Formation of facets in an equilibrium model of surface growth Dima - PowerPoint PPT Presentation

Formation of facets in an equilibrium model of surface growth Dima Ioffe 1 Technion December 2011 1 Based on joint works with Senya Shlosman Dima Ioffe (Technion) Microscopic facets December 2011 1 / 27

Plan of the talk Low temperature 3D Ising model, Wulff shapes and (unknown) structure of microscopic facets. Facets on SOS surfaces. Effective model of microscopic facets. Results and proofs. Dima Ioffe (Technion) Microscopic facets December 2011 2 / 27

3D Ising model The Gibbs State ∂ Λ N N = 1 � � − H − σ x σ y − σ x Λ N ⊂ Z 3 2 x ∼ y x ∈ ∂ Λ N N ,β ( σ ) ∼ e − β H − P − N | Λ N | = N 3 Low Temperature β ≫ 1 ⇒ m ∗ ( β ) > 0. Phase Segregation: Fix m > − m ∗ and consider � σ x = mN 3 � P m , − N ,β ( · ) = P − � � � · . N ,β Dima Ioffe (Technion) Microscopic facets December 2011 3 / 27

Microscopic Wulff shape Typical Picture under P m , − N ,β Volume of the microscopic Γ N Wulff droplet | Γ N | ≈ m + m ∗ N 3 2 Theorem (Bodineau, Cerf-Pisztora): As N → ∞ the scaled shape 1 N Γ N converges to the macroscopic Wulff shape. Dima Ioffe (Technion) Microscopic facets December 2011 4 / 27

Surface Tension and Macroscopic Wulff Shape n + log Z ± | sin n | M τ β ( n ) = − lim . + Z − M 2 M →∞ M − − h ∈ ∂ K β h · n τ β = max M n Dilated Wulff Shape h � 1 / 3 � m + m ∗ K m β = K β K β 2 | K β | Dima Ioffe (Technion) Microscopic facets December 2011 5 / 27

Bodineau, Cerf-Pisztora Result Define (on unit box Λ ⊂ R 3 ) Γ N φ N ( t ) = 1 I { Nt ∈ Γ N } − 1 I { Nt �∈ Γ N } . Nu Define χ m ( t ) = 1 β } − 1 I { t ∈ K m I { t �∈ K m β } N ( K m β + u ) P m , − � � Then, under , N ,β u � φ N ( · ) − χ m ( u + · ) � L 1 (Λ) = 0 N →∞ min lim Dima Ioffe (Technion) Microscopic facets December 2011 6 / 27

Macroscopic Facets n Fn τ β - support function of K β . Then F n = ∂τ β ( n ) . K β Set e i - lattice direction. Dobrushin ’72, Miracle-Sole ’94: For β ≫ 1 F e i is a proper 2D facet Dima Ioffe (Technion) Microscopic facets December 2011 7 / 27

Microscopic Facets Zooming Bodineau, Cerf-Pisztora picture, what happens? Γ N NF e 1 OR Γ N NF e 1 OR Γ N NF e 1 Dima Ioffe (Technion) Microscopic facets December 2011 8 / 27

SOS Model Bodineau, Schonmann, Shlosman ’05 Γ N P N (Γ N = γ ) ∼ e − β | γ | VN � V N ≥ mN 3 � P m � � N ( · ) = P N · N Result: There exists a ( β ) ց 0 such that ℓN k : A k ≥ a ( β ) N 2 � � ℓ N = max k Ak satisfies A ℓ N − 1 ≥ (1 − a ( β )) N 2 . 0 N Dima Ioffe (Technion) Microscopic facets December 2011 9 / 27

Effective Model of Microscopic Facets Configuration: � � Γ N , { ξ v � ξ s � i } i ∈ V N , . pv j j ∈ S N VN Total number of particles: � � ξ v ξ s Ξ N = i + Γ N j i ∈ V N j ∈ S N ps | Γ | - area of Γ SN B p ( ξ ) = p ξ (1 − p ) 1 − ξ N β large Probability Distribution: P N (Γ , ξ v , ξ s ) ∝ e − β | Γ | � B p v ( ξ v � B p s ( ξ s i ) j ) . i ∈ V N j ∈ S N Dima Ioffe (Technion) Microscopic facets December 2011 10 / 27

