Equilibrium States of Interacting Particle Systems Diana Conache - PowerPoint PPT Presentation

Equilibrium States of Interacting Particle Systems Diana Conache Technische Universit¨ at M¨ unchen A Talk in the Framework of ”Konstanz Women in Mathematics” February 2016 Diana Conache (TU M¨ unchen) Equilibrium States of IPS 1 / 22

Outline Basics of Statistical Mechanics 1 The Existence Problem for Gibbs Fields 2 The Uniqueness Problem for Gibbs Fields 3 Classical Systems in Continuum 4 Diana Conache (TU M¨ unchen) Equilibrium States of IPS 2 / 22

Basics of Statistical Mechanics Motivation A particular aim of statistical mechanics is to study the macroscopic behaviour of a system, knowing the behaviour of the microscopic states x = ( x ℓ ) ℓ ∈ Z d given by collections of Ξ -valued random variables. The equilibrium states of the system are heuristically described by x ℓ probability measures of the form L µ = ” 1 Z e − βH ( x ) dx ” . However, for an infinite configuration x = ( x ℓ ) ℓ ∈ Z d , the Hamiltonian H ( x ) is not well-defined and thus, the definition of µ makes no sense. Diana Conache (TU M¨ unchen) Equilibrium States of IPS 3 / 22

Basics of Statistical Mechanics Dobrushin-Lanford-Ruelle Approach Idea: construct probability measures µ on (Ξ) Z d with prescribed conditional probabilities given by the family of stochastic kernels 1 � � � π L ( A | y ) = (Ξ) Z d ✶ A ( x ) exp − βH L ( x L | y ) ⊗ ℓ ∈ L dx ℓ ⊗ ℓ ′ ∈ L c δ y ℓ ′ . Z L ( y ) Main problems (since 1970): • existence (Dobrushin, Ruelle, ...); • uniqueness/ non-uniqueness = ⇒ phase transitions ◦ finite or compact Ξ (Dobrushin, Ruelle, ...); ◦ non-compact Ξ ( Lebowitz and Presutti-for a particular model; Dobrushin and Pechersky (1983) - in the general case). Diana Conache (TU M¨ unchen) Equilibrium States of IPS 4 / 22

Basics of Statistical Mechanics Phase Transitions - The 2D-Ising Model • the configuration space is X := {− 1 , +1 } Z 2 . • the local energy of x with boundary condition y � � � H L ( x L | y ) := J x ℓ x ℓ ′ + J x ℓ y ℓ ′ + h x ℓ ℓ ∈ L ℓ ∼ ℓ ′ , ℓ ∼ ℓ ′ ,ℓ ∈ L , ℓ,ℓ ′ ∈ L ℓ ′ �∈ L • the system of conditional distributions Π = { π L ( ·| y ) } L ⋐ Z 2 ,y ∈ X , where L ( B | y ) := 1 � π β 1 B ( x L × y L c ) exp {− βH L ( x L | y ) } ν L ( dx L ) Z β X L L Theorem For all β large enough and h = 0 , there exist two pure limit Gibbs distributions for the 2D ferromagnetic( J > 0 ) Ising model. Diana Conache (TU M¨ unchen) Equilibrium States of IPS 5 / 22

The Existence Problem for Gibbs Fields Outline Basics of Statistical Mechanics 1 The Existence Problem for Gibbs Fields 2 The Uniqueness Problem for Gibbs Fields 3 Classical Systems in Continuum 4 Diana Conache (TU M¨ unchen) Equilibrium States of IPS 6 / 22

The Existence Problem for Gibbs Fields A Strategy for Solving the Existence Problem Main question: Given a specification Π , does there exist a Gibbs measure consistent with Π ? Strategy: Introduce a topology on P ( X ) , pick a boundary condition y ∈ X and show that (I) the net { π L ( ·| y ) } L has a cluster point with respect to the chosen topology; (II) each cluster point of { π L ( ·| y ) } L is consistent with Π . Trick: Choosing the best-suited topology on P ( X ) , which in this case turns our to be the topology of local convergence . Diana Conache (TU M¨ unchen) Equilibrium States of IPS 7 / 22

The Existence Problem for Gibbs Fields Dobrushin’s Existence Criterion If the one-point specification associated to Π satisfies the condition below, then (I) is satisfied. For (II) some continuity assumption is also needed . Compactness Condition: There exist a compact function h : Ξ → R + and nonnegative constants C and I ℓℓ ′ , ℓ � = ℓ ′ such that (i) The matrix I = ( I ℓℓ ′ ) ℓ,ℓ ′ ∈ V satisfies � � I � 0 := sup I ℓℓ ′ < 1 . ℓ ℓ ′ � = ℓ (ii) For all ℓ ∈ V and y ∈ X � � h ( x ℓ ) π ℓ ( dx ℓ | y ) ≤ C + I ℓℓ ′ h ( x ℓ ′ ) . X ℓ ′ � = ℓ Diana Conache (TU M¨ unchen) Equilibrium States of IPS 8 / 22

The Uniqueness Problem for Gibbs Fields Outline Basics of Statistical Mechanics 1 The Existence Problem for Gibbs Fields 2 The Uniqueness Problem for Gibbs Fields 3 Classical Systems in Continuum 4 Diana Conache (TU M¨ unchen) Equilibrium States of IPS 9 / 22

