Fractals : Spectral properties Statistical physics � Course 1 Eric Akkermans I N S O R A I T E A L D S N C U I O E F N C E 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, June 13-17, 2017
Benefitted from discussions and collaborations with: Technion : Elsewhere: � � Evgeni Gurevich (KLA-Tencor) Gerald Dunne (UConn.) Dor Gittelman Alexander Teplyaev (UConn.) Eli Levy (+ Rafael) Jacqueline Bloch (LPN, Marcoussis) Ariane Soret (ENS Cachan) Dimitri Tanese (LPN, Marcoussis) Or Raz (HUJI, Maths) Florent Baboux (LPN, Marcoussis) Omrie Ovdat Alberto Amo (LPN, Marcoussis) Yaroslav Don Eva Andrei (Rutgers) � Jinhai Mao (Rutgers) � Arkady Poliakovsky (Maths. BGU) Rafael : � Assaf Barak � Amnon Fisher � � �
Benefitted from discussions and collaborations with: Technion : Elsewhere: � � Evgeni Gurevich (KLA-Tencor) Gerald Dunne (UConn.) Dor Gittelman Alexander Teplyaev (UConn.) Eli Levy (+ Rafael) Jacqueline Bloch (LPN, Marcoussis) Ariane Soret (ENS Cachan) Dimitri Tanese (LPN, Marcoussis) Or Raz (HUJI, Maths) Florent Baboux (LPN, Marcoussis) Omrie Ovdat Alberto Amo (LPN, Marcoussis) Yaroslav Don Eva Andrei (Rutgers) � Jinhai Mao (Rutgers) � Arkady Poliakovsky (Maths. BGU) Rafael : � Assaf Barak � Amnon Fisher � � �
Plan of the 4 talks • Course 1 : Spectral properties of fractals - Application in statistical physics • Talk : quantum phase transition - scale anomaly and fractals • Course 2 : topology and fractals - measuring topological numbers with waves. • Elaboration : Renormalisation group and Efimov physics
Program for today • Introduction : spectral properties of self similar fractals. • Heat kernel - Asymptotic behaviour - Weyl expansion - Spectral volume. • Thermodynamics of the fractal blackbody. • Summary - Phase transitions.
Introduction : spectral properties of self similar fractals. attractive objects - Bear exotic names • Julia sets
� � � � � � � � � � � Hofstadter butterfly Sierpinski carpet Sierpinski gasket
Diamond fractals Triadic Cantor set Convey the idea of highly symmetric objects yet with an unusual type of symmetry and a notion of extreme subdivision
Fractal : Iterative graph structure n → ∞ Sierpinski gasket s s s s s s s s s s s s s s s Diamond fractals
Fractal : Iterative graph structure n → ∞ Sierpinski gasket s s s s s s s s s s s s s s s Diamond fractals
As opposed to Euclidean spaces characterised by translation symmetry, fractals possess a dilatation symmetry. Fractals are self-similar objects
Fractal ↔ Self-similar Discrete scaling symmetry
• But not all fractals are obvious, good faith geometrical objects. � Sometimes, the fractal structure is not geometrical but it is hidden at a more abstract level. Exemple : quasi-periodic stack of dielectric layers of 2 types A , B ⎡ ⎤ Fibonacci sequence : F 1 = B ; F 2 = A ; F j ≥ 3 = F j − 2 F j − 1 ⎣ ⎦ Defines a cavity whose mode spectrum is fractal.
• But generally, not all fractals are obvious, good faith geometrical objects. � Sometimes, the fractal structure is not geometrical but it is hidden at a more abstract level. Exemple : Quasi-periodic chain of layers of 2 types A , B ⎡ ⎤ Fibonacci sequence : F 1 = B ; F 2 = A ; F j ≥ 3 = F j − 2 F j − 1 ⎣ ⎦ Defines a cavity whose frequency spectrum is fractal.
D ensity of modes ρ ( ω ) : Discrete scaling symmetry Minicourse 2 - Tomorrow
Operators and fields on fractal manifolds Operators are often expressed by local differential equations relating the space-time behaviour of a field ∂ 2 u ∂ t 2 = Δ u Ex. Wave equation Such local equations cannot be defined on a fractal 16
But operators are essential quantities for physics! • Quantum transport in fractal structures : e.g. , networks, waveguides, ... electrons, photons • Density of states • Scattering matrix (transmission/reflection) 17
But operators are essential quantities for physics! • Quantum fields on fractals, e.g. , fermions (spin 1/2), photons (spin 1) - canonical quantisation (Fourier modes) - path integral quantisation : path integrals, Brownian motion. � • “curved space QFT” or quantum gravity � • Scaling symmetry (renormalisation group) - critical behaviour. 18
Recent new ideas >2000 Maths. Michel Lapidus Bob Strichartz Jun Kigami 19
Intermezzo : heat and waves
