple minimizers = non-Gibbsianness Perturbation of conditioning → discontinuous choice of trajectory

  44. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mechanism Alternative paradigm: graphical summary [ h = 0] ^ ^ m(t, � ) m(t, � ) Forbidden Region m * t 1 � � c c t t -m * One trajectory = Gibbs Many trajectories = non-Gibbs

  45. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition The program Prove rigorously the previous paradigm. Steps: (i) Mean-field models (ii) Kac models (iii) Finite-range models At present: (i) and (ii) for Ising under independent dynamics (i) Mean-field: ◮ No geometry – no notion of neighbourhood ◮ Everything in terms of empirical magnetization

  46. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition The program Prove rigorously the previous paradigm. Steps: (i) Mean-field models (ii) Kac models (iii) Finite-range models At present: (i) and (ii) for Ising under independent dynamics (i) Mean-field: ◮ No geometry – no notion of neighbourhood ◮ Everything in terms of empirical magnetization

  47. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition Mean-field Ising model N Ising spins ( ω i ∈ {− 1 , 1 } ) N N � � H N ( σ ) − J = σ i σ j − h σ i 2 N i,j =1 i =1 = NH ( m N ( σ )) where m N is the empirical magnetization � m N ( σ ) = 1 Nσ i N i =1 and, if m ∈ M N := {− 1 , − 1 + 2 N − 1 , . . . , +1 − 2 N − 1 , +1 } , 2 Jm 2 − hm H ( m ) := − 1

  48. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition Mean-field Ising model N Ising spins ( ω i ∈ {− 1 , 1 } ) N N � � H N ( σ ) − J = σ i σ j − h σ i 2 N i,j =1 i =1 = NH ( m N ( σ )) where m N is the empirical magnetization � m N ( σ ) = 1 Nσ i N i =1 and, if m ∈ M N := {− 1 , − 1 + 2 N − 1 , . . . , +1 − 2 N − 1 , +1 } , 2 Jm 2 − hm H ( m ) := − 1

  49. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition Mean-field measures and evolution The H N -Gibbs measure [ β absorbed] µ N ( dσ ) = e − H N ( σ ) dσ Z N induces a measure on M N � � e − NH ( m ) N µ N ( dm ) := dm 1+ m N N Z 2 [ µ N ← → µ N + permutation invariance]

  50. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition Mean-field measures and evolution The H N -Gibbs measure [ β absorbed] µ N ( dσ ) = e − H N ( σ ) dσ Z N induces a measure on M N � � e − NH ( m ) N µ N ( dm ) := dm 1+ m N N Z 2 [ µ N ← → µ N + permutation invariance]

  51. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition Mean-field evolution Independent ( T = ∞ ) dynamics on Ω N induces on M N a continuous-time Markov chain ( m N t ) t ≥ 0 with generator � � � � 1 + m f ( m − 2 N − 1 ) − f ( m ) L N f ( m ) = N 2 � � +1 − m f ( m + 2 N − 1 ) − f ( m ) N 2 This induces a dynamics on measures on M N t ( f ) = µ N � � e tL N f µ N

  52. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Definition Mean-field evolution Independent ( T = ∞ ) dynamics on Ω N induces on M N a continuous-time Markov chain ( m N t ) t ≥ 0 with generator � � � � 1 + m f ( m − 2 N − 1 ) − f ( m ) L N f ( m ) = N 2 � � +1 − m f ( m + 2 N − 1 ) − f ( m ) N 2 This induces a dynamics on measures on M N t ( f ) = µ N � � e tL N f µ N

  53. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Non-Gibbsianness Mean-field single-site specification (K¨ ulske and Le Ny) Consider the single-spin conditional probabilities γ N t ( σ 1 | α N − 1 ) := µ N t ( σ 1 | σ N − 1 ) , with ◮ σ 1 ∈ {− 1 , +1 } , ◮ α N − 1 ∈ M N − 1 , ◮ σ N − 1 ∈ Ω N − 1 any configuration s.t. m N − 1 ( σ N − 1 ) = α N − 1 [By permutation invariance RHS independ of choice of σ N − 1 ]

