Branched transport limit of the Ginzburg-Landau functional

Transcription

1 Branched transport limit of the Ginzburg-Landau functional Michael Goldman CNRS, LJLL, Paris 7 Joint work with S. Conti, F. Otto and S. Serfaty

2 Introduction Superconductivity was first observed by Onnes in 1911 and has nowadays many applications.

3 Meissner effect In 1933, Meissner understood that superconductivity was related to the expulsion of the magnetic field outside the material sample

4 Ginzburg Landau functional In the 50 s Ginzburg and Landau proposed a phenomenological model (later derived from the BCS theory): E(u, A) = Ω A u 2 + κ2 2 (1 ρ2 ) 2 dx + A B ex 2 dx R 3 where u = ρe iθ is the order parameter, B = A is the magnetic field, B ex is the external magnetic field, κ is the Ginzburg-Landau constant and A u = u iau is the covariant derivative. ρ 0 represents the normal phase and ρ 1 the superconducting one.

5 The various terms in the energy For u = ρe iθ, A u 2 = ρ 2 + ρ 2 θ A 2. In ρ > 0 first term wants A = θ = A = 0 That is ρ 2 B 0 (Meissner effect) and penalizes fast oscillations of ρ Second term forces ρ 1 (superconducting phase favored) Last term wants B B ex. In particular, this should hold outside the sample.

6 Coherence and penetration length Already two typical lengths, coherence length ξ and penetration length λ. B ξ ρ λ In our unites, λ = 1, κ = 1 ξ

7 Our setting We consider Ω = Q L,T = [ L, L] 2 [ T, T ] with periodic lateral boundary conditions and take B ex = b ex e 3. T b ex e 3 ρ 1 T L L We want to understand extensive behavior L 1.

8 First rescaling We let κt = 2α b ex = βκ 2 and then x = T 1 x Â( x) = A(x) û( x) = u(x) B( x) = Â( x) = TB(x) In these units, coherence length α 1 penetration length T 1 We are interested in the regime T 1, α 1, β 1.

9 The energy The energy can be written as E T (u, A) = 1 L 2 TA u 2 + ( B 3 α(1 ρ 2 ) ) 2 + B 2 Q L,1 + B 3 αβ 2 H 1/2 (x 3 =±1) First term: penalizes oscillations + ρ 2 B 0 (Meissner effect)

10 The energy The energy can be written as E T (u, A) = 1 L 2 TA u 2 + ( B 3 α(1 ρ 2 ) ) 2 + B 2 Q L,1 + B 3 αβ 2 H 1/2 (x 3 =±1) First term: penalizes oscillations + ρ 2 B 0 (Meissner effect) Second term: degenerate double well potential. If Meissner then: ( B3 α(1 ρ 2 ) ) 2 α 2 χ {ρ>0} (1 ρ 2 ) 2 Rk: wants B 3 = α in {ρ = 0} Similar features in mixtures of BEC (cf G. Merlet 15)

11 Crash course on optimal transportation For ρ 0, ρ 1 probability measures { } W2 2 (ρ 0, ρ 1 ) = inf x y 2 dπ(x, y) : Π 1 = ρ 0, Π 2 = ρ 1 Q L Q L Theorem (Benamou-Brenier) W 2 2 (ρ 0, ρ 1 ) = inf µ,b { 1 0 Q L B 2 dµ : 3 µ + div B µ = 0, µ(0, ) = ρ 0, µ(1, ) = ρ 1 } (Brenier) If ρ 0 dx, { } W2 2 (ρ 0, ρ 1 ) = min x T (x) 2 dρ 0 : T ρ 0 = ρ 1 Q L

12 The energy continued E T (u, A) = 1 L 2 Q L,1 TA u 2 + ( B 3 α(1 ρ 2 ) ) 2 + B 2 + B 3 αβ 2 H 1/2 (x 3 =±1) Third term: with Meissner and B 3 α(1 ρ 2 ) = χ, div B = 0 can be rewritten as 3 χ + div χb = 0 Benamou-Brenier = Wasserstein energy of x 3 χ(, x 3 )

13 The energy continued E T (u, A) = 1 L 2 Q L,1 TA u 2 + ( B 3 α(1 ρ 2 ) ) 2 + B 2 + B 3 αβ 2 H 1/2 (x 3 =±1) Third term: with Meissner and B 3 α(1 ρ 2 ) = χ, div B = 0 can be rewritten as 3 χ + div χb = 0 Benamou-Brenier = Wasserstein energy of x 3 χ(, x 3 ) Last term: penalizes non uniform distribution on the boundary but negative norm = allows for oscillations

14 A non-convex energy regularized by a gradient term If we forget the kinetic part of the energy, can make B = 0 and E T (u, A) = 1 L 2 Q L,1 ( B3 α(1 ρ 2 ) ) 2 + B3 αβ 2 H 1/2 (x 3 =±1) ρ=0 ρ=1 = infinitely small oscillations of phases {ρ = 0, B 3 = α} and {ρ = 1, B 3 = 0} with average volume fraction β. the kinetic term A u 2 fixes the lengthscale.

