Fractals : Spectral Properties Statistical Physics
Fractals : Spectral propertiesStatistical arc/showphoto.asp?photol.Course 1ASCDELISRABackTIONEric AkkermansIENCEFOUNTechnion logo English6th Cornell Conference on Analysis, Probability, andMathematical Physics on Fractals, June 13-17, 2017
Benefitted from discussions and collaborations with:Technion:Elsewhere:!Evgeni Gurevich (KLA-Tencor)Dor GittelmanEli Levy ( Rafael)Ariane Soret (ENS Cachan)Or Raz (HUJI, Maths)Omrie OvdatYaroslav Don!Rafael:Assaf BarakAmnon Fisher!!!!!Gerald Dunne (UConn.)Alexander Teplyaev (UConn.)Jacqueline Bloch (LPN, Marcoussis)Dimitri Tanese (LPN, Marcoussis)Florent Baboux (LPN, Marcoussis)Alberto Amo (LPN, Marcoussis)Eva Andrei (Rutgers)Jinhai Mao (Rutgers)Arkady Poliakovsky (Maths. BGU)!!
Benefitted from discussions and collaborations with:Technion:Elsewhere:!Evgeni Gurevich (KLA-Tencor)Dor GittelmanEli Levy ( Rafael)Ariane Soret (ENS Cachan)Or Raz (HUJI, Maths)Omrie OvdatYaroslav Don!Rafael:Assaf BarakAmnon Fisher!!!!!Gerald Dunne (UConn.)Alexander Teplyaev (UConn.)Jacqueline Bloch (LPN, Marcoussis)Dimitri Tanese (LPN, Marcoussis)Florent Baboux (LPN, Marcoussis)Alberto Amo (LPN, Marcoussis)Eva Andrei (Rutgers)Jinhai Mao (Rutgers)Arkady Poliakovsky (Maths. BGU)!!
Plan of the 4 talks Course 1 : Spectral properties of fractals Application in statistical physics Talk : quantum phase transition - scaleanomaly and fractals Course 2 : topology and fractals -measuring topological numbers withwaves. Elaboration : Renormalisation group andEfimov physics
Program for today Introduction : spectral properties of selfsimilar fractals. Heat kernel - Asymptotic behaviour - Weylexpansion - Spectral volume. Thermodynamics of the fractal blackbody. Summary - Phase transitions.
Introduction : spectral propertiesof self similar fractals. attractive objects - Bear exotic namesJulia sets
Anatomy of the Hofstadter butterflyel (1964)0anniern, B. Simonsard, R. Rammal0heless, Q. NiuThouless, Kohmoto, Nightingale, den NijsugaiHofstadter butterflyEnergy levels and wave functions of Bloch electronsin rational and irrational magnetic fields,Douglas Hofstadter, Phys. Rev. B 14 (1976) 2239Sierpinski carpetSierpinski gasket
Diamond fractalsFigure: Diamond fractals, non-p.c.f., but finitely ramifiedTriadic Cantor setConvey the idea of highly symmetric objects yet with anunusual type of symmetry and a notion of extreme subdivision
Fractal : Iterative graph structuren Sierpinski gasket!!!!!!!!!!!!!!!Diamond fractalsFigure: Diamond fractals, non-p.c.f., but finitely ramified
Fractal : Iterative graph structuren Sierpinski gasket!!!!!!!!!!!!!!!Diamond fractalsFigure: Diamond fractals, non-p.c.f., but finitely ramified
As opposed to Euclidean spacescharacterised by translationsymmetry, fractals possess adilatation symmetry.Fractals are self-similar objects
Fractal Self-similarDiscrete scaling symmetry
But not all fractals are obvious, good faith geometricalobjects.!Sometimes, the fractal structure is not geometricalbut it is hidden at a more abstract level.Exemple : quasi-periodic stack of dielectric layers of 2 types A, BFibonacci sequence : F1 B; F2 A; Fj 3 Fj 2 Fj 1 Defines a cavity whose mode spectrum is fractal.
