Spectral zeta function & quantum statistical mechanics on Sierpinski carpets Joe P. Chen Cornell University 6th Prague Summer School on Mathematical Statistical Physics / August 30, 2011 Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 1 / 15
Generalized Sierpinski carpets Sierpinski carpet Menger sponge ( m F = 8, d H = log 8 / log 3) ( m F = 20, d H = log 20 / log 3) Constructed via an IFS of affine contractions { f i } m F i =1 . They are infinitely ramified fractals. ( Translation: Analysis is hard.) Brownian motion and harmonic analysis on SCs have been studied extensively by Barlow & Bass (1989 onwards) and Kusuoka & Zhou (1992). Uniqueness of BM on SCs [Barlow, Bass, Kumagai & Teplyaev 2008]. Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 2 / 15
Outline What we know about diffusion on Sierpinski carpets 1 ◮ Hausdorff dimension, walk dimension, and spectral dimension ◮ Estimates of the heat kernel trace Spectral zeta function on Sierpinski carpets 2 ◮ Simple poles give the carpet’s ”complex dimensions” ◮ Meromorphic continuation to C Applications: Ideal quantum gas in Sierpinski carpets 3 ◮ Bose-Einstein condensation & its connection to Brownian motion Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 3 / 15
Dirichlet energy in the Barlow-Bass construction � Reflecting BM W 1 |∇ u ( x ) | 2 µ 1 ( dx ). t , Dirichlet energy E 1 ( u ) = F 1 Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 4 / 15
Dirichlet energy in the Barlow-Bass construction � Reflecting BM W 2 |∇ u ( x ) | 2 µ 2 ( dx ). t , Dirichlet energy E 2 ( u ) = F 2 Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 5 / 15
Dirichlet energy in the Barlow-Bass construction � |∇ u ( x ) | 2 µ n ( dx ). Time-scaled BM X n t = W n a n t , DF E n ( u ) = a n F n Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 6 / 15
Dirichlet energy in the Barlow-Bass construction � |∇ u ( x ) | 2 µ n ( dx ). Time-scaled BM X n t = W n a n t , DF E n ( u ) = a n F n µ n ⇀ µ = C ( d H -dim Hausdorff measure) on carpet F . � n � m F ρ F a n ≍ , ρ F = resistance scale factor. No closed form expression known. l 2 F BB showed that there exists subsequence { n j } such that, resp., the laws and the resolvents of X n j are tight. Any such limit process is a BM on the carpet F . If X is a limit process and T t its semigroup, define the Dirichlet form on L 2 ( F ) by 1 E BB ( u ) = sup t � u − T t u , u � with natural domain . t > 0 Denote by ∆ the corresponding Laplacian. Note E BB is self-similar: m F � E BB ( u ) = ρ F · E BB ( u ◦ f i ) . i =1 Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 7 / 15
Heat kernel estimates on GSCs Theorem (Barlow, Bass, · · · ) 1 � d W − 1 � � | x − y | d W � p t ( x , y ) ≍ C 1 t − d H / d W exp − C 2 . t Here d H = log m F / log l F (Hausdorff), d W = log( ρ F m F ) / log l F (walk). d S = 2 d H log m F d W = 2 log( m F ρ F ) is the spectral dimension of the carpet. For any carpet, d W > 2 and 1 < d S < d H , indicative of sub-Gaussian diffusion. Theorem (Hambly ’08, Kajino ’08) There exists a (log ρ F ) -periodic function G, bounded away from 0 and ∞ , such that the heat kernel trace � p t ( x , x ) µ ( dx ) = t − d H / d W [ G ( − log t ) + o (1)] K ( t ) := as t ↓ 0 . F Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 8 / 15
A better estimate of heat kernel trace Consider GSCs with Dirichlet b.c. on exterior boundary, and Neumann b.c. on the interior boundaries. Theorem (Kajino ’09, in prep ’11) For any GSC F ⊂ R d , there exist continuous, (log ρ F ) -periodic functions G k : R → R for k = 0 , 1 , · · · , d such that d 1 � � �� � t − d k / d W G k ( − log t ) + O − K ( t ) = exp − ct as t ↓ 0 , d W − 1 k =0 where d k := d H ( F ∩ { x 1 = · · · = x k = 0 } ) . Remark. G 0 > 0 and G 1 < 0. Numerics suggest that G 0 is nonconstant. Recall that the analogous result for manifolds M d is d � t − ( d − k ) / 2 G k + O (exp( − ct − 1 )) K ( t ) = as t ↓ 0 . k =0 Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 9 / 15
