Statistical mechanics of Bose gas in Sierpinski carpets

Transcription

1 Statistical mechanics of Bose gas in Sierpinski carpets Joe P. Chen Cornell University 4th Cornell Conference on Analysis, Probability & Mathematical Physics on Fractals Ithaca, NY September 13, 2011 Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 1 / 23

2 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 {φ i } m F i=1. Geometric conditions: Symmetry, connected interior, non-diagonality, bordering. l F : Length scale factor; m F : Mass scale factor. Goal: Study spectral geometry (of fractals) using physical probes (quantum gases). Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 2 / 23

3 Outline 1 The set-up for quantum statistical mechanics 2 What we know about diffusion on Sierpinski carpets 3 Spectral zeta function on Sierpinski carpets 4 Applications: Ideal quantum gas in Sierpinski carpets Photon gas: Blackbody radiation & Casimir effect Bose-Einstein condensation & its connection to Brownian motion Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 3 / 23

4 Quantum mechanics at zero temperature Single Particle H 1 = L 2 (F, µ): Hilbert space for a single particle in state space F. H : H 1 H 1: self-adjoint Hamiltonian (energy) operator. State ψ H 1 evolves in time under the Schrödinger equation i tψ(t, x) = Hψ(t, x). The observables of the system (position, momentum, energy, ) are represented by self-adjoint operators. The expectation value of an observable A in state ψ H 1 is ψ, Aψ. Many Particles H cl N = H N 1 : Hilbert space for N classical particles. However, when one has N identical particles, the N-particle wavefunction obeys certain quantum statistics under particle exchange: f 1 f i f j f N = ± (f 1 f j f i f N ). +: Bose-Einstein statistics. Hilbert space is H N = Sym(H N 1 ). : Fermi-Dirac statistics. Hilbert space is H N = Antisym(H N 1 ). Hamiltonian, Schrödinger equation, observables follow similarly. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 4 / 23

5 Quantum mechanics of many particles at positive temperature Working in the grand canonical ensemble: F = N 0 H N, with H 0 = C: Fock space. For any self-adjoint op K on H 1, define its second quantization by dγ(k) = N 0 K N, where K 0 0 and K N (f 1 f j f N ) = N (f 1 Kf j f N ), N 1. j=1 N = dγ(1): number operator. β > 0: inverse temperature. µ inf Spec(H): chemical potential. (Encodes information about particle number.) Definition The Gibbs grand canonical state at (β, µ) is a linear functional ω β,µ over the quasi-local C -algebras such that the expected value of operator A is ω β,µ (A) = 1 Ξ β,µ Tr F(e β(dγ(h) µn) A). where Ξ β,µ := Tr F(e β(dγ(h) µn) ) is the grand canonical partition function. A Gibbs state is also a Kubo-Martin-Schwinger state at τ = iβ. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 5 / 23

6 Example: Ideal Bose gas H N consists of one-body operators only. (No interaction amongst particles.) In this case the partition function can be expressed explicitly: log Ξ β,µ = Tr H1 log(1 e β(h µ) ). The expected values of particle number and energy can be obtained by taking derivatives of log Ξ: ω β,µ (N) = 1 β µ log Ξ = 1 TrH 1 e β(h µ) 1. ω β,µ (H) = log Ξ. β Rem. [, inf Spec(H)] µ ω β,µ (N) [0, ] is increasing and convex. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 6 / 23

7 Ideal Bose gas in generalized Sierpinski carpets log Ξ β,µ = Tr H1 log(1 e β(h µ) ). ω β,µ (N) = Tr H1 1 e β(h µ) 1. Statement of the problem Compute log Ξ β,µ and ω β,µ (N) when F is a GSC, and the many-body particle system is of the following types: 1 (Massless) photon gas: H =, µ = 0. 2 Massive Bose gas (cold atoms): H =, µ inf Spec(H). We don t know the exact eigenvalues of. But it turns out that all we need to compute these quantities is a sharp estimate of the trace of the heat kernel. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 7 / 23

8 Dirichlet energy in the Barlow-Bass construction Reflecting BM Wt 1, Dirichlet energy E 1(u) = u(x) 2 µ 1(dx). F 1 Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 8 / 23