But generally, not all fractals are obvious, good faithgeometrical objects.!Sometimes, the fractal structure is not geometricalbut it is hidden at a more abstract level.Exemple : Quasi-periodic chain of layers of 2 types A, BFibonacci sequence : F1 B; F2 A; Fj 3 Fj 2 Fj 1 Defines a cavity whose frequency spectrum is fractal.
Density of modes ρ(ω) :Discrete scaling symmetryMinicourse 2 - Tomorrow
Operators and fields on fractal manifoldsOperators are often expressed by local differentialequations relating the space-time behaviour of a fieldEx. Wave equation 2 u Δu2 tSuch local equations cannot be defined on a fractalFigure: Diamond fractals, non-p.c.f., but finitely ramified16
But operators are essentialquantities for physics! Quantum transport in fractal structures :e.g., networks, waveguides, .electrons, photons Density of states Scattering matrix (transmission/reflection)17
But operators are essentialquantities for physics! Quantum fields on fractals, e.g., fermions (spin 1/2),photons (spin 1) - canonical quantisation (Fouriermodes) - path integral quantisation : path integrals,Brownian motion.! “curved space QFT” or quantum gravity! Scaling symmetry (renormalisation group) - criticalbehaviour.18
Recent new ideas 2000Maths.Michel LapidusBob StrichartzJun Kigami19
Intermezzo : heat and waves
From classical diffusion to wave propagationImportantrelationbetween classicaldiffusion and waveGeneralitieson fractals propagation on a manifold.Many self-similar (fractal) structures in nature and many ways to model them:A random walk in free space or on a periodic lattice etc.Expresses the idea that it is possible to measure andFractals provide a usefultesting groundto investigateof disorderedcharacterisea manifoldusingwavesproperties(eigenvalueclassical or quantum systems, renormalization group and phase transitions,spectrumthe Laplacegravitationalsystems and ofquantumfield theory. onDifferential operator“propagating probe”physically:LaplacianSpectral dataHeat kernelZeta function
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 physics1910 Lorentz: why is the Jeans radiation law only dependenton the volume ?1911 Weyl : relation between asymptotic eigenvalues anddimension/volume.1966 Kac : can one hear the shape of a drum ?23
Important examples Heat equation u Δu t! Wave equation u Δu2 t! uSchr. equation. i Δu t2u ( x,t ) d µ ( y )Pt ( x, y )u ( y,0 )tPt ( x, y ) D xe0Brownian motionx( 0 ) x,x( t ) yPt ( x, y ) Pt ( x, y ) (i ) x! 2 d τ1td2 a (x, y)tnnn (i )S#e() classical ( x,y,t )geodesicsHeat kernel expansionGutzwiller - instantons24
Important examples Heat equation u Δu t! Wave equation u Δu2 t! uSchr. equation. i Δu t2u ( x,t ) d µ ( y )Pt ( x, y )u ( y,0 )tPt ( x, y ) D xe0Brownian motionx( 0 ) x,x( t ) yPt ( x, y ) Pt ( x, y ) (i ) x! 2 d τ1td2 a (x, y)tnnn (i )S#e() classical ( x,y,t )geodesicsHeat kernel expansionGutzwiller - instantons25
Important examples Heat equation u Δu t! Wave equation u Δu2 t! uSchr. equation. i Δu t2u ( x,t ) d µ ( y )Pt ( x, y )u ( y,0 )tPt ( x, y ) D xe0Brownian motionx( 0 ) x,x( t ) yPt ( x, y ) Pt ( x, y ) (i ) x! 2 d τ1td2 a (x, y)tnnn (i )S#e() classical ( x,y,t )geodesicsHeat kernel expansionGutzwiller - instantons26