1 ( − ∆+ γ ) s = � ∞ j =1 ( λ j + γ ) − s Spectral zeta function ζ ∆ ( s , γ ) := Tr (Mellin integral rep) � ∞ 1 t s e − γ t K ( t ) dt Re ( s ) > d S ζ ∆ ( s , γ ) = , 2 . Γ( s ) t 0 (Poles of ζ ∆ ( s , 0), for GSC ⊂ R 2 ) Using the Mittag-Leffler decomposition, we find Im ( s ) that the simple poles of ζ ∆ ( s ) are (at most) � d k d d � d W + 2 π pi � � � � := { d k , p } , log ρ F Re ( s ) k =0 k =0 p ∈ Z p ∈ Z ˆ G k , p with residues Res ( ζ ∆ , d k , p ) = Γ ( d k , p ). The poles of the spectral zeta fcn encode the dims of the relevant spectral volumes: � d 2 , ( d − 1) , · · · , 1 � On manifolds M d : 2 , 0 . 2 � d k d W + 2 π pi � On fractals: . (Complex dims, ` a la Lapidus) log ρ F k , p Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 10 / 15
Meromorphic continuation of ζ ∆ Im ( s ) Re ( s ) Theorem (Steinhurst & Teplyaev ’10) ζ ∆ ( · , γ ) has a meromorphic continuation to all of C . The exponential tail in the HKT estimate is essential for the continuation. In particular, ζ ∆ ( s , 0) is analytic for Re ( s ) < 0. Application: Casimir energy ∝ ζ ∆ ( − 1 / 2). Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 11 / 15
Application to quantum statistical mechanics on GSC Consider a gas of N bosons confined to a domain F . The N -body wavefunction is symmetric under particle exchange, so the Hilbert L 2 ( F ) ⊗ N � � space is H N = Sym . H N = � N j =1 K ( x j ) + � N i < j V ( x i − x j ) is the Hamiltonian on H N . In the grand canonical ensemble, F = ⊕ ∞ N =0 H N is the Fock space; d Γ( H ) = ⊕ ∞ N =0 H N the second quantization. The Gibbs state at inverse temp β > 0 and chemical potential µ is given by ω β,µ ( · ), where for any self-adjoint operator A one has ω β,µ ( A ) = Ξ − 1 Tr F ( e − β d Γ( H − µ 1 ) A ) , with GC part. fcn. Ξ = Tr F ( e − β d Γ( H − µ 1 ) ) . For ideal Bose gas ( V ≡ 0), log Ξ = − Tr H 1 log(1 − e − β ( H − µ ) ). For an ideal massive Bose gas ( K = − ∆) in a carpet of side length L , � σ + i ∞ � t � L 2 1 � � t , − L 2 µ log Ξ L ( β, µ ) = Γ( t ) ζ R ( t + 1) ζ ∆ dt . 2 π i β σ − i ∞ Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 12 / 15
Bose-Einstein condensation in GSC Consider the unbounded carpet F ∞ = � ∞ n =0 l n F F . We exhaust F ∞ by taking an increasing family of carpets { Λ n } n = { l n F F } n . Proposition As n → ∞ , the density of Bose gas in a GSC at ( β, µ ) is � m β ∞ � �� 1 m − d S / 2 + o (1) . � e m βµ G 0 ρ Λ n ( β, µ ) = − log (4 πβ ) d S / 2 ˆ ( l F ) 2 n G 0 , 0 m =1 In particular, ρ ( β ) := lim sup n →∞ ρ Λ n ( β, 0) < ∞ iff d S > 2. If the total density ρ tot > ρ ( β ), then the excess density must condense in the lowest eigenvector of the Hamiltonian → BEC. Observe also that ∞ � m β � 1 � e m βµ K ρ L ( β, µ ) = . L 2 CL d S m =1 � ∞ m =1 K ( mt ) < ∞ ⇐ ⇒ BM is transient. Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 13 / 15
Criterion for BEC in GSC Theorem For an ideal massive Bose gas in an unbounded GSC, the following are equivalent: Spectral dimension d S > 2 . 1 (The Brownian motion whose generator is) the Laplacian is transient. 2 BEC exists at positive temperature. 3 MS(3,1) MS(4,2) MS(5,3) MS(6,4) Rigorous bnds on 2 . 21 ∼ 2 . 60 2 . 00 ∼ 2 . 26 1 . 89 ∼ 2 . 07 1 . 82 ∼ 1 . 95 d S [Barlow & Bass ’99] Numerical d S [C.] 2 . 51 ... - 2 . 01 ... - BEC exists? Yes Yes Yes (?) No Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 14 / 15
Acknowledgements Many thanks to Robert Strichartz (Cornell) Ben Steinhurst (Cornell) Naotaka Kajino (Bielefeld) Alexander Teplyaev (UConn) Also discussions with Gerald Dunne (UConn) Matt Begu´ e (UConn → UMd) 2011 research & travel support by: NSF REU grant (Analysis on fractals project, 2011 Cornell Math REU) Cornell graduate school travel grant Technion & Israel Science Foundation (Taken at: AMS Fall Sectional Mtg, October 2-3, 2010, Syracuse) Charles University of Prague, ESF & NSF (Upcoming: ’Fractals4’ joint w/ AMS Mtg, Sept 10-13, 2011, Ithaca) Joe P. Chen (Cornell University) Spec zeta & Quantum statmech in SCs Prague 08/30/11 15 / 15
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