9 Dirichlet energy in the Barlow-Bass construction Reflecting BM Wt 2, Dirichlet energy E 2(u) = u(x) 2 µ 2(dx). F 2 Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 9 / 23

10 Dirichlet energy in the Barlow-Bass construction Time-scaled reflecting BM X n t = W n a nt, Dirichlet energy E n(u) = a n F n u(x) 2 µ n(dx). Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 10 / 23

11 Dirichlet energy in the Barlow-Bass construction Time-scaled reflecting BM X n t = W n a nt, Dirichlet energy E n(u) = a n F n u(x) 2 µ n(dx). µ n µ = C(d H -dim Hausdorff measure) on carpet F. ( ) n mf 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 energy on L 2 (F ) by 1 E BB (u) = sup u Ttu, u with natural domain. t>0 t It is strong local regular conservative, and is self-similar: m F E BB (u) = ρ F E BB (u φ i ). The Laplacian is the corresponding infinitesimal generator. i=1 (Barlow, Bass, Kumagai & Teplyaev 2008) Up to constant multiples,! DF on GSC. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 11 / 23

12 Heat kernel estimates on GSCs Theorem (Barlow, Bass, Kusuoka, Zhou, ) ( p t(x, y) C 1t d H/d W exp C 2 ( x y d W t ) 1 ) d W 1. Here d H = log m F / log l F (Hausdorff), d W = log(ρ F m F )/ log l F (walk). d S = 2 dh = 2 log m F is the spectral dimension of the carpet. d W log(m F ρ F ) 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 K(t) := p t(x, x)µ(dx) = t d H/d W [G( log t) + o(1)] as t 0. F Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 12 / 23

13 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 K(t) = d k=0 ( )) t d k /d W G k ( log t) + O exp ( ct d 1 W 1 as t 0, where d k := d H (F {x 1 = = x k = 0}). Rem 1. G 0 > 0 and G 1 < 0. Numerics suggest that G 0 is nonconstant. Rem 2. The analogous result for manifolds M d is K(t) = d t (d k)/2 G k + O(exp( ct 1 )) as t 0. k=0 Notation. G k (x) = ( 2πpix Ĝ k,p exp log ρ F p Z ). Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 13 / 23

14 The function G 0 (Numerics on level-5 standard Sierpinski carpet) t d s /2 K(t) t Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 14 / 23

15 Spectral zeta function ζ (s, γ) := Tr 1 (Mellin integral rep) ζ (s, γ) = 1 Γ(s) 0 ( +γ) s t s e γt K(t) dt t = j=1 (λ j + γ) s, Re(s) > ds 2. (Poles of ζ (s, 0), for GSC R 2 ) Im(s) Re(s) Using the Mittag-Leffler decomposition, we find that the simple poles of ζ (s) are (at most) d k=0 p Z { dk d W + 2πpi } =: log ρ F d k=0 p Z { dk,p ( with residues Res ζ, d ) k,p Ĝ k,p = 2 Γ (d k,p /2). 2 }, The poles of the spectral zeta fcn encode the dims of the relevant spectral volumes: { d On manifolds M d (d 1) :,,, 1 } , 0 { } dk,p On fractals:. (Complex dims, à la Lapidus) 2 k,p Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 15 / 23

16 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) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 16 / 23

17 Photon gas in GSC: H =, µ = 0 Proposition The grand canonical partition function Ξ for a non-interacting, spinless photon gas in a bounded domain F is given by log Ξ = 1 πi σ+i σ i β 2t Γ(2t)ζ R (2t + 1)ζ (t)dt, where σ R, σ > d S/2, ζ R is the Riemann zeta function, and ζ is the spectral zeta function associated with the dimensionful Laplacian on F. Lemma Let F be a GSC of side L. Then the grand canonical partition function Ξ for an ideal spinless photon gas confined in F is given by log Ξ(L) = 2 d k=0 p Z ( ) dk,p L Γ (dk,p ) β Γ (d k,p /2) ζ R (d k,p + 1) Ĝ k,p + β ( 2L ζ 1 ). 2 We interpret E Cas := 1 2L ζ ( 1/2) as the Casimir energy, the energy of (zero-temperature) vacuum fluctuations. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 17 / 23

