A nonlinear thermodynamical formalism J er ome BUZZI (CNRS Orsay) - PowerPoint PPT Presentation

A nonlinear thermodynamical formalism J´ erˆ ome BUZZI (CNRS Orsay) Joint work with Renaud LEPLAIDEUR Seminar Resistˆ encia Dinˆ amica July 10, 2020 J. Buzzi A nonlinear thermodynamical formalism July 2020 1 / 11

Outline Setup 1 A little statistical mechanics Thermodynamical formalism Results 2 Variational principle Equidistribution Equilibrium states Ingredients of proofs 3 Variational principle and Equidistribution Reduction of the nonlinear to the linear equilibrium states Conclusion 4 J. Buzzi A nonlinear thermodynamical formalism July 2020 2 / 11

Setup A little statistical mechanics A little statistical mechanics: Ising model One-dimensional spin system: X = { +1 , − 1 } Z , n ≥ 1 with σ : ( x n ) n ∈ Z �→ ( x n +1 ) n ∈ Z Each spin interacts with its immediate neighbors : Energy of x ∈ X N = { x ∈ X : σ N x = x } : � φ ◦ σ p ( x ) with φ ( x ) = − x 0 ( x − 1 + x +1 ) E N ( x ) := 0 < p < N A Gibbs ensemble is µ N ∈ P ( X N ) such that µ N ( x ) = 1 � Z N e − β E N ( x ) where Z N := e − β E N ( z ) ( β = ( temperature ) − 1 ) z ∈ X N Theorem (No phase transition for 1-D short-range) The Gibbs ensembles converge as n → ∞ to the unique µ 0 ∈ P ( σ ) which maximizes the free energy : P ( µ ) := h ( µ ) − βµ ( φ ) µ 0 is the equilibrium state . Moreover, P ( µ 0 ) = 1 β lim n →∞ − 1 n log Z n � Thermodynamical formalism for Subshifts of Finite Type (Sinai, Ruelle, Bowen) See: Ruelle, Thermodynamical formalism, Cambridge Mathematical Library, 1978, 2004 J. Buzzi A nonlinear thermodynamical formalism July 2020 3 / 11

Setup A little statistical mechanics A little statistical mechanics: Ising model One-dimensional spin system: X = { +1 , − 1 } Z , n ≥ 1 with σ : ( x n ) n ∈ Z �→ ( x n +1 ) n ∈ Z Each spin interacts with its immediate neighbors : Energy of x ∈ X N = { x ∈ X : σ N x = x } : � φ ◦ σ p ( x ) with φ ( x ) = − x 0 ( x − 1 + x +1 ) E N ( x ) := 0 < p < N A Gibbs ensemble is µ N ∈ P ( X N ) such that µ N ( x ) = 1 � Z N e − β E N ( x ) where Z N := e − β E N ( z ) ( β = ( temperature ) − 1 ) z ∈ X N Theorem (No phase transition for 1-D short-range) The Gibbs ensembles converge as n → ∞ to the unique µ 0 ∈ P ( σ ) which maximizes the free energy : P ( µ ) := h ( µ ) − βµ ( φ ) µ 0 is the equilibrium state . Moreover, P ( µ 0 ) = 1 β lim n →∞ − 1 n log Z n � Thermodynamical formalism for Subshifts of Finite Type (Sinai, Ruelle, Bowen) See: Ruelle, Thermodynamical formalism, Cambridge Mathematical Library, 1978, 2004 J. Buzzi A nonlinear thermodynamical formalism July 2020 3 / 11