Contour Representation of Γ Orientation of contours: Positive and negative (holes) α ( γ ) - signed area. | γ | - length. Compatibility γ ∼ γ ′ For Γ = { γ i } ∆ � � | Γ | ∼ | γ i | , α (Γ) = α ( γ i ) Dima Ioffe (Technion) Microscopic facets December 2011 11 / 27

Creation of Facets Ξ N - total number of particles E N (Ξ N ) = p s + p v pv N 3 ∆ = pN 3 2 Consider α (Γ N ) ps � Ξ N = pN 3 + aN 2 � P a � � N ( · ) = P N · δ = 2( ps − pv ) > 0 α (Γ N ) = bN 2 � � Surface Tension: log P ≈ − N . = pN 3 + δ N 2 α (Γ N ). � � � Bulk Fluctuations: E N Ξ N � α (Γ N ) = − ( aN 2 − δ bN 2 ) 2 � α (Γ N ) = bN 2 � ∼ Ξ N = pN 3 + aN 2 � � log P N . N 3 D where D = p s (1 − p s ) + p v (1 − p v ). Dima Ioffe (Technion) Microscopic facets December 2011 12 / 27

Reduction to Large Contours Fix β ≫ 1. Bulk fluctuations simplify analysis of P a N . Recall the contour representation Γ = { γ i } . Lemma 1 (No intermediate contours). ∀ a > 0 there exists ǫ = ǫ ( a ) > 0 such that � � ∃ γ i : 1 P a ǫ log N ≤ | γ i | ≤ ǫ N = o (1) . N Lemma 2 (Irrelevance of small contours) �� � � � P a α ( γ i )1 I {| γ i |≤ ǫ − 1 log N } ≫ N = o (1) . N Definition: γ is large if | γ | ≥ ǫ N . Dima Ioffe (Technion) Microscopic facets December 2011 13 / 27

Cluster Expansion and Reduced Model A. Fix a > 0 and forget about intermediate contours 1 ǫ log N ≤ | γ | ≤ ǫ N . B. Expand with respect to small contours | γ | ≤ 1 ǫ log N . For Γ = { γ i } collection of large contours the effective weight is − β � | γ i | − � � � ˆ P N (Γ) ∝ exp C�∼ Γ Φ β ( C ) . The family of clusters C depends on N and a . However the cluster weights Φ β ( C ) remain the same. The corrections are negligible: For all β sufficiently large ∃ ν ( β ) ր ∞ such that sup C� = ∅ e ν |C| | Φ β ( C ) | ≤ 1 . Reduced Model of Large Contours and Bulk Particles: P N (Γ , ξ v , ξ s ) = ˆ ˆ � � B p v ( ξ v B p s ( ξ s P N (Γ) i ) j ) i ∈ ˆ j ∈ ˆ V N S N Dima Ioffe (Technion) Microscopic facets December 2011 14 / 27

Surface Tension and Variational Problem � G β ( Nx ) = w β ( γ ) . C γ :0 → Nx Nx 1 τ β ( x ) = − lim N log G β ( Nx ) . N →∞ γ 0 � τ β ( γ ) = τ β ( n s ) d s . w β ( γ ) = e − β | γ |− P C�∼ γ Φ β ( γ ) γ Macroscopic Variational Problem B = [0 , 1] 2 unit box. γ 1 , . . . , γ n is a nested family of loops inside B : If for i � = j either γ i ⊆ γ j or γ j ⊆ γ i or γ i ∩ γ j = ∅ . Recall δ = 2( p s − p v ) and D = p v (1 − p v ) + p s (1 − p s ). � ( a − δ b ) 2 � � (VP) a min + min τ β ( γ i ) . D b α ( γ 1 )+ ··· + α ( γ n )= b Dima Ioffe (Technion) Microscopic facets December 2011 15 / 27

