The quasi-periodic Frenkel-Kontorova model

Transcription

1 The quasi-periodic Frenkel-Kontorova model Philippe Thieullen (Bordeaux) joint work with E. Garibaldi (Campinas) et S. Petite (Amiens) Korean-French Conference in Mathematics Pohang, August 24-28,2015 Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 1/32

2 Summary of the talk I. The general problem of minimizing configurations II. Almost periodic and quasi-crystalline models III. Some ideas of the proof Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 2/32

3 I. The general problem of minimizing configurations Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 3/32

4 I.1. General problem Notations Consider a discrete path in R d : (x n ) x Z (chain of atoms) Choose a discrete action E(x, y) : R d R d R (energy interaction between two consecutive atoms) Define the total action of the path E tot := n Z E(x n, x n+1 ) Problem Find a bi-infinite path which minimizes the total action Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 4/32

5 I.2. Three examples The periodic 1D Frenkel-Kontorova model E(x, y) = 1 2 y x λ 2 + K ( 1 cos(2πx) ) = W λ (y x) + V (x) λ: distance at rest when there is no external interaction W λ (x, y) = W 0 (x, y) λ(y x) + cte: elastic internal interaction V (x): periodic external interaction The almost periodic 1D Frenkel Kontorova E(x, y) = 1 2 y x λ 2 + K 1 ( 1 cos(2πx1) ) + K2 ( 1 cos(2πx 2) ) The quasicrystalline 1D model To be explained later Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 5/32

6 1.3. Minimizing configurations In the weak sense A configuration (x n ) n Z is said to be minimizing (in the weak sense) if for any finite sub-path (x m, x m+1,..., x n ) define the total action of the sub-path: E(x m,..., x n ) := n k=m+1 E(x k 1, x k ) by fixing the two endpoints x m and x n, the total action of the sub-path can only increase: let (y m, y m+1,..., y n ) be another finite path, with y m = x m and y n = x n, then E(x m,..., x n ) E(y m,..., y n ) Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 6/32

7 1.4. Calibrated configurations Remark The notion of minimizing configurations is close to the notion of minimal geodesics E(x, y) plays the role of the distance (or a cost) But there is no reason to ask E(x, y) 0 and E(x, y) = 0 x = y a normalizing factor Ē R has to be introduce Minimizing in the strong sense A configuration (x n ) n Z is minimizing if for any finite path (x m, x m+1,..., x n ) by fixing the two endpoints x m and x n if (y k, y k+1,..., y l ) is another path, possibly not having the same length, but with the same endpoints y k = x m and y l = x n E(x m,..., x n ) (n m)ē E(y k,..., y l ) (l k)ē A minimizing configuration in the strong sense will be called calibrated Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 7/32

8 I.5. Mañé potential The effective action Ē (or Mañé critical value, or mean energy per site, or ground energy) Ē := lim n + 1 inf x 0,...,x n R d n E(x 0,..., x n ) The Mañé potential S(x, y) Given two points x, y R d S(x, y) = inf inf n 1 x=x 0,...,x n=y [ E(x0,..., x n ) nē] Minimizing in the strong sense (x n ) n Z is said to be calibrated if m n, S(x m, x n ) = E(x m,..., x n ) (n m)ē Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 8/32

9 I.6. Assumptions for 1D periodic models General hypotheses E(x, y) : R R R is C 2 E(x, y) is translation periodic: E(x, y) is superlinear: E(x, y) is twist E(x + 1, y + 1) = E(x, y) lim inf R + y x R E(x, y) y x = + x, y, E(x, y) x y (x, ) < 0 and E(x, y) (, y) < 0 x y a.e. (check that if E(x, y) = 1 2 (y x)2 then E x y = 1) Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 9/32

10 I.7. Main question again Remark Under the hypotheses of translation periodicity, superlineraty and twist Ē = lim n + The Mañé potential S(x, y) = inf 1 inf x 0,...,x n R d n E(x 0,..., x n ) inf n 1 x=x 0,...,x n=y is finite and satisfies S(x, z) S(x, y) + S(y, z) (sub-cocycle) S(x, y) E(x, y) Ē S(x, x) 0 exists and is finite [ E(x0,..., x n ) nē] Question Do there exist calibrated configurations (x n ) n Z? m n E(x m,..., x n ) (n m)ē = S(x m, x n ) Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 10/32