Important examples Heat equation u Δu t! Wave equation u Δu2 t! uSchr. equation. i Δu t2u ( x,t ) d µ ( y )Pt ( x, y )u ( y,0 )tPt ( x, y ) D xe0Brownian motionx( 0 ) x,x( t ) yPt ( x, y ) Pt ( x, y ) (i ) x! 2 d τ1td2 a (x, y)tnnn (i )S#e() classical ( x,y,t )geodesicsHeat kernel expansionGutzwiller - instantons27
Spectral functionsPt ( x, y ) y e Δtx ψ (y)ψ λ (x)e λλZ(t) Tre Δt dx x e Δt x e λtλ λtHeat kernel 1s 1ζ Z (s) dtt Z ( t ) Γ (s) 0Mellin transform11ζ Z ( s ) Tr s sΔλ λSmall t behaviour of Z(t) poles of ζZ(s)Weylexpansion28
Spectral functionsPt ( x, y ) y e Δtx ψ (y)ψ λ (x)e λλZ(t) Tre Δt dx x e Δt x e λtλ λtHeat kernel 1s 1ζ Z (s) dtt Z ( t ) Γ (s) 0Mellin transform11ζ Z ( s ) Tr s sΔλ λSmall t behaviour of Z(t) poles of ζZ(s)Weylexpansion29
Spectral functionsPt ( x, y ) y e Δtx ψ (y)ψ λ (x)e λλZ(t) Tre Δt dx x e Δt x e λtλ λtHeat kernel 1s 1ζ Z (s) dtt ReturnZ (t ) Γ (s) 0probabilityMellin transform11ζ Z ( s ) Tr s sΔλ λSmall t behaviour of Z(t) poles of ζZ(s)Weylexpansion30
Spectral functionsPt ( x, y ) y e Δtx ψ (y)ψ λ (x)e λλZ(t) Tre Δt dx x e Δt x e λtλ λtHeat kernel 1s 1ζ Z (s) dtt Z ( t ) Γ (s) 0Mellin transform11ζ Z ( s ) Tr s sΔλ λSmall t behaviour of Z(t) poles of ζZ(s)Weylexpansion31
Spectral functionsPt ( x, y ) y e Δtx ψ (y)ψ λ (x)e λλZ(t) Tre Δt dx x e Δt x e λtλ λtHeat kernel 1s 1ζ Z (s) dtt Z ( t ) Γ (s) 0Mellin transform11ζ Z ( s ) Tr s sΔλ λSmall t behaviour of Z(t) poles of ζZ(s)32
Spectral functionsPt ( x, y ) y e Δtx ψ (y)ψ λ (x)e λλZ(t) Tre Δt dx x e Δt x e λtλ λtHeat kernel 1s 1ζ Z (s) dtt Z ( t ) Γ (s) 0Mellin transform11ζ Z ( s ) Tr s sΔλ λSmall t behaviour of Z(t) poles of ζZ(s)Weylexpansion33
The heat kernel is related to the density ofstates of the LaplacianThere are “Laplace transform” of each other:From the Weyl expansion, it is possible to obtain the densityof states.!
How does it work ?Diffusion (heat) equation in d 1whose spectral solution is Pt ( x, y ) 1( 4π Dt ) 21e2x y )( 4 DtProbability of diffusing from x to y in a time t.!In d space dimensions:Pt ( x, y ) 1( 4π Dt )de2x y )( 4 Dt2We can characterise the “spatial geometry” by watching how the heat flows.The heat kernel Z d ( t ) isZd (t ) Vol.d x Pt ( x, x ) dVolume( 4π Dt )d2access the volumeof the manifold
How does it work ?Diffusion (heat) equation in d 1whose spectral solution is Pt ( x, y ) 1( 4π Dt ) 21e2x y )( 4 DtProbability of diffusing from x to y in a time t.!In d space dimensions:Pt ( x, y ) 1( 4π Dt )de2x y )( 4 Dt2We can characterise the “spatial geometry” by watching how the heat flows.The heat kernel Z d ( t ) isZd (t ) Vol.d x Pt ( x, x ) dVolume( 4π Dt )d2access the volumeof the manifold
How does it work ?Diffusion (heat) equation in d 1whose spectral solution is Pt ( x, y ) 1( 4π Dt ) 21e2x y )( 4 DtProbability of diffusing from x to y in a time t.!In d space dimensions:Pt ( x, y ) 1( 4π Dt )de2x y )( 4 Dt2We can characterise the “spatial geometry” by watching how the heat flows.The heat kernel Z d ( t ) isZd (t ) Vol.d x Pt ( x, x ) dVolume( 4π Dt )d2access the volumeof the manifold