15 Branching is energetically favored ρ 1 B 3 αβ 2 H 1/2 (x 3 =±1) 0 but interfacial energy x 3 = 1 x 3 = 1 ρ 1 interfacial energy but Q L,1 B 2. Landau 43

16 Experimental results Complex patterns at the boundary Experimental pictures from Prozorov and al. Limitations: Difficult to see the pattern inside the sample Hysteresis

17 Branching patterns in other related models Shape memory alloys (Kohn-Müller model) (Left, picture from Chu and James) Uniaxial ferromagnets (Right, picture from Hubert and Schäffer) Schematic difference: in our problem W 2 2 replaces H 1 norm See works of Kohn, Müller, Conti, Otto, Choksi... Related functional: Ohta-Kawasaki

18 Scaling law Theorem (Conti, Otto, Serfaty 15, See also Choksi, Conti, Kohn, Otto 08) In the regime T 1, α 1, β 1, min E T min(α 4/3 β 2/3, α 10/7 β) ρ 1 First regime: E T α 4/3 β 2/3 Uniform branching, B 3 αβ 2 H 1/2 (x 3 =±1) = 0 ρ 1 Second regime: E T α 10/7 β Non-Uniform branching, B 3 αβ 2 H 1/2 (x 3 =±1) > 0 fractal behavior

19 Scaling law Theorem (Conti, Otto, Serfaty 15, See also Choksi, Conti, Kohn, Otto 08) In the regime T 1, α 1, β 1, min E T min(α 4/3 β 2/3, α 10/7 β) We concentrate on the first regime (uniform branching) ρ 1 = α 2/7 β.

20 Multiscale problem sample size ρ 1 B coherence length ρ domain size penetration length B From the upper bound construction, we expect penetration length coherence length domain size sample size which amounts in our parameters to T 1 α 1 α 1/3 β 1/3 L.

21 Crash course in Γ-convergence F n sequence of functionals on a metric space (X, d). We say that F n Γ converges to F if x n X, F n (x n ) C = Compactness + x X, x n x with It implies inf F n inf F lim F n (x n ) F (x) n + lim F n(x n ) F (x) n + if x n are minimizers of F n = x is a minimizer of F.

22 Compactness and Lower bounds

23 First limit, T + Recall: E T (u, A) = 1 L 2 TA u 2 + ( B 3 α(1 ρ 2 ) ) 2 + B 2 Q L,1 + B 3 αβ 2 H 1/2 (x 3 =±1) Proposition If E T (u T, A T ) C then ρ T = u T ρ, B T = A T B and ρ 2 B = 0, div B = 0 (Meissner effect) lim T E T (u T, A T ) F α,β (ρ, B) where F α,β (ρ, B) = 1 L 2 ρ 2 + ( B 3 α(1 ρ 2 ) ) 2 + B 2 Q L,1 + B 3 αβ 2 H 1/2 (x 3 =±1)

24 Second rescaling In this limit, penetration length = 0, coherence length α 1, domain size α 1/3 β 1/3. In order to get sharp interface limit with finite domain size, we make the anisotropic rescaling ) ( (ˆx α = 1/3 x ), Fα,β = α ˆx 4/3 F 3 x α,β. ( ) ( 3 ˆB α (ˆx) = 2/3 B ) ˆB 3 α 1 (x), ˆρ(ˆx) = ρ(x), B 3 In these variable: coherence length α 2/3 1 and normal domain size β 1/3

25 Second limit, α + Dropping the hats L 2 F α,β (ρ, B) = Q L,1 α 2/3 and the Meissner condition still holds Proposition ( ) ρ 2 α 1/3 +α 2/3 B 3 ρ 3 (1 ρ 2 ) 2 + B 2 div B = 0 and ρ 2 B = 0 + α 1/3 B 3 β 2 H 1/2 (x 3 =±1) If F α,β (ρ α, B α ) C, then 1 ρ 2 α χ {0, 1} B α B and χ(, ±1) = β, χb = B, 3 χ + div χb = 0 lim α F α,β (ρ α, B α ) G β (χ, B ) where G β (χ, B ) = 1 L 2 Q L,1 4 3 χ + B 2

26 Comments on the proof Anisotropic rescaling = control only on the horizontal derivative. Thanks to Meissner, double well potential ( α 2/3 ) ρ 2 α 1/3 + α 2/3 B 3 ρ 3 (1 ρ 2 ) 2 α 2/3 ρ 2 + α 2/3 χ {ρ>0} (1 ρ 2 ) 2 Recall Modica-Mortola ε ρ ε 2 + ε 1 ρ 2 ε(1 ρ 2 ε) C χ