Setup A little statistical mechanics A little statistical mechanics: Ising model One-dimensional spin system: X = { +1 , − 1 } Z , n ≥ 1 with σ : ( x n ) n ∈ Z �→ ( x n +1 ) n ∈ Z Each spin interacts with its immediate neighbors : Energy of x ∈ X N = { x ∈ X : σ N x = x } : � φ ◦ σ p ( x ) with φ ( x ) = − x 0 ( x − 1 + x +1 ) E N ( x ) := 0 < p < N A Gibbs ensemble is µ N ∈ P ( X N ) such that µ N ( x ) = 1 � Z N e − β E N ( x ) where Z N := e − β E N ( z ) ( β = ( temperature ) − 1 ) z ∈ X N Theorem (No phase transition for 1-D short-range) The Gibbs ensembles converge as n → ∞ to the unique µ 0 ∈ P ( σ ) which maximizes the free energy : P ( µ ) := h ( µ ) − βµ ( φ ) µ 0 is the equilibrium state . Moreover, P ( µ 0 ) = 1 β lim n →∞ − 1 n log Z n � Thermodynamical formalism for Subshifts of Finite Type (Sinai, Ruelle, Bowen) See: Ruelle, Thermodynamical formalism, Cambridge Mathematical Library, 1978, 2004 J. Buzzi A nonlinear thermodynamical formalism July 2020 3 / 11

Setup A little statistical mechanics A little statistical mechanics: Curie-Weiss mean-field model Each spin interacts with the average spin :   2 − x p · 1  1 � � � φ ◦ σ p ( x ) E N ( x ) := x q = N where φ ( x ) := x 0 .  N N 0 ≤ p < N 0 ≤ q < N 0 ≤ p < N A Gibbs ensemble is µ N ∈ P ( X N ) such that, for some y ∈ X , µ n ( x ) = 1 ζ N e − β E N ( x ) where ζ N := � e − β E N ( z ) z ∈ X N Theorem (Phase transition in 1-D mean-field) For β large, the Gibbs ensembles converge as n → ∞ to the average of two equilibrium states µ + , µ − ∈ P ( σ ) which are the two maximizers of the nonlinear free energy : Π( µ ) := h ( µ ) − βµ ( φ ) 2 Moreover, Π( µ 0 ) = lim n →∞ 1 n log ζ n See: R. Ellis, Entropy, large deviation and statistical mechanics, Springer-Verlag, 1985 � Leplaideur-Watbled (2019): quadratic thermodynamical formalism for SFTs Is there a nonlinear thermodynamical formalism for dynamical systems? J. Buzzi A nonlinear thermodynamical formalism July 2020 4 / 11

Setup A little statistical mechanics A little statistical mechanics: Curie-Weiss mean-field model Each spin interacts with the average spin :   2 − x p · 1  1 � � � φ ◦ σ p ( x ) E N ( x ) := x q = N where φ ( x ) := x 0 .  N N 0 ≤ p < N 0 ≤ q < N 0 ≤ p < N A Gibbs ensemble is µ N ∈ P ( X N ) such that, for some y ∈ X , µ n ( x ) = 1 ζ N e − β E N ( x ) where ζ N := � e − β E N ( z ) z ∈ X N Theorem (Phase transition in 1-D mean-field) For β large, the Gibbs ensembles converge as n → ∞ to the average of two equilibrium states µ + , µ − ∈ P ( σ ) which are the two maximizers of the nonlinear free energy : Π( µ ) := h ( µ ) − βµ ( φ ) 2 Moreover, Π( µ 0 ) = lim n →∞ 1 n log ζ n See: R. Ellis, Entropy, large deviation and statistical mechanics, Springer-Verlag, 1985 � Leplaideur-Watbled (2019): quadratic thermodynamical formalism for SFTs Is there a nonlinear thermodynamical formalism for dynamical systems? J. Buzzi A nonlinear thermodynamical formalism July 2020 4 / 11