  54. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Non-Gibbsianness Gibbs and Non-Gibbs mean-field models (K¨ ulske and Le Ny) For fixed t ≥ 0: (a) A magnetization α ∈ [ − 1 , 1] is good for µ t if γ N γ t ( · | � α ) := lim t ( · | α N − 1 ) , N →∞ α N → � α ◮ exists and is independent of the sequence α N → � α ◮ it is continuous in � α for � α in a neighbourhood of α (b) A magnetization α ∈ [ − 1 , +1] is bad if it is not good (c) µ t is Gibbs if it has no bad magnetizations

  55. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Non-Gibbsianness Gibbs and Non-Gibbs mean-field models (K¨ ulske and Le Ny) For fixed t ≥ 0: (a) A magnetization α ∈ [ − 1 , 1] is good for µ t if γ N γ t ( · | � α ) := lim t ( · | α N − 1 ) , N →∞ α N → � α ◮ exists and is independent of the sequence α N → � α ◮ it is continuous in � α for � α in a neighbourhood of α (b) A magnetization α ∈ [ − 1 , +1] is bad if it is not good (c) µ t is Gibbs if it has no bad magnetizations

  56. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations Large-deviations: General definition Informally: A family of measures ( ν N ) satisfies a large-deviation principle if ν N ( A ) ∼ e − N I ( A ) ◮ N is the LDP speed , I the rate function ◮ As a consequence, supp( ν N ) → argmin( I ) Formally: ( ν N ) on a Borel space satisf. LDP with rate fcn I and speed N if 1 N log ν N ( A ) lim inf ≥ − inf x ∈ A I ( x ) for A open N →∞ 1 N log ν N ( A ) lim sup ≤ − sup I ( x ) for A closed N →∞ x ∈ A

  57. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations Large-deviations: General definition Informally: A family of measures ( ν N ) satisfies a large-deviation principle if ν N ( A ) ∼ e − N I ( A ) ◮ N is the LDP speed , I the rate function ◮ As a consequence, supp( ν N ) → argmin( I ) Formally: ( ν N ) on a Borel space satisf. LDP with rate fcn I and speed N if 1 N log ν N ( A ) lim inf ≥ − inf x ∈ A I ( x ) for A open N →∞ 1 N log ν N ( A ) lim sup ≤ − sup I ( x ) for A closed N →∞ x ∈ A

  58. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations Large-deviations: General definition Informally: A family of measures ( ν N ) satisfies a large-deviation principle if ν N ( A ) ∼ e − N I ( A ) ◮ N is the LDP speed , I the rate function ◮ As a consequence, supp( ν N ) → argmin( I ) Formally: ( ν N ) on a Borel space satisf. LDP with rate fcn I and speed N if 1 N log ν N ( A ) lim inf ≥ − inf x ∈ A I ( x ) for A open N →∞ 1 N log ν N ( A ) lim sup ≤ − sup I ( x ) for A closed N →∞ x ∈ A

  59. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations LDP for mean-field Ising: “Static” part: The family ( µ N ) satisfies a LDP with speed N and rate I S − inf( I S ) with I S ( m ) := H ( m ) + 1 + m log(1 + m ) + 1 − m log(1 − m ) . 2 2 Independent evolutions: Let P N = law of ( m N t ) t ≥ 0 Defined on the space of c` adl` ag trajectories; Skorohod topology

  60. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations LDP for mean-field Ising: “Static” part: The family ( µ N ) satisfies a LDP with speed N and rate I S − inf( I S ) with I S ( m ) := H ( m ) + 1 + m log(1 + m ) + 1 − m log(1 − m ) . 2 2 Independent evolutions: Let P N = law of ( m N t ) t ≥ 0 Defined on the space of c` adl` ag trajectories; Skorohod topology

