Gradient interfaces with and without disorder Codina Cotar - PowerPoint PPT Presentation

Gradient interfaces with and without disorder Gradient interfaces with and without disorder Codina Cotar University College London September 09, 2014, Toronto

Gradient interfaces with and without disorder Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective interface models 2 The model Dimension d = 1 Generalization to dimension d ≥ 2 3 Questions 4 Known results Results: Strictly Convex Potentials Techniques: Strictly Convex Potentials Results: Non-convex potentials Interfaces with disorder 5 Open questions: non-convex potentials

Gradient interfaces with and without disorder Physics motivation Microscopic model ↔ emerging macroscopic structures. Macroscopic phases → microscopic interfaces Approach: Microscopic modelling of the interface itself.

Gradient interfaces with and without disorder Physics motivation Example 1: Elasticity Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective interface models 2 The model Dimension d = 1 Generalization to dimension d ≥ 2 3 Questions 4 Known results Results: Strictly Convex Potentials Techniques: Strictly Convex Potentials Results: Non-convex potentials Interfaces with disorder 5 Open questions: non-convex potentials

Gradient interfaces with and without disorder Physics motivation Example 1: Elasticity Crystals are macroscopic objects, with ordered arrangements of atoms or molecules in microscopic scale Mechanical model of a crystal: little balls connected by springs, where heat causes the jiggling Configuration: snapshot of the atoms’ positions at a given time.

Gradient interfaces with and without disorder Physics motivation Example 1: Elasticity In thermal equilibrium, the jigglings explore samples of a probability measure on the configurations. This is the Gibbs measure: Prob ( Configuration ) ∝ exp ( − β Energy of Configuration ) , where β = 1 / temperature > 0. Moving every atom in the same direction the same amount does not change the energy, and hence the probability, of the configuration (shift-invariance). If Hook’s law holds, the elastic energy between two atoms with displacements x , y is given by c ( x − y ) 2 (the force F needed to extend or compress a spring by some distance | x − y | is proportional to that distance). Then the measure on the atoms’ configurations is multi-dimensional Gaussian.

Gradient interfaces with and without disorder Physics motivation Recap-Gaussian Measure Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective interface models 2 The model Dimension d = 1 Generalization to dimension d ≥ 2 3 Questions 4 Known results Results: Strictly Convex Potentials Techniques: Strictly Convex Potentials Results: Non-convex potentials Interfaces with disorder 5 Open questions: non-convex potentials

Gradient interfaces with and without disorder Physics motivation Recap-Gaussian Measure 1D Gaussian random variables Recall: A standard 1D Gaussian random variable X has distribution given by the density P ( X ∈ [ x , x + dx ]) = exp ( − x 2 / 2 ) √ dx . 2 π

Gradient interfaces with and without disorder Physics motivation Recap-Gaussian Measure Gaussian random variables in R n If If � x , y � is an inner product in R n , then � � x , x � � ( 2 π ) − n / 2 exp 2 is the density of an associated multidimensional Gaussian. This is the same as taking n � z j e j j = 1 where { e j } is an orthonormal basis and { z j } are independent 1D Gaussians.

Gradient interfaces with and without disorder Physics motivation Example 2: Effective interface models Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective interface models 2 The model Dimension d = 1 Generalization to dimension d ≥ 2 3 Questions 4 Known results Results: Strictly Convex Potentials Techniques: Strictly Convex Potentials Results: Non-convex potentials Interfaces with disorder 5 Open questions: non-convex potentials

Gradient interfaces with and without disorder Physics motivation Example 2: Effective interface models The interface for the Ising model - simplest description of ferromagnetism The spontaneous magnetization on cooling down the substance below a critical temperature, the so-called Curie temperature. The Ising model on a domain Ω ⊂ Z d with free boundary condition, at inverse temperature β = 1 / T > 0 and external field h ∈ R , is given by the following Gibbs measure on spin configurations ( σ x ) x ∈ Ω ∈ {± 1 } Ω � � 1 � � P Ω , h ,β ( σ ) := β σ x σ y + h σ x P ( σ ) , exp Z Ω , h ,β x , y ∈ Ω x ∈ Ω | x − y | = 1 where P is the uniform distribution on {± 1 } Ω .