How does it work ?Diffusion (heat) equation in d 1whose spectral solution is Pt ( x, y ) 1( 4π Dt ) 21e2x y )( 4 DtProbability of diffusing from x to y in a time t.!In d space dimensions:Pt ( x, y ) 1( 4π Dt )de2x y )( 4 Dt2We can characterise the “spatial geometry” by watching how the heat flows.The heat kernel Z d ( t ) isZd (t ) Vol.d x Pt ( x, x ) dVolume( 4π Dt )d2volume of themanifold
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0L
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0L
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formula
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formula
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formulaWeyl expansion(1d)
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formulaWeyl expansion (2d) :Vol. L 11Z d 2 (t) 4π t 4 4π t 6
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formulaWeyl expansion (2d) :Vol. L 11Z d 2 (t) 4π t 4 4π t 6bulk
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formulaWeyl expansion (2d) :Vol. L 11Z d 2 (t) 4π t 4 4π t 6bulksensitive to boundary
Boundary terms- Hearing the shape of a drum Mark Kac (1966)0LPoisson formulaWeyl expansion (2d) :Vol. L 11Z d 2 (t) 4π t 4 4π t 6bulksensitive to boundaryintegral of bound.curvature
ζ -functionZeta11ζ Z ( s ) Tr s sfunctionΔλ λhas a simple pole atso that,
How does it work on a fractal ?Differently No access to the eigenvalue spectrum but we know howto calculate the Heat Kernel.Z(t) Tre Δt dx x eand thus, the density of states, Δtx eλ λt
How does it work on a fractal ?Differently No simple access to the eigenvalue spectrum but weknow how to calculate the heat kernel.Z(t) Tre Δt dx x eand thus, the density of states, Δtx eλ λt
More precisely,is the total length upon iteration of the elementary stepwhich has poles at
ds 2iπ nsn 2 dw ln aInfinite number of complex poles : complex fractal dimensions.They control the behaviour of the heat kernel which exhibits oscillations.K!t"#K leading!t"1.00.8Z diamond ( t ) 0.61.005Figure: Diamond fractals, non-p.c.f., but finitely .0510!30.10t2.10!30.150.20t0.25A new fractal dimension : spectral dimensionds
Notion of spectral volume
From the previous expression we obtainZ (t )Consider for simplicity n 1, namely s1 ds 2iπ ds iδ2to compare withdw ln a2
From the previous expression we obtainZ (t )Consider for simplicity n 1, namely s1 ds 2iπ ds iδ2so thatto compare withdw ln a2
From the previous expression we obtainZ (t )Consider for simplicity n 1, namely s1 ds 2iπ ds iδ2so thatto compare withdw ln aSpectralvolume2
From the previous expression we obtainZ (t )Consider for simplicity n 1, namely s1 ds 2iπ ds iδ2Zd (t ) 2Spectralvolumeso thatto compare withdw ln a dVol.dx Pt ( x, x ) Volume( 4π Dt )d2
Spectral volume ?Geometric volume described bythe Hausdorff dimension is large(infinite)Spectral volumeis the finite volume occupied by themodesNumerical solution of Maxwell eqs. in the Sierpinski gasket(courtesy of S.F. Liew and H. Cao, Yale)
Spectral volume ?Geometric volume described bythe Hausdorff dimension is large(infinite)Spectral volumeis the finite volume occupied by themodesNumerical solution of Maxwell eqs. on the Sierpinski gasket
Physical application :Thermodynamics of photons on fractalsElectromagnetic field in a waveguide fractal structure.How to measure the spectral volume ?
