Limit shapes and crystal formation: a probabilist s point of view

Transcription

1 Limit shapes and crystal formation: a probabilist s point of view Marek Biskup (UCLA) Mathematical Facets of Crystallization Lorentz Center, Leiden 2014

2 Outline Some physics (via picture gallery) Mathematical models for crystal formation Phenomenological theory of equilibrium droplets Results on microscopic droplet models

3 PT phase diagram

4 Ordinary salt NaCl solid/vapor system T 650 C T 650 C J.C. Heyraud, J.J. Métois, J. Cryst. Growth 84 (1987)

5 Ordinary gold Au on graphite substrate T 1000 C

6 Morphological instablity Unstable conditions, depletion Webpage of Sharon Cooper, Department of Chemistry, University of Durham

7 Two types of models Two mechanisms affecting crystal shapes Growth dominated: DLA (IDLA, rotor-router model), PNG model, abelian sandpiles, Williams-Bjerkness tumor growth model,... Equilibration dominated: droplets in stat-mech models, Young tableaux, lozenge tiling height function, first-passage percolation, isoperimetric sets in random environment,...

8 Key mechanisms Surface vs bulk Bulk evolves only (mostly) via changes in surface (i.e., shape) Surface static/dynamics governed by bulk considerations

9 Growth-dominated models Diffusion limited aggregation T.A. Witten, L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981) Dynamics: Start with one particle on Z 2 New particle attached with probability proportional to harmonic measure from Also an Internal DLA version 10 6 particles

10 Growth-dominated models Polynuclear growth in 1+1 dimensions Dynamics: drop & grow Cube-root fluctuations & KPZ: K. Johansson, P. Ferrari, T. Sasamoto, H. Spohn, T. Seppäläinen, I. Corwin, A. Hammond,...

11 Growth-dominated models 2D liquid crystal model stirred by laser pulse K.A. Takeuchi, M. Sano, T. Sasamoto, H. Spohn, Science Reports 1, no. 34 (2011)

12 Growth-dominated models Abelian sandpile Sandpile of chips in Z 2 Dynamics: (on Z 2 ) Particles drop at origin If > 4 particles topple P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) D. Dhar, Phys. Rev. Lett. 64 (1990), no. 14, L. Levine, W. Pegden, C. Smart (arxiv: ) 10 6 particles Lionel Levine Abelian Networks

13 Equilibrated systems Unformly-random Young tableaux Counting Young tableaux of size N counting partitions of N Shape Theorem: A uniformly-random Young tableaux of size N, rescaled by N, has asymptotic shape described by e x + e y = 1 B.F. Logan and L.A. Shepp, Adv. Math. 26 (1977) A. Vershik and S. Kerov, Soviet Math. Dokl. 18 (1977)

14 Equilibrated systems Lozenge tiling height function Also: Ising (111) inteface perfect sample: courtesy D.B. Wilson (CFTP) phase classification: R. Kenyon, A. Okounkov, S. Sheffield, Ann. Math 163 (2006), no. 3,

15 Wulff s phenomenological theory Isoperimetric problem in disguise Gibbs-Wulff Theorem [Gibbs (1878), Wulff (1901)] Equilibrium crystal shape is that which minimizes the interface/surface energy amongst all sets of a given volume Wulff s ansatz: Interface/surface (free) energy given by where n := unit normal to A F (A) := A τ(n)dn τ(n) := surface tension for flat interface w/ normal n dn := surface (Hausdorff) measure on A So this boils down to an isoperimetric problem

16 Wulff construction Some convex duality Minimizing shape: W := { x R d : x n τ(n) } n =1 If n τ(n) convex, then τ = support function of W NOTE: Facets of W caused by exposed inner cusps of τ

17 Main questions Principal questions to address: Determine material constant τ Justify phenomenology from stat-mech point of view Need to develop a microscopic theory of droplet equilibrium Key issue: Both bulk and surface fluctuate so need to exhibit averaging in spatially inhomogeneous setting

18 Microscopic theory 2D percolation & Ising model Two competing attempts in early 1990s: 2D bond percolation: Let p > p c (Z 2 ). Denote C (x) := component of x and condition on {N C (0) < }. What s the asymptotic shape of C (0)? K.S. Alexander, L. Chayes, J. Chayes, Commun. Math. Phys. 131 (1990) D Ising model: For Ising model on N N box with spins σ, inverse temperature β β c (2), equilibrium magnetization m. Set M N := x σ x and condition on {M N = m N 2 a N }. What does σ typically look like? R.L. Dobrushin, R. Kotecký, S. Shlosman, Wulff construction: A global shape from local interaction, AMS 1992 NOTE: Conditioning on unlikely events!