11 1.8. Aubry theory for 1D periodic models One-parameter family E λ (x, y) := E(x, y) λ(y x) Aubry theory (1983) λ R Ē(λ) is C1 concave for every λ, there exist calibrated configurations for E λ ; and all calibrated configurations have the same rotation number ρ = x n x m lim n +, m n m = dē dλ (λ) conversely, every ρ is the rotation number of a calibrated configuration for some E λ Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 11/32

12 I.9. The original 1D Frenkel-Kontorova E λ,k (x, y) = 1 2 y x 2 λ(y x) + ρ = Ē (λ, K) λ K ( ) (2π) 2 1 cos(2πx) K=3 The system is locked at rational rotation number ρ ρ λ λ Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 12/32

13 II. Almost periodic and quasi-crystalline models Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 13/32

14 II.1. Example of almost periodic models The almost periodic 1D Frenkel-Kontorova E ω (x, y) = W (y x) + V ω (x) W (y x) = 1 2 y x λ 2 ( V ω (x) = K 1 1 cos 2π(ω1 + x) ) ( + K 2 1 cos(ω2 + x 2) ) Remark In the periodic case there is no need to introduce the space Ω of all phases. In the almost periodic case, it is more appropriate to work with a full family of discrete actions Notations Ω = T 2 = {(ω 1, ω 2 ) : ω i mod 1} the set of all environments τ t : Ω Ω, τ t (ω 1, ω 2 ) := (ω 1 + t, ω 2 + t 2) is minimal V ω (x) = V (τ x (ω)) where V : Ω R and t R, E ω (x+t, y+t) = E τt(ω)(x, y) (translation invariance property) Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 14/32

15 II.2.A general almost periodic model Definition An almost periodic model (Ω, {τ t } t R d, L) Ω a compact space τ t : Ω Ω a minimal flow τ s+t = τ s τ t a family of translation-invariant discrete actions x, y, t, R d E ω (x + t, y + t) = E τt(ω)(x, y) we assume actually that E ω (x, y) has a Lagrangian form E ω (x, y) := L(τ x (ω), y x) L(ω, t) : Ω R d R is C 0 in (ω, t) and superlinear in t lim R + inf ω Ω inf L(ω, t) = + t R t Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 15/32

16 II.3. A general quasicrystalline model Definition A quasicrystalline model (Ω, {τ t } t R, L) (Ω, {τ t } t R, L) is an almost periodic model (Ω, {τ t }) is minimal and uniquely ergodic d = 1 and (to simplify) L(ω, t) = W (t) + V (ω) E ω (x, y) = W (y x) + V ω (x) and V ω (x) = V (τ x (ω)) V : Ω R is C 0 W is twist: W is C 2 and convex W > 0 a.e. x, y R E ω x y (x, ) < 0 a.e. and E ω (, y) < 0 a.e. x y W (t) W is superlinear: lim = + t + t In addition to the above properties V is locally transversally constant Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 16/32

17 II.4. Locally transversally constant The periodic case V L x q 1 V ω (x) V L V L V L x x x x x q 0 0 q 1 q 2 q 3 q 4 α L L L L L x 1 tile L 1 function V L : [0, L] R The quasicrystalline case V S V L V L V L V S V S V S x x x x x x x x q 0 0 q 1 q 2 q 3 q q 5 q x 4 6 q 1 V ω (x) V S S L L S L S S S 2 tiles L,S 2 functions V L : [0, L] R V S : [0, S] R ω = (q n ) n Z ordered along R with minimal complexity Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 17/32