Solutions to (VP) a γ ∗ = ( γ ∗ 1 , . . . , γ ∗ n ) form regular stacks: γ ∗ 1 ⊇ γ ∗ 2 ⊇ · · · ⊇ γ ∗ All solutions ¯ n . Optimal loops γ ∗ i are of two types: B B r r W r T r β β Wulff shape of radius r Wulff TV of radius r Radius r ≤ 1 γ ∗ : Either (a) γ ∗ 1 = · · · = γ ∗ n = T r 2 is fixed for ¯ β or (b) γ ∗ 1 = · · · = γ ∗ n − 1 = T r β and γ ∗ n = W r β . Dima Ioffe (Technion) Microscopic facets December 2011 16 / 27

1st Order Transition in the Variational Problem γ ∗ = ( γ ∗ 1 , . . . , γ ∗ Let ¯ n ) be a solution to (VP) a . i α ( γ ∗ Define n = n ( a ), b = b ( a ) = � i ) and r = r ( a ). Then: n 3 2 1 a b 2 1 a r 1 2 a a 0 a 1 a 3 a 4 Dima Ioffe (Technion) Microscopic facets December 2011 17 / 27

1st Order Transition in the Microscopic Model Theorem. Fix β large. Then there exist 0 < a 1 < a 2 < a 3 < . . . such that ∀ a ∈ ( a n , a n +1 ) typical configurations under P a N N ; where a N = ⌊ N 3 a ⌋ , contain exactly n large contours, which are close in shape to N γ ∗ 1 , . . . , N γ ∗ n . Remark: 1st order transition - spontaneous appearance of a droplet of linear size N 2 / 3 in the context of the 2D Ising model was originally established by Biskup, Chayes and Kotecky CMP’03. Because of large bulk fluctuations in our model, their result is more difficult for n = 1, but for n = 2 , 3 , 4 , . . . large contours in our model start to interact, and a refined control is needed for deriving appropriate upper bounds. There are two levels of difficulty: (a) Controlling interactions between two large contours. (b) For β fixed, controlling interactions for arbitrary fixed number of large contours as N → ∞ . Dima Ioffe (Technion) Microscopic facets December 2011 18 / 27

Interaction Between 2 Contours C γ 1 γ 2 � � � � Φ β ( C ) = Φ β ( C ) + Φ β ( C ) − Φ β ( C ) C�∼ γ 1 ∪ γ 2 C�∼ γ 1 C�∼ γ 2 C�∼ γ 1 ∩ γ 2 Dima Ioffe (Technion) Microscopic facets December 2011 19 / 27

Interaction Between ℓ Contours γ 1 γ 2 γ 3 γℓ Dima Ioffe (Technion) Microscopic facets December 2011 20 / 27

Effective Random Walk Representation of G β Portion of a Contour Between x and y y x ˆ γm ˆ γ 1 ˆ γ 2 ξm ξ 1 y ξ 2 x � ρ β (ˆ e τ β ( y − x ) G β ( y − x ) ∼ � � γ i ) = m ˆ γ 1 ,... ˆ γ m { ρ β ( · ) } is a probability distribution on the set of irreducible animals. ξ 1 = ( T 1 , X 1 ) , ξ 2 = ( T 2 , X 2 ) , . . . steps of the effective random walk. Dima Ioffe (Technion) Microscopic facets December 2011 21 / 27

Attraction vrs Repulsion: Two Walks • S ( n ) = S (0) + � n 1 X ℓ , where X ℓ ∈ Z are i.i.d. with exponential tails. • S 1 ( · ) , S 2 ( · ) are two independent copies starting at x = ( x 1 , x 2 ) and ending (time n ) at y = ( y 1 , y 2 ). • Repulsion: Via event R + n = { S 1 ( ℓ ) ≥ S 2 ( ℓ ) ∀ ℓ = 0 , 1 , 2 , . . . , n } • Attraction: Via potential n � � S 1 Φ β, n ( S ) = φ β ( | I | ) , y 1 x 1 ℓ =1 I ⊃ S ( ℓ ) y 2 S 2 and φ β ( m ) ≤ e − c ( β ) m with x 2 I c ( β ) ր ∞ . Lemma. For all β large enough e Φ β, n ( S ) ; R + � � n ; S ( n ) = y ≤ 1 E x uniformly in x , y and n ≥ n 0 . Dima Ioffe (Technion) Microscopic facets December 2011 22 / 27