Gradient interfaces with and without disorder Physics motivation Example 2: Effective interface models Assume d = 2 and Ω = [ 0 , N ] × [ 0 , N ] . Spin configuration σ = { σ x } x ∈{ 0 ,..., N }×{ 0 ,..., N } , spins σ x ∈ {− 1 , 1 } Goal: Modelling and analysis of the interface phase boundary

Gradient interfaces with and without disorder The model Dimension d = 1 Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective interface models 2 The model Dimension d = 1 Generalization to dimension d ≥ 2 3 Questions 4 Known results Results: Strictly Convex Potentials Techniques: Strictly Convex Potentials Results: Non-convex potentials Interfaces with disorder 5 Open questions: non-convex potentials

Gradient interfaces with and without disorder The model Dimension d = 1 Interface — transition region that separates different phases Λ n := {− n , − n + 1 , . . . , n − 1 , n } , ∂ Λ n = {− n − 1 , n + 1 } Height Variables (configurations) φ i ∈ R , i ∈ Λ n Boundary condition 0, such that φ i = 0 , when i ∈ ∂ Λ n . The energy H ( φ ) := � n + 1 i = − n V ( φ i − φ i − 1 ) , with V ( s ) = s 2 for Hooke’s law.

Gradient interfaces with and without disorder The model Dimension d = 1 The finite volume Gibbs measure 1 ν 0 Λ n ( φ − n , . . . , φ 1 , . . . , φ n ) = exp ( − β H ( φ )) d φ Λ n = Z 0 Λ n n + 1 n 1 � � ( φ i − φ i − 1 ) 2 ) exp ( − β d φ i , Z 0 Λ n i = − n i = − n where β = 1 / T > 0, φ − n − 1 = φ n + 1 = 0 and n + 1 n � � � Z 0 ( φ i − φ i − 1 ) 2 ) Λ n := R 2 n + 1 exp ( − β d φ i , i = − n i = − n is a multidimensional centered Gaussian measure. We can replace the 0-boundary condition in ν 0 Λ n by a ψ -boundary condition in ν ψ Λ n with φ − n − 1 := ψ − n − 1 , φ n + 1 := ψ n + 1 .

Gradient interfaces with and without disorder The model Generalization to dimension d ≥ 2 Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective interface models 2 The model Dimension d = 1 Generalization to dimension d ≥ 2 3 Questions 4 Known results Results: Strictly Convex Potentials Techniques: Strictly Convex Potentials Results: Non-convex potentials Interfaces with disorder 5 Open questions: non-convex potentials

Gradient interfaces with and without disorder The model Generalization to dimension d ≥ 2 Replace the discrete interval {− n , − n + 1 , . . . , 1 , 2 , . . . , n } by a discrete box Λ n := {− n , − n + 1 , . . . , 1 , . . . , n − 1 , n } d , with boundary ∂ Λ n := { i ∈ Z d \ Λ n : ∃ j ∈ Λ n with | i − j | = 1 } . V ( φ i − φ j ) , where V ( s ) = s 2 The energy H ( φ ) := � i , j ∈ Λ n ∪ ∂ Λ n | i − j | = 1 and φ i = 0 for i ∈ ∂ Λ n . The corresponding finite volume Gibbs measure on R Λ n is given by 1 � ν 0 Λ n ( φ ) := exp ( − β H ( φ )) d φ i . Z Λ n i ∈ Λ n It is a Gaussian measure, called the Gaussian Free Field (GFF).

Gradient interfaces with and without disorder The model Generalization to dimension d ≥ 2 For GFF If x , y ∈ Λ n Λ n ( φ x , φ y ) = G Λ n ( x , y ) , cov ν 0 where G Λ n ( x , y ) is the Green’s function, that is, the expected number of visits to y of a simple random walk started from x killed when it exits Λ n . GFF appears in many physical systems; two-dimensional GFF has close connections to Schramm-Loewner Evolution (SLE). Random, fractal curve in Ω ⊆ C simply connected. Introduced by Oded Schramm as a candidate for the scaling limit of loop erased random walk (and the interfaces in critical percolation). Contour lines of the GFF converge to SLE (Schramm-Sheffield 2009).

Gradient interfaces with and without disorder The model Generalization to dimension d ≥ 2 General potential V , general boundary condition ψ , general Λ V : R → R , V ∈ C 2 ( R ) with V ( s ) ≥ As 2 + B , A > 0 , B ∈ R for large s . The finite volume Gibbs measure on R Λ Λ ( φ ) := 1 ν ψ � � exp ( − β V ( φ i − φ j )) d φ i , Z ψ Λ i , j ∈ Λ ∪ ∂ Λ i ∈ Λ | i − j | = 1 where φ i = ψ i for i ∈ ∂ Λ . tilt u = ( u 1 , . . . , u d ) ∈ R d and tilted boundary condition ψ u i = i · u , i ∈ ∂ Λ . Finite volume surface tension (free energy) σ Λ ( u ) : macroscopic energy of a surface with tilt u ∈ R d . σ Λ ( u ) := 1 | Λ | log Z ψ u Λ . Gradients ∇ φ : ∇ φ b = φ i − φ j for b = ( i , j ) , | i − j | = 1