Branched transport limit of the Ginzburg-Landau functional Michael - PowerPoint PPT Presentation

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

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

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

Ginzburg Landau functional In the 50’s Ginzburg and Landau proposed a phenomenological model (later derived from the BCS theory): � � |∇ A u | 2 + κ 2 2 (1 − ρ 2 ) 2 dx + R 3 |∇ × A − B ex | 2 dx E ( u , A ) = Ω where u = ρ e i θ 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.

The various terms in the energy For u = ρ e i θ , |∇ 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.

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

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.

First rescaling We let √ b ex = βκ κ T = 2 α √ 2 and then x = T − 1 x � u ( � � x ) = u ( x ) � � x ) = ∇ × � A ( � x ) = A ( x ) B ( � A ( � 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.

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

The energy The energy can be written as � � � 2 + | B ′ | 2 E T ( u , A ) = 1 |∇ TA u | 2 + B 3 − α (1 − ρ 2 ) L 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: � � 2 ≃ α 2 χ { ρ> 0 } (1 − ρ 2 ) 2 B 3 − α (1 − ρ 2 ) Rk: wants B 3 = α in { ρ = 0 } Similar features in mixtures of BEC (cf G. Merlet ’15)

Crash course on optimal transportation For ρ 0 , ρ 1 probability measures �� � W 2 | x − y | 2 d Π( x , y ) : Π 1 = ρ 0 , Π 2 = ρ 1 2 ( ρ 0 , ρ 1 ) = inf Q L × Q L Theorem ◮ (Benamou-Brenier) �� 1 � W 2 | B ′ | 2 d µ : ∂ 3 µ + div ′ B ′ µ = 0 , 2 ( ρ 0 , ρ 1 ) = inf µ, B ′ 0 Q L µ (0 , · ) = ρ 0 , µ (1 , · ) = ρ 1 } ◮ (Brenier) If ρ 0 ≪ dx, �� � W 2 | x − T ( x ) | 2 d ρ 0 : T ♯ρ 0 = ρ 1 2 ( ρ 0 , ρ 1 ) = min Q L

The energy continued � � � 2 + | B ′ | 2 E T ( u , A ) = 1 |∇ TA u | 2 + B 3 − α (1 − ρ 2 ) L 2 Q L , 1 + � 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 ′ χ B ′ = 0 Benamou-Brenier = ⇒ Wasserstein energy of x 3 → χ ( · , x 3 )

The energy continued � � � 2 + | B ′ | 2 E T ( u , A ) = 1 |∇ TA u | 2 + B 3 − α (1 − ρ 2 ) L 2 Q L , 1 + � 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 ′ χ B ′ = 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

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

Branching is energetically favored � B 3 − αβ � 2 H − 1 / 2 ( x 3 = ± 1) ↓ 0 ρ ≃ 1 but interfacial energy ↑ ∞ x 3 = − 1 x 3 = 1 ρ ≃ 1 interfacial energy ↓ � Q L , 1 | B ′ | 2 ↑ . but Landau ’43

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

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

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 ) 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 ρ ≃ 1 fractal behavior

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 . =

Multiscale problem sample size coherence length ρ ≃ 1 B 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 .

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 + lim F n ( x n ) ≥ F ( x ) n → + ∞ ◮ ∀ x ∈ X , ∃ x n → x with n → + ∞ F n ( x n ) ≤ F ( x ) lim It implies ◮ inf F n → inf F ◮ if x n are minimizers of F n = ⇒ x is a minimizer of F .

Compactness and Lower bounds

First limit, T → + ∞ Recall: � � � 2 + | B ′ | 2 E T ( u , A ) = 1 |∇ TA u | 2 + B 3 − α (1 − ρ 2 ) L 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 � � � 2 + | B ′ | 2 F α,β ( ρ, B ) = 1 |∇ ρ | 2 + B 3 − α (1 − ρ 2 ) L 2 Q L , 1 + � B 3 − αβ � 2 H − 1 / 2 ( x 3 = ± 1)

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 � ˆ � � � x ′ α 1 / 3 x ′ � F α,β = α − 4 / 3 F α,β . = , x 3 ˆ x 3 � ˆ � � � B ′ α − 2 / 3 B ′ (ˆ x ) = ( x ), ρ (ˆ ˆ x ) = ρ ( x ), ˆ α − 1 B 3 B 3 In these variable: coherence length ≃ α − 2 / 3 ≪ 1 and normal domain size ≃ β 1 / 3

Second limit, α → + ∞ Dropping the hats � � �� � 2 � � ∇ ′ ρ L 2 F α,β ( ρ, B ) = α − 2 / 3 � � + α 2 / 3 | B 3 − (1 − ρ 2 ) | 2 + | B ′ | 2 � � α − 1 / 3 ∂ 3 ρ Q L , 1 + α 1 / 3 � B 3 − β � 2 H − 1 / 2 ( x 3 = ± 1) and the Meissner condition ρ 2 B = 0 div B = 0 and still holds Proposition α ⇀ B ′ and If F α,β ( ρ α , B α ) ≤ C, then 1 − ρ 2 α → χ ∈ { 0 , 1 } B ′ ◮ χ ( · , ± 1) = β , χ B ′ = B ′ , ∂ 3 χ + div ′ χ B ′ = 0 ◮ lim α F α,β ( ρ α , B α ) ≥ G β ( χ, B ′ ) where � G β ( χ, B ′ ) = 1 4 3 |∇ ′ χ | + | B ′ | 2 L 2 Q L , 1

Comments on the proof ◮ Anisotropic rescaling = ⇒ control only on the horizontal derivative. ◮ Thanks to Meissner, double well potential � �� � 2 � ∇ ′ ρ � + α 2 / 3 | B 3 − (1 − ρ 2 ) | 2 ≥ α − 2 / 3 � � � � α − 1 / 3 ∂ 3 ρ α − 2 / 3 � � � 2 + α 2 / 3 χ { ρ> 0 } | (1 − ρ 2 ) | 2 � ∇ ′ ρ Recall Modica-Mortola � � ε |∇ ′ ρ ε | 2 + ε − 1 ρ 2 ε (1 − ρ 2 |∇ ′ χ | ε ) → C