The Uniqueness Problem for Gibbs Fields Approaches in Solving the Uniqueness Problem • Dobrushin classical criterion, Dobrushin-Pechersky, Dobrushin-Shlosman; • exponential decay of correlations; • exponential relaxation of the corresponding Glauber dynamics, expressed e inequalities for π ( dx | y ) ; by means of the log-Sobolev and Poincar´ • Ruelle’s superstability method. Most methods work only in the case of a compact spin space. We will focus on the Dobrushin-Pechersky criterion, which can be applied also to more general spin spaces. Diana Conache (TU M¨ unchen) Equilibrium States of IPS 10 / 22

The Uniqueness Problem for Gibbs Fields Contraction Condition Assume that π satisfies ℓ , π y � d ( π x ℓ ) ≤ κ ℓℓ ′ ✶ � = ( x ℓ ′ , y ℓ ′ ) , (CC) ℓ ′ ∈ ∂ℓ for all ℓ ∈ V and x, y ∈ X ℓ ( h, K ) , where κ = ( κ ℓℓ ′ ) ℓ,ℓ ′ ∈ V has positive entries and null diagonal such that � κ := sup ¯ κ ℓℓ ′ < 1 . ℓ ∈ V ℓ ′ ∈ ∂ℓ For a constant K > 0 , ℓ ∈ V and a measurable function h : Ξ → R + := [0 , + ∞ ) , we set X ℓ ( h, K ) = { x ∈ X : h ( x ℓ ) ≤ K for all ℓ ∈ ∂ℓ } . Diana Conache (TU M¨ unchen) Equilibrium States of IPS 11 / 22

The Uniqueness Problem for Gibbs Fields Integrability Condition Moreover, suppose h satisfies the following integrability condition � π x ℓ ( h ) ≤ 1 + c ℓℓ ′ h ( x ℓ ′ ) , (IC) ℓ ′ ∈ ∂ℓ for all ℓ ∈ V and x ∈ X , where c = ( c ℓℓ ′ ) ℓ,ℓ ′ ∈ V has positive entries and null diagonal such that � c := sup ¯ c ℓℓ ′ < C ( graph ) < 1 . ℓ ∈ V ℓ ′ ∈ ∂ℓ We introduce the set of tempered measures M ( π, h ) consisting of all measures µ ∈ M ( π ) for which � sup h ( x ℓ ) µ ( dx ) < ∞ . ℓ X Diana Conache (TU M¨ unchen) Equilibrium States of IPS 12 / 22

The Uniqueness Problem for Gibbs Fields The Uniqueness Result Theorem For each K > K ∗ ( graph ) and π ∈ Π( h, K, κ, c ) , the set M ( π, h ) contains at most one element. The proof of the theorem follows immediately from Lemma Let µ 1 , µ 2 ∈ M ( π, h ) and ν ∈ C ( µ 1 , µ 2 ) such that � � ✶ � = ( x 1 ℓ , x 2 ℓ ) ν ( dx 1 , dx 2 ) = 0 . γ ( ν ) := sup ℓ ∈ V X X Then µ 1 = µ 2 . Diana Conache (TU M¨ unchen) Equilibrium States of IPS 13 / 22

The Uniqueness Problem for Gibbs Fields The Uniqueness Result Theorem For each K > K ∗ ( graph ) and π ∈ Π( h, K, κ, c ) , the set M ( π, h ) contains at most one element. The proof of the theorem follows immediately from Lemma Let µ 1 , µ 2 ∈ M ( π, h ) and ν ∈ C ( µ 1 , µ 2 ) such that � � ✶ � = ( x 1 ℓ , x 2 ℓ ) ν ( dx 1 , dx 2 ) = 0 . γ ( ν ) := sup ℓ ∈ V X X Then µ 1 = µ 2 . Diana Conache (TU M¨ unchen) Equilibrium States of IPS 13 / 22

The Uniqueness Problem for Gibbs Fields Comparison with Dobrushin’s Classical Criterion An earlier uniqueness result due to Dobrushin (1968), for Ξ Polish, compact with ρ a metric that makes Ξ complete, requires that the following interdependence matrix be ℓ ∞ − contractive, i.e. W ρ ( π y 1 ℓ , π y 2 � � ℓ ) < 1 , ℓ � = ℓ ′ . D ℓℓ ′ := sup ρ ( y 1 ℓ ′ , y 2 ℓ ′ ) y 1 ,y 2 ∈ X y 1 = y 2 off ℓ ′ Advantages of the DP approach: • one needs to check the condition of weak dependence not for all boundary conditions (like here), but only for such y ∈ X whose components y ℓ lye in a certain ball in Ξ ; • it can also be applied for non-compact spins and for pair-potentials with more than quadratic growth. Diana Conache (TU M¨ unchen) Equilibrium States of IPS 14 / 22

The Uniqueness Problem for Gibbs Fields Decay of Correlations for Gibbs measures Theorem Let π and K be as in the previous theorem and M ( π, h ) be nonempty, hence containing a single state µ . Consider bounded functions f, g : X → R + , such that f is B (Ξ ℓ 1 ) -measurable and g is B (Ξ ℓ 2 ) -measurable. Then there exist positive C K and α K , dependent on K only, such that | Cov µ ( f ; g ) | ≤ C K � f � ∞ � g � ∞ exp [ − α K δ ( ℓ 1 , ℓ 2 )] , ℓ 1 , ℓ 2 ∈ L Diana Conache (TU M¨ unchen) Equilibrium States of IPS 15 / 22

Classical Systems in Continuum Outline Basics of Statistical Mechanics 1 The Existence Problem for Gibbs Fields 2 The Uniqueness Problem for Gibbs Fields 3 Classical Systems in Continuum 4 Diana Conache (TU M¨ unchen) Equilibrium States of IPS 16 / 22