From classical diffusion to wave propagation Important relation between classical diffusion and wave propagation on a manifold. Expresses the idea that it is possible to measure and characterise a manifold using waves (eigenvalue spectrum of the Laplace operator) Differential operator geometry “propagating probe” Spectral data curvature Heat kernel volume physically: Zeta function dimension Laplacian
Use propagating waves/particles to probe : • spectral information: density of states, transport, heat kernel, ... • geometric information: dimension, volume, boundaries, shape, ... 22
Use propagating waves/particles to probe : • spectral information: density of states, transport, heat kernel, ... • geometric information: dimension, volume, boundaries, shape, ... Mathematical physics 1910 Lorentz: why is the Jeans radiation law only dependent on the volume ? 1911 Weyl : relation between asymptotic eigenvalues and dimension/volume. 1966 Kac : can one hear the shape of a drum ? 23
Important examples ∂ u • Heat equation ∂ t = Δ u � ∂ 2 u ( ) = ( ) P ( ) u y ,0 ( ) ∫ d µ y • Wave equation u x , t t x , y ∂ t 2 = Δ u � i ∂ u ∂ t = Δ u Schr. equation. t ∫ x 2 d τ − ( i ) ! ( ) = ∫ D P t x , y xe 0 Brownian motion ( ) = x , x t ( ) = y x 0 ) ∼ 1 ∑ ( a n ( x , y ) t n P t x , y Heat kernel expansion d 2 t n ∑ ( ) ∼ ( ) e − ( i ) S classical ( x , y , t ) P t x , y # Gutzwiller - instantons geodesics 24
Important examples ∂ u • Heat equation ∂ t = Δ u � ∂ 2 u ( ( ) = ) = ( ) P ( ) P ( ( ) u y ,0 ) u y ,0 ( ( ) ) ∫ ∫ d µ y d µ y • Wave equation u x , t u x , t t x , y t x , y ∂ t 2 = Δ u � i ∂ u ∂ t = Δ u Schr. equation. t ∫ x 2 d τ − ( i ) ! ( ) = ∫ D P t x , y xe 0 Brownian motion ( ) = x , x t ( ) = y x 0 ) ∼ 1 ∑ ( a n ( x , y ) t n P t x , y Heat kernel expansion d 2 t n ∑ ( ) ∼ ( ) e − ( i ) S classical ( x , y , t ) P t x , y # Gutzwiller - instantons geodesics 25
Important examples ∂ u • Heat equation ∂ t = Δ u � ∂ 2 u ( ( ) = ) = ( ) P ( ) P ( ( ) u y ,0 ) u y ,0 ( ( ) ) ∫ ∫ d µ y d µ y • Wave equation u x , t u x , t t x , y t x , y ∂ t 2 = Δ u � i ∂ u ∂ t = Δ u Schr. equation. t t ∫ ∫ x 2 d τ x 2 d τ − ( i ) − ( i ) ! ! ( ( ) = ) = ∫ D ∫ D P P t x , y t x , y xe xe 0 0 Brownian motion Brownian motion ( ) = x , x t ( ) = x , x t ( ) = y ( ) = y x 0 x 0 ) ∼ 1 ) ∼ 1 ∑ ∑ ( ( a n ( x , y ) t n a n ( x , y ) t n P P t x , y t x , y Heat kernel expansion Heat kernel expansion d 2 d 2 t t n n ∑ ∑ ( ( ) ∼ ) ∼ ( ) ( ) e − ( i ) S classical ( x , y , t ) e − ( i ) S classical ( x , y , t ) P P t x , y t x , y # # Gutzwiller - instantons Gutzwiller - instantons geodesics geodesics 26
Important examples ∂ u • Heat equation ∂ t = Δ u � ∂ 2 u ( ( ) = ) = ( ) P ( ) P ( ( ) u y ,0 ) u y ,0 ( ( ) ) ∫ ∫ d µ y d µ y • Wave equation u x , t u x , t t x , y t x , y ∂ t 2 = Δ u � i ∂ u ∂ t = Δ u Schr. equation. t t ∫ ∫ x 2 d τ x 2 d τ − ( i ) − ( i ) ! ! ( ( ) = ) = ∫ D ∫ D P P t x , y t x , y xe xe 0 0 Brownian motion Brownian motion ( ) = x , x t ( ) = x , x t ( ) = y ( ) = y x 0 x 0 ) ∼ 1 ) ∼ 1 ∑ ∑ ( ( a n ( x , y ) t n a n ( x , y ) t n P P t x , y t x , y Heat kernel expansion Heat kernel expansion d 2 d 2 t t n n ∑ ∑ ( ( ) ∼ ) ∼ ( ) ( ) e − ( i ) S classical ( x , y , t ) e − ( i ) S classical ( x , y , t ) P P t x , y t x , y # # Gutzwiller - instantons Gutzwiller - instantons geodesics geodesics 27
Spectral functions ∑ ( ) = y e −Δ t x = ∗ ( y ) ψ λ ( x ) e − λ t ψ λ P t x , y λ Z ( t ) = Tre −Δ t = ∑ dx x e −Δ t x ∫ = e − λ t Heat kernel λ ∞ 1 ( ) ≡ ( ) ∫ ζ Z s ( ) dtt s − 1 Z t Γ s Mellin transform 0 ( ) = Tr 1 1 ∑ ζ Z s Δ s = λ s λ ⇔ ( ) Small t behaviour of Z(t) poles of ζ Z s Weyl expansion 28
Spectral functions ∑ ( ) = y e −Δ t x = ∗ ( y ) ψ λ ( x ) e − λ t ψ λ P t x , y λ Z ( t ) = Tre −Δ t = ∑ dx x e −Δ t x ∫ = e − λ t Heat kernel λ ∞ 1 ( ) ≡ ( ) ∫ ζ Z s ( ) dtt s − 1 Z t Γ s Mellin transform 0 ( ) = Tr 1 1 ∑ ζ Z s Δ s = λ s λ ⇔ ( ) Small t behaviour of Z(t) poles of ζ Z s Weyl expansion 29
Spectral functions ∑ ( ) = y e −Δ t x = ∗ ( y ) ψ λ ( x ) e − λ t ψ λ P t x , y λ Z ( t ) = Tre −Δ t = ∑ dx x e −Δ t x ∫ = e − λ t Heat kernel λ ∞ 1 ( ) ≡ ( ) ∫ ζ Z s ( ) dtt s − 1 Z t Return Γ s Mellin transform 0 probability ( ) = Tr 1 1 ∑ ζ Z s Δ s = λ s λ ⇔ ( ) Small t behaviour of Z(t) poles of ζ Z s Weyl expansion 30
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