Setup A little statistical mechanics A little statistical mechanics: Curie-Weiss mean-field model Each spin interacts with the average spin :   2 − x p · 1  1 � � � φ ◦ σ p ( x ) E N ( x ) := x q = N where φ ( x ) := x 0 .  N N 0 ≤ p < N 0 ≤ q < N 0 ≤ p < N A Gibbs ensemble is µ N ∈ P ( X N ) such that, for some y ∈ X , µ n ( x ) = 1 ζ N e − β E N ( x ) where ζ N := � e − β E N ( z ) z ∈ X N Theorem (Phase transition in 1-D mean-field) For β large, the Gibbs ensembles converge as n → ∞ to the average of two equilibrium states µ + , µ − ∈ P ( σ ) which are the two maximizers of the nonlinear free energy : Π( µ ) := h ( µ ) − βµ ( φ ) 2 Moreover, Π( µ 0 ) = lim n →∞ 1 n log ζ n See: R. Ellis, Entropy, large deviation and statistical mechanics, Springer-Verlag, 1985 � Leplaideur-Watbled (2019): quadratic thermodynamical formalism for SFTs Is there a nonlinear thermodynamical formalism for dynamical systems? J. Buzzi A nonlinear thermodynamical formalism July 2020 4 / 11

Setup Thermodynamical formalism Nonlinear Thermodynamical formalism Weighted dynamics ( X , T , φ ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: h T : P ( X ) → R ∪ {−∞} linear pressure: P φ ( T , µ ) = h T ( µ ) + µ ( φ ) nonlinear pressure: Π φ, F ( T , µ ) = h T ( µ ) + F ( µ ( φ )) Equilibrium states µ ∈ P ( T ) such that P φ ( T , µ ) = sup ν ∈ P ( T ) P φ ( T , ν ) (linear) µ ∈ P ( T ) such that Π φ, F ( T , µ ) = sup ν ∈ P ( T ) Π φ, F ( T , ν ) (nonlinear) Counting B T ( x , ǫ, n ) := { y ∈ X : ∀ 0 ≤ k < n d ( f k y , f k x ) < ǫ } ; ( ǫ, n )-cover of X : � x ∈ C B T ( x , ǫ, n ) = X := S n φ ( x ) � �� � weight of order n : w φ ( C , n ) := � � 0 ≤ k < n φ ( σ k x ) x ∈ C exp nonlinear weight of order n : ω φ ( C , n ) := � n exp � x ∈ C nF ( 1 0 ≤ k < n φ ( σ k x )) Topological pressure P top 1 φ ( T ) := lim ǫ → 0 lim sup n →∞ n log min C ( ǫ, n )-cover w φ ( C , n ) Π top 1 φ, F ( T ) := lim ǫ → 0 lim sup n →∞ n log min C ( ǫ, n )-cover ω φ, F ( C , n ) J. Buzzi A nonlinear thermodynamical formalism July 2020 5 / 11

Setup Thermodynamical formalism Nonlinear Thermodynamical formalism Weighted dynamics ( X , T , φ ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: h T : P ( X ) → R ∪ {−∞} linear pressure: P φ ( T , µ ) = h T ( µ ) + µ ( φ ) nonlinear pressure: Π φ, F ( T , µ ) = h T ( µ ) + F ( µ ( φ )) Equilibrium states µ ∈ P ( T ) such that P φ ( T , µ ) = sup ν ∈ P ( T ) P φ ( T , ν ) (linear) µ ∈ P ( T ) such that Π φ, F ( T , µ ) = sup ν ∈ P ( T ) Π φ, F ( T , ν ) (nonlinear) Counting B T ( x , ǫ, n ) := { y ∈ X : ∀ 0 ≤ k < n d ( f k y , f k x ) < ǫ } ; ( ǫ, n )-cover of X : � x ∈ C B T ( x , ǫ, n ) = X := S n φ ( x ) � �� � weight of order n : w φ ( C , n ) := � � 0 ≤ k < n φ ( σ k x ) x ∈ C exp nonlinear weight of order n : ω φ ( C , n ) := � n exp � x ∈ C nF ( 1 0 ≤ k < n φ ( σ k x )) Topological pressure P top 1 φ ( T ) := lim ǫ → 0 lim sup n →∞ n log min C ( ǫ, n )-cover w φ ( C , n ) Π top 1 φ, F ( T ) := lim ǫ → 0 lim sup n →∞ n log min C ( ǫ, n )-cover ω φ, F ( C , n ) J. Buzzi A nonlinear thermodynamical formalism July 2020 5 / 11