  61. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations LDP for mean-field evolutions (Ermolaev and K¨ ulske) ( P N ) restricted to [0 , T ] satisfies LDP with speed N and rate I T − inf( I T ) given by I T ( φ ) := I S ( φ (0)) + I T D ( φ ) , where � � T 0 L ( φ ( s ) , ˙ if ˙ φ ( s )) ds φ exists I T D ( φ ) := ∞ otherwise is the action integral with Lagrangian � − 1 4 (1 − m 2 ) + ˙ m 2 L ( m, ˙ m ) = 2 �� � m 2 + ˙ 4 (1 − m 2 ) + ˙ m +1 2 ˙ m log + 1 2(1 − m )

  62. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations LDP for mean-field evolutions (Ermolaev and K¨ ulske) ( P N ) restricted to [0 , T ] satisfies LDP with speed N and rate I T − inf( I T ) given by I T ( φ ) := I S ( φ (0)) + I T D ( φ ) , where � � T 0 L ( φ ( s ) , ˙ if ˙ φ ( s )) ds φ exists I T D ( φ ) := ∞ otherwise is the action integral with Lagrangian � − 1 4 (1 − m 2 ) + ˙ m 2 L ( m, ˙ m ) = 2 �� � m 2 + ˙ 4 (1 − m 2 ) + ˙ m +1 2 ˙ m log + 1 2(1 − m )

  63. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations LDP for conditioned mean-field evolutions The family of measures on trajectory space t,α ( · ) := P N � � � � m N ( t ) = α Q N ( m N ( s )) 0 ≤ s ≤ t = · satisfies LDP with speed N and rate I t,α − inf( I t,α ), with � I t ( φ ) if φ t = α I t,α ( mφ ) = ∞ otherwise Hence, conditioned optimal trajectories correspond to I t ( φ ) argmin φ : φ ( t )= α

  64. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Large deviations LDP for conditioned mean-field evolutions The family of measures on trajectory space t,α ( · ) := P N � � � � m N ( t ) = α Q N ( m N ( s )) 0 ≤ s ≤ t = · satisfies LDP with speed N and rate I t,α − inf( I t,α ), with � I t ( φ ) if φ t = α I t,α ( mφ ) = ∞ otherwise Hence, conditioned optimal trajectories correspond to I t ( φ ) argmin φ : φ ( t )= α

  65. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results The mean-field computational advantage Simplifying feature: ◮ There is an explicit expression for I t ( φ ) C t,α ( m ) := inf φ : φ (0)= m, φ ( t )= α ◮ We have the identity φ : φ ( t )= α I t ( φ ) m ∈ [ − 1 , +1] C t,α ( m ) = inf inf Hence, multiple conditioned trajectories ⇐ ⇒ multiple global minima of C t,α ( m )

  66. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results The mean-field computational advantage Simplifying feature: ◮ There is an explicit expression for I t ( φ ) C t,α ( m ) := inf φ : φ (0)= m, φ ( t )= α ◮ We have the identity φ : φ ( t )= α I t ( φ ) m ∈ [ − 1 , +1] C t,α ( m ) = inf inf Hence, multiple conditioned trajectories ⇐ ⇒ multiple global minima of C t,α ( m )

  67. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results I. Single optimal trajectory = Gibbsianness α �→ γ t ( σ | α ) is continuous at α 0 if and only if ◮ I t ( φ ) has a unique minimizing path � φ ◮ or, equivalently, C t,α 0 ( m ) has a unique minimizing m . Furthermore, in this case, the specification kernel equals � φ (0)+ h ] p t ( x, z ) x ∈{− 1 , +1 } e x [ J � γ t ( z | α ) = � φ (0)+ h ] p t ( x, y ) x,y ∈{− 1 , +1 } e x [ J � p t ( · , · ) = kernel of Markov jump process on {− 1 , +1 } with ◮ jumping rate 1 ◮ jump probabilities p t ( i, ± i ) = e − t � cosh( t ) sinh( t ) “If” part and for of γ t proven by Ermolaev and K¨ ulske