usual approach: count modes in momentum spaceThe radiating fractal blackbodythermal equilibrium:Equationof stateat thermodynamicequationof stateequilibrium relating pressure, volume1P V energy: Uand internalddET2 V 2 3 d 2 cpressurevolumeinternal energyIn an enclosure with a perfectly reflecting surface there can form standing electromagnetic wavesanalogous to tones of an organ pipe; we shall confine our attention to very high overtones. asks for the energy in the frequency interval dν . It is here that there arises the mathematicalproblem to prove that the number of sufficiently high overtones that lies in the interval ν to ν dνis independent of the shape of the enclosure and is simply proportional to its volume.1 ln ZP VU ln Z 1H. Lorentz, 1910T
usual approach: count modes in momentum spaceThe radiating fractal blackbodythermal equilibrium:Equationof stateat thermodynamicequationof stateequilibrium relating pressure, volume1P V energy: Uand internalddET2 V 2 3 d 2 cpressurevolumeinternal energyIn an enclosure with a perfectly reflecting surface there can form standing electromagnetic wavesanalogous to tones of an organ pipe; we shall confine our attention to very high overtones. asks for the energy in the frequency interval dν . It is here that there arises the mathematicalproblem to prove that the number of sufficiently high overtones that lies in the interval ν to ν dνis independent of the shape of the enclosure and is simply proportional to its volume.1 ln ZP VU ln Z Spectralvolume ?1H. Lorentz, 1910T
proach: count modes in momentum spaceUsual approach : count modes in momentum spacethermalequilibrium:Calculate the partition (generating)is a dimensionlessequationof statefunctionfunction z (T ,V ) for a blackbody of1a large volume! Black-body radiationP V in Ud is a dimensionlessfunctionlarge volume V in dimensiond( 2π )dVe decompositionof the volumefield:Black-bodyradiationin a largeenergyvolumepressureinternalMode decompositionof thefield:Mode decompositionof thefieldensional integer-valued vector-elementary momentum space cellsction (generating function)hatnZVln Z(T, V )d-dimensional integer-valued vector-elementary momentum space cellsso thatU ln Z withβ 1k TBis thephoton thermal wavelength.is the photon thermalwavelength.Lβ β !c1 thermal(photonTwavelength)
Thermodynamics :so thatStefan-BoltzmannAdiabatic expansion(The exact expression of Q is unimportant)is a consequence of
Thermodynamics :so thatStefan-BoltzmannAdiabatic expansion(The exact expression of Q is unimportant)is a consequence of
Thermodynamics :so thatStefan-BoltzmannAdiabatic expansion(The exact expression of Q is unimportant)is a consequence of
On a fractal there is no notion of Fourier mode decomposition.!Dimensions of momentum and position spaces are usuallydifferent : problem with the conventional formulation in terms ofphase space cells.!Volume of a fractal is usually infinite.!Nevertheless,is the “spectral volume”.
On a fractal there is no notion of Fourier mode decomposition.!Dimensions of momentum and position spaces are usuallydifferent : problem with the conventional formulation in terms ofphase space cells.!Volume of a fractal is usually infinite.!Nevertheless,is the “spectral volume”.
On a fractal there is no notion of Fourier mode decomposition.!Dimensions of momentum and position spaces are usuallydifferent : problem with the conventional formulation in terms ofphase space cells.!Volume of a fractal is usually infinite.!Nevertheless,is the “spectral volume”.
On a fractal there is no notion of Fourier mode decomposition.!Dimensions of momentum and position spaces are usuallydifferent : problem with the conventional formulation in terms ofphase space cells.!Volume of a fractal is usually infinite.!Nevertheless,is the “spectral volume”.
Re-phrase the thermodynamic problem interms of heat kernel and zeta function.