27 Last rescaling We want to send β 0 and get 1 dimensional trees. We make another anisotropic rescaling: ) ( (ˆx β = 1/6 x ), ˆχ(ˆx) = β ˆx 1 χ(x) {0, β 1 } 3 x 3 ˆB (ˆx) = β 1/6 B (x) Ĝ β = β 2/3 G β Now, domain width β 1/2 1, L = 1 and (dropping hats) G β (χ, B 4 ) = Q 1,1 3 β1/2 χ + χ B 2 with 3 χ + div (χb ) = 0, χb = β 1 B and χ(, x 3 ) dx when x 3 ±1.

28 Limiting functional Because of isoperimetric effects, on every slice χ β 1 i χ B(xi,β 1/2 r i ) β 1/2 r i If φ i = πr 2 i, [ 1,1] 2 χ i φ i x i and φi [ 1,1] 2 β 1/2 χ 2π 1/2 i Hence χ i φ iδ xi

29 The limiting functional II For µ a probability measure with µ x3 = i φ iδ xi (x 3 ) for a.e. x 3 and µ x3 dx when x 3 ±1, and m µ, with 3 µ + div m = 0, 1 8π 1/2 ( I (µ, m) = µx3 (x ) ) ( ) 1/2 dm 2 dx3 + dµ 3 dµ Formally, 1 x Q 1 1 I (µ) = inf I (µ, m) = m 1 i 8π 1/2 3 Q 1,1 φ 1/2 i + φ i ẋ 2 i dx 3

30 Last lower bound Proposition If G β (χ β, B β ) C χ β µ, χ β B β m and lim G β (χ β, B β ) I (µ, m) β

31 The limiting functional I (µ) reminiscent of branched transport models (see Bernot-Caselles-Morel). Our result, similar in spirit to Oudet-Santambrogio 11. Minimizers of I, contain no loop, finite number of branching points away from boundary (with quantitative estimate), branches are linear between two branching points Every measure can be irrigated

32 Definition of regular measures Definition For ε > 0, we denote by M ε R (Q 1,1) the set of regular measures, i.e., of measures such that: (i) µ is finite polygonal. (ii) All branching points are triple points. This means that any x Q 1,1 belongs to no more than three segments. (iii) there is ε 1/2 λ ε ε with 1/λ ε N, such that for all x 3 [1 ε, 1] one has µ x3 = x λ εz 2 Q 1 ϕ x δ x, with ϕ x = λ 2 ε, and the same in [ 1, 1 + ε].

33 A crucial density result Proposition For every µ with I (µ) <, µ ε M ε R (Q 1,1) with µ ε µ and lim ε I (µ ε ) I (µ). = enough to make the construction for finite polygonal measures.

34 Idea of the proof 1 Rescale µ to [ 1 + 2ε, 1 2ε] 1 2ε 1 + 2ε 1

35 Idea of the proof 1 Rescale µ to [ 1 + 2ε, 1 2ε] in the boundary layer plug in a uniform branching construction 1 2ε 1 + 2ε 1

36 Idea of the proof 1 Rescale µ to [ 1 + 2ε, 1 2ε] in the boundary layer plug in a uniform branching construction remove small branches. Tool: notion of subsystem cf. Ambrosio-Gigli-Savare 1 2ε 1 + 2ε 1

37 Idea of the proof Rescale µ to [ 1 + 2ε, 1 2ε] in the boundary layer plug in a uniform branching construction remove small branches. Tool: notion of subsystem cf. Ambrosio-Gigli-Savare discretize and minimize 1 1 2ε 1 + 2ε 1

38 Recovery sequences

39 Recovery sequence, from trees to sharp interface Want to enlarge the 1D trees. Far from branching points, easy (take a tube). At a branching point: Need to transform a rectangle into disk with controlled energy

40 Recovery sequence, from sharp to diffuse interface Need to reintroduce a smooth transition + vertical derivative. 1 1 To get smooth transition: use optimal profile (keeping Meissner) + careful estimate of the error terms

41 Recovery sequence, from diffuse interface to full GL Can define A with A = B. Need to define θ. Would be easy if quantization of flux φ i 2πZ. 0 2πk i Γ If Γ 0 x, θ(x) = Γ A τ Quantization+Stokes = well defined + θ = A x = Need to modify the fluxes to get quantization

42 Ongoing work and perspective Understand the cross-over regime α 2/7 β Go from GL to sharp interface when coherence penetration (more complex/non-local optimal profile problem) Understand minimizers of the limiting functional (self-similarity à la Conti) Investigate the non-uniform branching fractal behavior

43 What in the name of Sir Isaac H. Newton happened here? Dr. Emmett Doc Brown Thank you! Any Question?