  68. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results I. Single optimal trajectory = Gibbsianness α �→ γ t ( σ | α ) is continuous at α 0 if and only if ◮ I t ( φ ) has a unique minimizing path � φ ◮ or, equivalently, C t,α 0 ( m ) has a unique minimizing m . Furthermore, in this case, the specification kernel equals � φ (0)+ h ] p t ( x, z ) x ∈{− 1 , +1 } e x [ J � γ t ( z | α ) = � φ (0)+ h ] p t ( x, y ) x,y ∈{− 1 , +1 } e x [ J � p t ( · , · ) = kernel of Markov jump process on {− 1 , +1 } with ◮ jumping rate 1 ◮ jump probabilities p t ( i, ± i ) = e − t � cosh( t ) sinh( t ) “If” part and for of γ t proven by Ermolaev and K¨ ulske

  69. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results I. Single optimal trajectory = Gibbsianness α �→ γ t ( σ | α ) is continuous at α 0 if and only if ◮ I t ( φ ) has a unique minimizing path � φ ◮ or, equivalently, C t,α 0 ( m ) has a unique minimizing m . Furthermore, in this case, the specification kernel equals � φ (0)+ h ] p t ( x, z ) x ∈{− 1 , +1 } e x [ J � γ t ( z | α ) = � φ (0)+ h ] p t ( x, y ) x,y ∈{− 1 , +1 } e x [ J � p t ( · , · ) = kernel of Markov jump process on {− 1 , +1 } with ◮ jumping rate 1 ◮ jump probabilities p t ( i, ± i ) = e − t � cosh( t ) sinh( t ) “If” part and for of γ t proven by Ermolaev and K¨ ulske

  70. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results II. Short-term Gibbsianness Theorem If J ≤ 1 the evolved measures µ t are Gibbs for all t ≥ 0 ◮ Proven by K¨ ulske and Le Ny and K¨ ulske and Ermolaev ◮ Note that 1 = β MF cr

  71. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results II. Short-term Gibbsianness Theorem If J ≤ 1 the evolved measures µ t are Gibbs for all t ≥ 0 ◮ Proven by K¨ ulske and Le Ny and K¨ ulske and Ermolaev ◮ Note that 1 = β MF cr

  72. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results III. Case J > 1 , h = 0 Consider the critical time � 1 if 1 < J ≤ 3 2 acoth(2 J − 1) 2 , Ψ c ( J ) := if J > 3 t ∗ ( J ) implicitly calculable 2 , Then: ◮ t < Ψ c : Evolved measure µ t is Gibbs ◮ t > Ψ c : Discontinuity at α = 0; two optimal trajectories ± � φ ◮ If Λ t, 0 ( J ) = cone between the trajectories ± ˆ φ ◮ No trajectory can penetrate Λ t, 0 ( J ) ◮ For J ≤ 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous ◮ For J > 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous except at t = Ψ c where it exhibits a right-continuous jump

  73. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results III. Case J > 1 , h = 0 Consider the critical time � 1 if 1 < J ≤ 3 2 acoth(2 J − 1) 2 , Ψ c ( J ) := if J > 3 t ∗ ( J ) implicitly calculable 2 , Then: ◮ t < Ψ c : Evolved measure µ t is Gibbs ◮ t > Ψ c : Discontinuity at α = 0; two optimal trajectories ± � φ ◮ If Λ t, 0 ( J ) = cone between the trajectories ± ˆ φ ◮ No trajectory can penetrate Λ t, 0 ( J ) ◮ For J ≤ 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous ◮ For J > 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous except at t = Ψ c where it exhibits a right-continuous jump

  74. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results III. Case J > 1 , h = 0 Consider the critical time � 1 if 1 < J ≤ 3 2 acoth(2 J − 1) 2 , Ψ c ( J ) := if J > 3 t ∗ ( J ) implicitly calculable 2 , Then: ◮ t < Ψ c : Evolved measure µ t is Gibbs ◮ t > Ψ c : Discontinuity at α = 0; two optimal trajectories ± � φ ◮ If Λ t, 0 ( J ) = cone between the trajectories ± ˆ φ ◮ No trajectory can penetrate Λ t, 0 ( J ) ◮ For J ≤ 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous ◮ For J > 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous except at t = Ψ c where it exhibits a right-continuous jump