18 Blackbody radiation in GSC Recall that the appropriate thermodynamic volume is the spectral volume (Akkermans, Dunne & Teplyaev 10) ( ) ( ) V S(Λ L ) = (4π) ds/2 ds Γ Res ζ, ds L d S. 2 2 Let E(L) := 1 V S (Λ L ) ω β,0(h) be the energy density of the photon gas in a GSC of side L. Proposition For large L, E(L) = 1 d S β d S+1 π Γ (d S+1)/2 + 1 β d S+1 8 Ĝ 0,0(4π) d S/2 ( ds p=1 ) ζ R (d S + 1) { log(l/β) 4πpi Re e log R } Γ(d 0,p) Γ(d ζ R(d 0,p + 0,p/2) 1)Ĝ0,p + o(1). First term is the classical Stefan-Boltzmann law. (cf. Landau-Lifshitz, Vol. 5) Second term is contributed by the tower of poles at Re(s) = d S/2. Equation of state: P = 1 d S E. Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 18 / 23

19 Massive Bose gas in GSC: H =, µ inf Spec(H) Proposition The grand canonical partition function Ξ for a non-interacting, spinless massive Bose gas in a bounded domain F is given by log Ξ(β, µ) = 1 2πi σ+i σ i ( ) L 2 t Γ(t)ζ R (t + 1)ζ (t, µ)dt. β 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. Lemma As Λ n, the density of Bose gas in a GSC at (β, µ) is 1 ( )) ρ Λn (β, µ) = e mβµ mβ G 0 ( log m ds/2 + o(1). (4πβ) d S/2 Ĝ 0,0 (l F ) 2n m=1 In particular, the upper critical density ρ c (β) := lim sup Λn ρ Λn (β, 0) < iff d S > 2. Caveat: Numerics suggests that the infinite-volume limit does not exist! Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 19 / 23

20 Bose-Einstein condensation (BEC) in GSC Alternatively one could write ρ L (β, µ) = 1 V S(Λ L ) e mβµ K m=1 ( ) mβ. L 2 m=1 K(mt) < BM is transient. What is the significance of finite ρ c (β)? Theorem (C. 11) Assume d S > 2. For each ρ tot > 0, choose µ n to be the unique root of ρ Λn (β, µ n) = ρ tot. If ρ tot > ρ c (β), then lim Λn µ n = 0 and the lower bound on the density in the ground state ρ (β) := lim inf Λ n V S (Λ n) e β(e 0(Λ n) µ n) 1 = ρtot ρ c (β). Interpretation: Any Bose gas in excess of ρ c (β) must condense in the lowest eigenstate of the Laplacian. This is the fractal version of Bose-Einstein condensation, a genuine quantum phase transition. (In Euclidean setting: theory 1924, experimentally realized in R , Nobel Physics Prize 2001) Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 20 / 23

21 Criterion for BEC in GSC Theorem (C. 11) For an ideal massive Bose gas in an unbounded GSC, the following are equivalent: 1 Spectral dimension d S > 2. 2 (The BM whose infinitesimal generator is) the Laplacian is transient. 3 BEC exists at positive temperature. Transience BEC has been mentioned in the context of inhomogeneous graphs (Fidaleo, Guido, Isola; Matsui). Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 21 / 23

22 Probing BEC in non-integer dimensions: Menger sponges MS(3,1) MS(4,2) MS(5,3) MS(6,4) Rigorous bnds on d S [Barlow & Bass 99] Numerical d S [C.] BEC exists? Yes Yes Yes (?) No Open problems (in increasing order of difficulty) Show BEC for a hardcore Bose gas on fractal lattices. ( phase transition in XY-model: need reflection positivity) Show BEC for an interacting dilute Bose gas in fractals. (Need to develop some notion of field theory on fractals, e.g. Gaussian free fields) Show superfluidity of BEC in fractals. (Mermin-Wagner theorem rotational symmetry breaking for d 2, but what d?) Dynamics of BEC in fractals. (Gross-Pitaevskii?) Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 22 / 23

23 Acknowledgements Many thanks to Robert Strichartz (Cornell) Ben Steinhurst (Cornell) Naotaka Kajino (Bielefeld) Alexander Teplyaev (UConn) Also discussions with Gerald Dunne (UConn) Matt Begue (Maryland) 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 Charles University of Prague & European Science Foundation Joe P. Chen (Cornell University) Bose gas in Sierpinski carpets Fractal Cornell 09/13/11 23 / 23