19 Ising model Contour representation J. Math. Phys., Vol. 41, No. 3, March 2000 Equilibrium crystal shapes FIG. 9. The DKS picture under the 1/N scaling: On the left the microscopic N box with the unique K log N lar. On the right the continuous box K 1 with the scaled image of. Contour representation outside. In particular, the average magnetization inside respectively, outside is clos respectively, m*), and the area encircled by can be thus recovered from the c constraint, Scaling picture m* int m* N 2 int m*n 2 a N int a N 2m*. Under the scaling of N by 1/N, that is into the normalized continuous shape K R 2, th

20 DKS theorem P + N,β := Ising measure in N N box with + b.c. Theorem (DKS) Let d = 2, β β c (2) and suppose {v N } is a sequence with lim N v N /N 2 > 0 small. Then, with high probability (N ) w.r.t. ( M N = m N 2 2m ) v N P + N,β there is a unique contour Γ such that 1 N inf x R d d H(Γ,x + v N W 1 ) N 0 where W 1 is the Wulff shape of unit volume for given β, while all other contours have diameter O(log N). In fact: true for all β > β c D. Ioffe, R. Schonmann, Commun. Math. Phys. 199 (1998)

21 Proof of DKS theorem Definition of surface tension 8 INTRODUCTION Fig. 1.2 Definition. The surface tension with respect 1 to an interface orthogonal to a vector n 1 is the limit N τ(n) := lim N τ β(n) = lim lim 1 N Z(VN,M Z +, β, n) log N N M βd(n,n) Z(V N,M, β, +), (1.5.6) where d(n,n) is the length of the segment log Z (±,n) {t 2 :(t, n) =0,t 1 [ N,N]}. (1.5.7)

22 Proof of DKS theorem Calculus of skeletons PLAN OF THE PROOF 15 s-skeleton of contour Γ := polygon on s-grid interpolating Γ Lemma Fig Skeleton of a configuration with two large contours. As can be seen on the figure, not all intersections of the curves with the grid are taken. The employed algorithm assures that the distance between neighbouring intersections diverges in the thermodynamic limit. For any s > 0, any β > 0 and any collection S of s-skeletons, P + L,β (S) exp { C.-E. Pfister, Helv. Phys. Acta. 64 (1991) S S for every δ > 0. The claim (1.9.2) then immediately follows. The derivation of bounds (1.12.2) and (1.12.3) begins by picking up large contours. Simplifying slightly, we may describe it in the following way. We first fix ωn a sequence ωn such that log N and ωn 0. Among all contours of a configuration we choose } those of diameter larger then ωn; on each of these contours N we are then choosing (in an algorithmic way) a sequence of points, a skeleton, so that F the (S) distance of neighbouring points approximately equals ωn (see Fig. 1.4). Further, we make a partial integration by summing up the probabilities of all configurations having a fixed skeleton. First, we evaluate the contribution of an isolated i-th fragment of the contour joining two neighbouring vertices of the skeleton. This contribution can be measured by the ratio of the partition functions entering in the argument of the logarithm in (1.5.6) with n = ni, where ni is the unit vector orthogonal to the segment i joining the considered neighbouring vertices of the skeleton. Since the length i of this segment goes to, the considered contribution asymptotically equals exp{ βnτβ(ni) i}, and the total contribution from all the skeleton equals the product of contributions corresponding to separated segments and yields thus exp{ βn τβ(ni) i }. (1.12.4) i