  75. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results III. Case J > 1 , h = 0 Consider the critical time � 1 if 1 < J ≤ 3 2 acoth(2 J − 1) 2 , Ψ c ( J ) := if J > 3 t ∗ ( J ) implicitly calculable 2 , Then: ◮ t < Ψ c : Evolved measure µ t is Gibbs ◮ t > Ψ c : Discontinuity at α = 0; two optimal trajectories ± � φ ◮ If Λ t, 0 ( J ) = cone between the trajectories ± ˆ φ ◮ No trajectory can penetrate Λ t, 0 ( J ) ◮ For J ≤ 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous ◮ For J > 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous except at t = Ψ c where it exhibits a right-continuous jump

  76. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results III. Case J > 1 , h = 0 Consider the critical time � 1 if 1 < J ≤ 3 2 acoth(2 J − 1) 2 , Ψ c ( J ) := if J > 3 t ∗ ( J ) implicitly calculable 2 , Then: ◮ t < Ψ c : Evolved measure µ t is Gibbs ◮ t > Ψ c : Discontinuity at α = 0; two optimal trajectories ± � φ ◮ If Λ t, 0 ( J ) = cone between the trajectories ± ˆ φ ◮ No trajectory can penetrate Λ t, 0 ( J ) ◮ For J ≤ 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous ◮ For J > 3 / 2 the map t �→ Λ t, 0 ( J ) is continuous except at t = Ψ c where it exhibits a right-continuous jump

  77. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results Graphic summary: h = 0 , α = 0 ^ ^ ^ m(t, � ) m(t, � ) m(t, � ) m � Forbidden Region m * t 1 � � � t 2 c c c t t t -m * -m � C t,0 C t,0 C t,0 -m * m * m m m t < Ψ c t = Ψ c t > Ψ c First row: Minimizing trajectories for ( J, h ) = (1 . 6 , 0) Second row: Corresponding plots of m �→ C t, 0 ( m )

  78. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Mean-field models: Results IV. Bad magnetizations as function of time J=1.4, h=0 J=2.5, h=0 � c UB t B � � t c t U -UB J=1.4, h=0.29 J=2.5, h=0.1 � T � U � * U � � * B U M B t t M T t B s B U L B t B B � L 1 < J ≤ 3 J > 3 2 2 S B = trifurcation point, rest of the line = bifurcation

  79. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition Kac models: Basic definitions n := Z d /n Z d = the discrete torus of size n ◮ ∆ d ◮ Ω n := {− 1 , +1 } ∆ d n = Ising-spin configurations on ∆ d n ◮ Kac-type Hamiltonian: � � � x − y � H n ( σ ) := − 1 h ( x J σ ( x ) σ ( y ) − n ) σ ( x ) 2 n d n x,y ∈ ∆ d x ∈ ∆ d n n [ J ≥ 0 symmetric] ◮ Gibbs measure associated with H n : µ n ( dσ ) := e − βH n ( σ ) dσ Z n

  80. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition Kac models: Basic definitions n := Z d /n Z d = the discrete torus of size n ◮ ∆ d ◮ Ω n := {− 1 , +1 } ∆ d n = Ising-spin configurations on ∆ d n ◮ Kac-type Hamiltonian: � � � x − y � H n ( σ ) := − 1 h ( x J σ ( x ) σ ( y ) − n ) σ ( x ) 2 n d n x,y ∈ ∆ d x ∈ ∆ d n n [ J ≥ 0 symmetric] ◮ Gibbs measure associated with H n : µ n ( dσ ) := e − βH n ( σ ) dσ Z n

  81. Intro Gibbs Non-Gibbs Dynamics Mean field Kac End Kac models: Definition Continuum limit ◮ T d := R d / Z d , the d -dimensional unit torus ◮ T d n := 1 n ∆ d n = (1 /n )-discretization of T d ◮ M ( T d n ) [ M ( T d )] = signed measures on T d n [ T d ] (TV ≤ 1) The empirical density of σ ∈ Ω n inside Λ ⊆ ∆ d n is π n Λ : Ω n → M ( T d n ) ⊆ M ( T d ) � Λ ( σ ) := 1 π n σ ( x ) δ x/n | Λ | x ∈ Λ