18 II.5. Quasicrystals Delone set An ordered subset ω = (qn) n Z R which is uniformly discrete and relatively dense Finite complexity There is a finite number of jumps n Z, n := q n+1 q n {L 1,..., L r } Repetitive For instance assume there are 2 tiles {L, S}. A pattern P = LSL is any finite sub-word. An occurrence of a pattern is some n n = L, n+1 = S, n+2 = L repetitive = for every pattern the set of occurrences is relatively dense Uniformly distributed For every pattern P, uniformly in x 1 Card{occurrences of P in [x T, x + T ]} ν(p ) > 0 2T Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 18/32

19 II.6. A model of Sturmian type Construction A model of type cut and projection let be α Q and B = {(x, y) R 2 : αx α < y αx + 1} we construct a sequence of points (q n u ) n 0 on the line D = R u, u = (1, α), by projecting orthogonally on D the set Z 2 B 3 2 0<α<1 D 1 q 4 0 q q q α ω = (q n) n Z is a quasicrystal with 2 tiles q n+1 q n { 1/ 1 + α 2, α/ 1 + α 2} = {L, S} A particular case For α = 5 1 2, one obtains the Fibonacci substitution S L et L LS Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 19/32

20 II.7. A quasicrystalline model Definition (first definition) ω a quasicrystal (an ordered subset of finite complexity, which is repetitive, with uniformly distributed patterns) a discrete action E ω (x, y) = W (y x) + V ω (x) V ω : R R a C 0 transversally constant function ( a juxtaposition of of potentials according to the tiles) W : R R is C 2 and satisfies W > 0 a.e. W (t) W is superlinear lim = + t + t Main theorem In a quasicrystalline model there exist calibrated configurations (x n ) n Z that is satisfying m n E ω (x m,..., x n ) (n m)ēω = S ω (x m, x n ) Ē ω := lim n + 1 inf x 0,...,x n R d n E ω (x 0,..., x n ) Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 20/32

21 III. Some ideas of the proof Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 21/32

22 III.1. Why locally transversally constant? Assume V ω is given by the Fibonacci sequence: ω (, S, L L, S, L, L, S, ) {L, S} Z V ω (x) V L V L V L V L V S V S V S x x x x x x x x q 1 q 0 =0 q 1 q 2 q 3 q 4 q 5 S L L S L L S E ω (x, y) = W (y x) + V ω (x) Denote σ : {L, S} Z {L, S} Z the left shift. Then we obtain infinitely many similar Fibonacci sequences Σ := {σ n (ω ) : n Z} Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 22/32

23 III.2. Why locally transversally constant? There is a natural R-action of the set of quasicrystals: ω ω t 0 L S ω t q 0 q 1 q 2 τ t (ω ) L τ L (ω ) q 0 L q 1 L Ω := Hull(ω ) = {τ t (ω ) : t R} compact-hausdorff Σ Σ S τ L (ω ) Ω suspension over Σ of a locally constant return map or Σ can be seen as a global transverse section to the flow Σ L 0 ω S t L { VL (t) if ω Σ V ω (t) = V (τ t ω)) = L V S (t) if ω Σ S Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 23/32

24 III.3. Why twist? Aubry lemma The twist property E(x, y) x y (x, ) < 0 and E(x, y) (, y) < 0 a.e. x y implies E(x 0, x 1 ) + E(y 0, y 1 ) > E(x 0, y 1 ) + E(y 0, x 1 ) x 0 y 0 y 1 x 1 x 0 y 0 y 1 x 1 Corollary Consider a finite minimizing path (x k ) n k=0. Then the following configuration is forbidden. x m x m+1 x m x m+1 x k 1 x k x k +1 Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 24/32

25 Proof III.4. Why twist? x m x m+1 x m x m+1 x k 1 x k x k +1 x m x k 1 x k x k +1 x m+1 x x m+1 m x k x m x k x m+1 x k 1 x k +1 Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 25/32

26 III.5. Why twist? Corollary Similarly the following configuration is forbidden x m 1 x m x m+1 x m 1 x m x m+1 x k 2 x k 1 x k x k +1 Consequence If (x k ) n k=0 is a finite minimizing path, for every pattern, say LSL, the number of points x k inside this pattern differs at most by 2 Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 26/32