23 Proof of DKS theorem Large-deviation theory Set skeleton cutoff s logn P + L,β ( σ : Γ only s-large contour in σ ) e F (Γ) Size restriction to get M N = m N 2 2m v N is V (Γ) v N Scale Γ by v N to have a unit volume. Best shape determined by w := min { F (A) : A = 1} All minimizers = shifts of W 1

24 Beyond 2 dimensions d 3 open for a long time L 1 -theory: Distance between contours measured by d(a,b) := 1 A 1 B 1 applied to coarse-grained interiors of contours T. Bodineau, Commun. Math. Phys. 207 (1999) R. Cerf, Astérisque 267 (2000) vi+177 R.Cerf, A. Pisztora, Ann.Probab. 28 (2000) REMARKS 13 Open problem: Prove hairs are not there! Fig A hair attached to a Wulff surface does not contribute

25 Droplet formation Critical regime for droplets to appear Question: What happens when v N N 2? Assume: For χ := susceptibility, suppose := 2(m ) 2 χw exists; i.e., v N N 4/3 (still d = 2) v 3/2 N lim N N 2 (0, ) Theorem Set Φ (λ ) := λ + (1 λ ) 2. Then lim N 1 logp + ( vn N,β MN = m N 2 2m ) v N = w inf Φ (λ ) 0 λ 1 M. Biskup, L. Chayes, R. Kotecký, Commun. Math. Phys. 242 (2003) Note inf 0 λ 1 Φ (λ ) 1 so doing better than before!

26 Droplet formation (continued) Analysis of Φ (λ ) := λ + (1 λ ) 2 Calculus: Set c := 1 2 ( 3 2 )3/2 1.0 (a) 2.0 (b) Two regimes Minimizers: < c > c 1.0 λ c λ Note the jump! 1.0

27 Droplet formation (continued) Theorem Let > 0. The following holds with high probability in measure ( M N = m N 2 2m ) v N P + N,β (1) if < c, all contours are O(logN) (2) if > c, there is a unique logn-large contour Γ with and V (Γ) = λ v N (1 + o(1)) 1 inf d H(Γ,x + λ v N W 1 ) 0 vn x R d N All other contours are O(logN) in size. M. Biskup, L. Chayes, R. Kotecký, Commun. Math. Phys. 242 (2003) NOTE: λ λ c > 0 once > c so droplet appears discontinuously!

28 Back of envelope calculation Interpretation: λ := fraction of excess spins ending up in a droplet, rest dissolves in bulk fluctuations (that have Gaussian tail) Probability for a given λ-fraction is This is recast as { exp { exp w (λv N ) d 1 d surface cost w (v N ) d 1 d ( λ d 1 d 2(m ) 2 Now optimize over λ to get result [(1 λ )2m v N ] 2 2χL d bulk fluctuations } (v N ) d+1 ) } d χw L }{{ d (1 λ ) 2 } NOTE: In general dimension: c = 1 d ( d+1 2 ) d+1 d and λ c = 2 d+1.

29 Experimentally checked Ising model, β 1.5β c, fixed magnetization λ 0.4 (a) L = L =80 L =160 L =320 L = magnetisation m /3 λ (b) c 1.5 =2 m2 0 χτw v 3/2 L L 2 2 analytic L =40 L =80 L =160 L =320 L = A. Nussbaumer, E. Bittner, W. Janke, Phys. Rev. E 77 (2008) FIG. 14: Fraction λ for the two-dimensional n.n. Ising model on square lattices of size L =

30 Droplet formation Challenging questions Question 1: Finite-size scaling effects O. Hryniv, D. Ioffe, R. Kotecký, in (perpetual) preparation Question 2: Dimensions d 3 M. Biskup, L. Chayes, R. Kotecký, Europhys. Lett. 60 (2002), no. 1, Question 3: Metastability (conserved dynamics)

31 Some more recent work Isoperimetric sets on supercritical percolation cluster M. Biskup, O. Louidor, E. Procaccia, R. Rosenthal, Commun. Pure App. Math. (to appear) Folding polymer model M. Biskup, E. Procaccia, R. Rosenthal (in preparation)

32 THE END