Recommend

A Large Deviation Principle for Gibbs States on countable Markov shifts at zero temperature

A Large Deviation Principle for Gibbs States on countable Markov shifts at zero temperature

A Large Deviation Principle for Gibbs States on countable Markov shifts at zero temperature Rodrigo Bissacot - IME USP Jointly with Edgardo P erez (IME-USP, Brazil) and Jairo K. Mengue (UFRGS, Brazil) Workshop on Functional Analysis and

897 views • 56 slides
The Gibbs Sampler CSE 527 Lecture 9 Lawrence, et al. Detecting Subtle Sequence

The Gibbs Sampler CSE 527 Lecture 9 Lawrence, et al. Detecting Subtle Sequence

The Gibbs Sampler CSE 527 Lecture 9 Lawrence, et al. Detecting Subtle Sequence Signals: A The Gibbs Sampler Gibbs Sampling Strategy for Multiple Sequence Alignment, Science 1993 The Double Helix Some DNA Binding Domains

62 views • 5 slides
CSE 527 Lecture 9 The Gibbs Sampler Talk Today Zasha Weinberg Combi HSB K-069, 1:30

CSE 527 Lecture 9 The Gibbs Sampler Talk Today Zasha Weinberg Combi HSB K-069, 1:30

CSE 527 Lecture 9 The Gibbs Sampler Talk Today Zasha Weinberg Combi HSB K-069, 1:30 Fast, accurate annotation of non-coding RNAs The Gibbs Sampler Lawrence et al. Detecting Subtle Sequence Signals: A Gibbs Sampling

972 views • 18 slides
College P Planning N Night GIBBS GIBBS HIGH IGH SCHOOL SC SCHO HOOL COUNSE SELING OFFICE

College P Planning N Night GIBBS GIBBS HIGH IGH SCHOOL SC SCHO HOOL COUNSE SELING OFFICE

College P Planning N Night GIBBS GIBBS HIGH IGH SCHOOL SC SCHO HOOL COUNSE SELING OFFICE INTRODUCTIONS Gibbs H High S School C Counselors: Kim Kim Ga Galyon on: s students l last n names A A-F Ma Male lee Hogan: s

490 views • 28 slides
Poisson-Minibatching for Gibbs Sampling with Convergence Rate Guarantees Ruqi Zhang and

Poisson-Minibatching for Gibbs Sampling with Convergence Rate Guarantees Ruqi Zhang and

Poisson-Minibatching for Gibbs Sampling with Convergence Rate Guarantees Ruqi Zhang and Christopher De Sa Cornell University Scale Gibbs Sampling by Subsampling Gibbs sampling is one of the most popular Markov chain Monte Carlo (MCMC) methods +

381 views • 16 slides
Mathematics in the Sciences Eggeling et al. Gibbs sampling for parsMMs with latent variables 1

Mathematics in the Sciences Eggeling et al. Gibbs sampling for parsMMs with latent variables 1

Introduction Gibbs sampling algorithm Case studies Gibbs sampling for parsimonious Markov models with latent variables Ralf Eggeling 1 , Pierre-Yves Bourguignon 2 , Andr e Gohr 1 , Ivo Grosse 1 1 Martin Luther University Halle-Wittenberg 2 Max

522 views • 38 slides
Factors of Gibbs measures on subshifts What is a Gibbs measure? Two-ish definitions Equivalence

Factors of Gibbs measures on subshifts What is a Gibbs measure? Two-ish definitions Equivalence

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction Factors of Gibbs measures on subshifts What is a Gibbs measure? Two-ish definitions Equivalence DLR theorems Sophie MacDonald Preservation of Gibbsianness UBC

1.23k views • 94 slides
CH301 GIBBS FREE ENERGY UNIT 4 Day 8 What are we going to learn today? Quantify change in Gibbs