27 III.6. Why uniquely ergodic? Proposition If (x (n) k )n k=0 realizes the minimum of E ω (x(n) 0,..., x(n) n ) among all finite path of size n, then x (n) k+1 x(n) k x (n) n x (n) 0 lim 1 n + n ν L N L + ν S N S Proof R (is uniformly bounded) (lim inf > 0 and lim sup < + ) x 0 x 1 x 2 x n T := x (n) n x (n) 0 L S L L S L ν L := (nb of occurrences of L )/T N L := nb of x k inside each tile L Then n = (T ν L )N L + (T ν S )N S Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 27/32

28 III.7. Why a quasicrystalline model? Beginning of the proof ω a Fibonacci quasicrystal V ω a locally transversally constant potential E ω a discrete action E ω (x, y) = W (y x) + V ω (x) (x (n) k )n k=0 = arg min x 0,...,x n E ω (x 0,..., x n ) a finite minimizing path Remark There is no reason that this path is calibrated, that there exists a normalizing constant Ēω such that But we already know that sup x (n) n,k E ω (x (n) 0,..., x(n) n ) nēω = S ω (x(n) 0, x(n) n ) k+1 x(n) k x (n) n R and lim n + x (n) 0 > 0 n Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 28/32

29 III.8. Why a quasicrystalline model? Idea Include this specific quasicrystal into a larger family Ω = Hull(ω ) the Hull of ω τ t : Ω Ω the flow acting on subsets by translation V : Ω R such that V ω (x) = V (τ x (ω )) L(ω, x) : Ω R R the Lagrangian form of E L(ω, x) = W (x) + V (τ x (ω)) and E ω (x, y) = L(τ x (ω), y x) Main observation Σ 0 A flow box [0, R] Ξ Ξ= L S S n 1 Define µ n := 1 n k=0 δ τ xk (ω ) and µ = lim µ n n + Then µ gives positive mass to any flow box of sufficiently long length t L proof Say Ξ = LS 1 µ ([0, R] Ξ) = lim nb occurrences of LS n + n x n x 0 nb occurrences of LS = lim > 0 n + n x n x 0 x 0 x 1 x 2 L S L L S L Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 29/32 x n

30 III.8. Why a quasicrystalline model? Corollary of the main observation { supp(µ ) meets every trajectory Ξ µ ([0, R] Ξ) > 0 = in time less than R µ appears to be the projection of a specific measure in Ω := Ω R n 1 1 µ = pr( µ ) µ = lim n + n Second main observation µ is minimizing: L(ω, t) d µ (ω, t) = µ is holonomic φ C 0 k=0 δ (τxk (ω ),x k+1 x k ) µ satisfies the two following properties 1 lim n + n inf x 0,...,x n E ω (x 0,..., x n ) = Ēω φ(ω) d µ (ω, t) = φ(τ t (ω)) d µ (ω, t) Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 30/32

31 Definition We call Mather set Theorem III.9. Main result again We say that a probability measure in Ω R is minimizing if { } µ = arg min L(ω, t) dν : ν is holonomic Mather(L) := {supp(µ) : µ is minimizing } In the almost periodic case, the Mather set is a non empty compact set In the quasicrystaline case, the projected Mather set meets every trajectory Ēω = inf{ Ldµ : µ is minimizing } is independant of ω for every ω Mather(L), there exists a calibrated configuration passing through x 0 = 0 Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 31/32

32 Bibliographie I S. Aubry and P.Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground states, Physica D 8 (1983), A. Fathi, The weak KAM theorem in Lagrangian dynamics, book to appear, Cambridge University Press. J. M. Gambaudo, P. Guiraud and S. Petite, Minimal configurations for the Frenkel-Kontorova, model on a quasicrystal, Communications in Mathematical Physics 265 (2006), E. Garibaldi, S. Petite, Ph. Thieullen, Calibrated configurations for Frenkel-Kontorova type models in almost-periodic environments, arxiv (2014). Pohang, August 24-28,2015 quasi-periodique Frenkel Kontorova 32/32