Partition function of equilibrium quantum radiation 212 ln z (T ,V ) lnDet M V 2 c Δ 2 τ Looks (almost) like a bona fide wave equationbut proper time.This expression does not rely on mode decomposition.!Rescale by Lβ β !c
Partition function of equilibrium quantum radiation 2 12ln z (T ,V ) lnDet M V 2 Lβ Δ 2 u M : circle of radius Lβ β !cSpatial manifold (fractal)Thermal equilibrium of photons on a spatial manifold V attemperature T is described by the (scaled) wave equationon M V
2 12ln z (T ,V ) lnDet M V 2 Lβ Δ 2 u can be rewritten 1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τ(Z L τ2β)Heat kernelLarge volume limit (a high temperature limit)Weyl expansion:()Z L τ 2β(V4π L2β τ)d2
2 12ln z (T ,V ) lnDet M V 2 Lβ Δ 2 u can be rewritten 1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τLarge volume limit (a high temperature limit)Weyl expansion:()Z L τ 2β(V4π L2β τ)d2
2 12ln z (T ,V ) lnDet M V 2 Lβ Δ 2 u can be rewritten 1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τLarge volume limit (a high temperature limit)Weyl expansion:()Z L τ 2β(V4π L2β τ)d2
2 12ln z (T ,V ) lnDet M V 2 Lβ Δ 2 u can be rewritten 1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τ(Z L τ2β)Heat kernelLarge volume limit (a high temperature limit)Weyl expansion:()Z L τ 2β(V4π L2β τ)d2
2 12ln z (T ,V ) lnDet M V 2 Lβ Δ 2 u can be rewritten 1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τ(Z L τ2β)Heat kernelLarge volume limit (a high temperature limit)Weyl expansion:()Z L τ 2β(V4π L2β τ)d2
1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τ Weyl expansion Vln z (T ,V ) dLβ
1 dτ τ L2β Δln z (T ,V ) f (τ ) TrV e20 τ Weyl expansion Vln z (T ,V ) dLβThermodynamics :Thermodynamics measures the spectral volumeso thatStefan-Boltzmann(The exact expression of Q is unimportant)is a co
Generalities on fractals Many self-similar (fractal) structures in nature and many ways to model them: A random walk in free space or on a periodic lattice etc. Fractals provide a useful testing ground to investigate properties of disordered classical or quantum systems, renormalization group and phase transitions,
A commonly asked question is: What are fractals useful for Nature has used fractal designs for at least hundreds of millions of years. Only recently have human engineers begun copying natural fractals for inspiration to build successful devices. Below are just a few examples of fractals being used in engineering and medicine.
Fractals and 3D printing 1. What are fractals? A fractal is a geometric object (like a line or a circle) that is rough or irregular on all scales of length (invariant of scale). A Fractal has a broken dimension. By zooming-in and zooming-out the new object is similar to the original object: fractals have a self-similar structure.
the scaling laws of disorder averages of the con gurational properties of SAWs, and clearly indicate a multifractal spectrum which emerges when two fractals meet each other. 1. INTRODUCTION Self-avoiding walks (SAWs) on regular lattices provide a successful description of the universal con gurational properties of polymer chains in good solvent .
Physics 20 General College Physics (PHYS 104). Camosun College Physics 20 General Elementary Physics (PHYS 20). Medicine Hat College Physics 20 Physics (ASP 114). NAIT Physics 20 Radiology (Z-HO9 A408). Red River College Physics 20 Physics (PHYS 184). Saskatchewan Polytechnic (SIAST) Physics 20 Physics (PHYS 184). Physics (PHYS 182).
fractals involve chance, their regularities and irregularities being statistical. Finally, they engage the . Fractal dimension Fractals can be constructed through limits of iterative schemes involving generators of iterative . Its form is extremely irregular or fragmente
Edge midpoints, ::: Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 38/41. Height Maps: Clouds and Coloring Fractals in Nature and Mathematics R. L. Herman OLLI STEM Society, Oct 13, 2017 39/41. Other Applications Video Games Fracture - link Ceramic Material - link
Fractals and triangles What is a fractal? A fractal is a pattern created by repeating the same process on a different scale. One of the most famous fractals is the Mandelbrot set, shown below. This was first printed out on dot matrix paper! (ask your teacher ). Created by Wolfgang Beyer with the program Ultra Fractal 3. - Own work
Scrum 1 Agile has become one of the big buzzwords in the software development industry. But what exactly is agile development? Put simply, agile development is a different way of executing software development teams and projects. To understand what is new, let us recap the traditional methods. In conventional software development, the product requirements are finalized before proceeding with .