CH301 GIBBS FREE ENERGY UNIT 4 Day 8 What are we going to learn today? Quantify change in Gibbs

Sparks CH301 GIBBS FREE ENERGY UNIT 4 Day 8 What are we going to learn today? Quantify change in Gibbs Free Energy Predict Spontaneity at Specific Temperatures QUIZ: iClicker Questions S H2 = 131 J/K mol S O2 = 205 J/ K mol S H2O

472 views • 21 slides
Gibbs sampling Dr. Jarad Niemi Iowa State University March 29, 2018 Jarad Niemi (Iowa State)

Gibbs sampling Dr. Jarad Niemi Iowa State University March 29, 2018 Jarad Niemi (Iowa State)

Gibbs sampling Dr. Jarad Niemi Iowa State University March 29, 2018 Jarad Niemi (Iowa State) Gibbs sampling March 29, 2018 1 / 32 Outline Two-component Gibbs sampler Full conditional distribution K -component Gibbs sampler Blocked Gibbs

538 views • 32 slides
Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity Bjoern Bringmann University of California, Los Angeles October 6th, 2020 Baby equation Nonlinear wave equations (main result) The Gibbs

457 views • 29 slides
Gradient Gibbs measures with disorder Codina Cotar University College London April 16, 2015,

Gradient Gibbs measures with disorder Codina Cotar University College London April 16, 2015,

Gradient Gibbs measures with disorder Gradient Gibbs measures with disorder Codina Cotar University College London April 16, 2015, Providence Partly based on joint works with Christof Klske Gradient Gibbs measures with disorder Outline 1

616 views • 35 slides
Gradient Gibbs measures with disorder Codina Cotar University College London June 25, 2015, GGI

Gradient Gibbs measures with disorder Codina Cotar University College London June 25, 2015, GGI

Gradient Gibbs measures with disorder Gradient Gibbs measures with disorder Codina Cotar University College London June 25, 2015, GGI Partly based on joint works with Christof Klske Gradient Gibbs measures with disorder Outline 1 The model

987 views • 35 slides
Gibbs-Markov-Young structures Jos e F. Alves International Workshop on Differentiable

Gibbs-Markov-Young structures Jos e F. Alves International Workshop on Differentiable

Gibbs-Markov-Young structures Jos e F. Alves International Workshop on Differentiable Dynamical Systems 5-9 August, 2019 Jilin University, China http://www.fc.up.pt/pessoas/jfalves/slides.pdf http://www.fc.up.pt/pessoas/jfalves/notes.pdf

1.23k views • 112 slides
Merrill Field (MRI)- Past, Present, and Future By D. Ralph Gibbs, Ph. D. Airport Manager

Merrill Field (MRI)- Past, Present, and Future By D. Ralph Gibbs, Ph. D. Airport Manager

Merrill Field (MRI)- Past, Present, and Future By D. Ralph Gibbs, Ph. D. Airport Manager March 7, 2019 donald.gibbs@anchorageak.gov Overview Who are we? In the beginning Today Operations Fiscal Airport

548 views • 39 slides
A large deviation approach to computing rare transitions in multi-stable stochastic turbulent

A large deviation approach to computing rare transitions in multi-stable stochastic turbulent

A large deviation approach to computing rare transitions in multi-stable stochastic turbulent flows Jason Laurie and Freddy Bouchet Laboratoire de Physique, ENS de Lyon, France ENS de Lyon, 13 June 2012 Bistability in Rotating Tank Experiment

224 views • 20 slides
Gibbs Sampling Bayesian Networks: A First Attempt with Cilk++ Alexander Dubbs May 13, 2010

Gibbs Sampling Bayesian Networks: A First Attempt with Cilk++ Alexander Dubbs May 13, 2010

Gibbs Sampling Bayesian Networks: A First Attempt with Cilk++ Alexander Dubbs May 13, 2010 Alexander Dubbs Gibbs Sampling Bayesian Networks: A First Attempt with Cilk++ Introduction I Gibbs sample large Bayesian Networks using C and Cilk++,

305 views • 15 slides
How well do the Hermite-Pad e approximants reduce the Gibbs phenomenon? Ana C. Matos joint

How well do the Hermite-Pad e approximants reduce the Gibbs phenomenon? Ana C. Matos joint

The Gibbs phenomenon Hermite-Pad e approach Particular sequences of HP approximants Numerical experiments How well do the Hermite-Pad e approximants reduce the Gibbs phenomenon? Ana C. Matos joint work with B. Beckermann, F. Wielonsky

671 views • 36 slides
Review Gibbs sampling MH with proposal Q( X | X ) = P( X B(i) | X B(i) ) I( X

Review Gibbs sampling MH with proposal Q( X | X ) = P( X B(i) | X B(i) ) I( X

Review Gibbs sampling MH with proposal Q( X | X ) = P( X B(i) | X B(i) ) I( X B(i) = X B(i) ) / #B failure mode: lock-down Relational learning (properties of sets of entities) document clustering,

715 views • 35 slides
College Planning Night GI GIBBS BBS HI HIGH GH SCHOOL SCHOOL CO COUNSE UNSELING O NG OFFI

College Planning Night GI GIBBS BBS HI HIGH GH SCHOOL SCHOOL CO COUNSE UNSELING O NG OFFI

College Planning Night GI GIBBS BBS HI HIGH GH SCHOOL SCHOOL CO COUNSE UNSELING O NG OFFI FFICE CE Int Introduct oductions ions Gibbs H High S h Scho hool C Counselors: Kim Kim Ga Galyon on: s students l last n

345 views • 32 slides
Residuals and Goodness-of-fit tests for marked Gibbs point processes Fr ed eric Lavancier

Residuals and Goodness-of-fit tests for marked Gibbs point processes Fr ed eric Lavancier

Residuals and Goodness-of-fit tests for marked Gibbs point processes Fr ed eric Lavancier (Laboratoire Jean Leray, Nantes) Joint work with J.-F. Coeurjolly (Grenoble) 01/02/2010 F. Lavancier () Residuals and GoF tests for Gibbs pp

713 views • 59 slides
Sampling Gibbs : Maximization Expectation Scribes Jered McInerney : 2- hang Xiongyi

Sampling Gibbs : Maximization Expectation Scribes Jered McInerney : 2- hang Xiongyi

Lecture 9 Sampling Gibbs : Maximization Expectation Scribes Jered McInerney : 2- hang Xiongyi Mixture Gaussian ( Gibbs Sampling 2) Homework : Model Generative Grapical Model let K , Eh I ) Mh pipe ~ . , , , . . b

459 views • 30 slides
Nonlinear Gibbs measure and equilibrium Bose gases Phan Th` anh Nam (LMU Munich) Joint work

Nonlinear Gibbs measure and equilibrium Bose gases Phan Th` anh Nam (LMU Munich) Joint work

Nonlinear Gibbs measure and equilibrium Bose gases Phan Th` anh Nam (LMU Munich) Joint work with Mathieu Lewin and Nicolas Rougerie ICMP Montreal, July 27, 2018 1 / 10 Goal Nonlinear Gibbs measure d ( u ) = z

120 views • 10 slides
More on Gibbs Energy 2NO 2 (g) N 2 O 4 (g) What do you

More on Gibbs Energy 2NO 2 (g) N 2 O 4 (g) What do you

More on Gibbs Energy 2NO 2 (g) N 2 O 4 (g) What do you predict for the sign of S? S = 175.8 J/K.mol H = 57.2 kJ/mol What is G?

504 views • 5 slides
Dynamic and Distributed Algorithms for Sampling from Gibbs Distributions Yitong Yin Nanjing

Dynamic and Distributed Algorithms for Sampling from Gibbs Distributions Yitong Yin Nanjing

Dynamic and Distributed Algorithms for Sampling from Gibbs Distributions Yitong Yin Nanjing University Gibbs Distribution G = ( V , E ) spin states q 2 each , distribution v V b v : [ q ] [0,1] v b v e A e : [ q

473 views • 30 slides

More recommend