joe p. chenteplyaev/cornell4/...outline 1 the set-up for quantum statistical mechanics 2 what we...

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 4 @ Cornell 09/13/11 1 / 23

Generalized Sierpinski carpets

Sierpinski carpet Menger sponge(mF = 8, dH = log 8/ log 3) (mF = 20, dH = log 20/ log 3)

Constructed via an IFS of affine contractions {φi}mFi=1.

Geometric conditions: Symmetry, connected interior, non-diagonality, bordering.

lF : Length scale factor; mF : Mass scale factor.

Goal: Study spectral geometry (of fractals) using physical probes (quantum gases).


Outline

1 The set-up for quantum statistical mechanics

2 What we know about diffusion on Sierpinski carpets

3 Spectral zeta function on Sierpinski carpets4 Applications: Ideal quantum gas in Sierpinski carpets

I Photon gas: Blackbody radiation & Casimir effectI Bose-Einstein condensation & its connection to Brownian motion


Quantum mechanics at zero temperature

Single Particle

H1 = L2(F , µ): Hilbert space for a single particle in state space F .

H : H1 → H1: self-adjoint Hamiltonian (energy) operator.

State ψ ∈ H1 evolves in time under the Schrodinger equation i∂tψ(t, x) = Hψ(t, x).

The observables of the system (position, momentum, energy, · · · ) are represented byself-adjoint operators. The expectation value of an observable A in state ψ ∈ H1 is〈ψ,Aψ〉.

Many Particles

HclN = H⊗N

1 : Hilbert space for N classical particles.

However, when one has N identical particles, the N-particle wavefunction obeyscertain quantum statistics under particle exchange:

f1 ⊗ · · · ⊗ fi ⊗ · · · ⊗ fj ⊗ · · · ⊗ fN = ± (f1 ⊗ · · · ⊗ fj ⊗ · · · ⊗ fi ⊗ · · · ⊗ fN) .

I +: Bose-Einstein statistics. Hilbert space is HN = Sym(H⊗N1 ).

I −: Fermi-Dirac statistics. Hilbert space is HN = Antisym(H⊗N1 ).

Hamiltonian, Schrodinger equation, observables follow similarly.


Quantum mechanics of many particles at positive temperatureWorking in the grand canonical ensemble:

F =⊕

N≥0HN , with H0 = C: Fock space.

For any self-adjoint op K on H1, define its second quantization bydΓ(K) =

⊕N≥0 KN , where K0 ≡ 0 and

KN(f1 ⊗ · · · ⊗ fj ⊗ · · · ⊗ fN) =N∑j=1

(f1 ⊗ · · · ⊗ Kfj ⊗ · · · ⊗ fN) , N ≥ 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 ωβ,µ over the quasi-localC∗-algebras such that the expected value of operator A is

ωβ,µ(A) =1

Ξβ,µTrF (e−β(dΓ(H)−µN)A).

where Ξβ,µ := TrF (e−β(dΓ(H)−µN)) is the grand canonical partition function.

A Gibbs state is also a Kubo-Martin-Schwinger state at τ = iβ.


Example: Ideal Bose gas

HN consists of one-body operators only. (No interaction amongst particles.)

In this case the partition function can be expressed explicitly:

log Ξβ,µ = −TrH1 log(1− e−β(H−µ)).

The expected values of particle number and energy can be obtained by takingderivatives of log Ξ:

ωβ,µ(N) =1

β

∂

∂µlog Ξ = TrH1

1

eβ(H−µ) − 1.

ωβ,µ(H) = − ∂

∂βlog Ξ.

Rem. [−∞, inf Spec(H)] 3 µ 7→ ωβ,µ(N) ∈ [0,∞] is increasing and convex.


Ideal Bose gas in generalized Sierpinski carpets

log Ξβ,µ = −TrH1 log(1− e−β(H−µ)).

ωβ,µ(N) = TrH1

1

eβ(H−µ) − 1.

Statement of the problem

Compute log Ξβ,µ and ωβ,µ(N) when F is a GSC, and the many-body particle system isof 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 tocompute these quantities is a sharp estimate of the trace of the heat kernel.


Dirichlet energy in the Barlow-Bass construction

Reflecting BM W 1t , Dirichlet energy E1(u) =

∫F1

|∇u(x)|2µ1(dx).



Reflecting BM W 2t , Dirichlet energy E2(u) =

∫F2

|∇u(x)|2µ2(dx).



Time-scaled reflecting BM X nt = W n

ant , Dirichlet energy En(u) = an

∫Fn

|∇u(x)|2µn(dx).



Time-scaled reflecting BM X nt = W n

ant , Dirichlet energy En(u) = an

∫Fn

|∇u(x)|2µn(dx).

µn ⇀ µ = C(dH -dim Hausdorff measure) on carpet F .

an �(

mFρFl2F

)n

, ρF = resistance scale factor. No closed form expression known.

BB showed that there exists subsequence {nj} such that, resp., the laws and theresolvents of X nj are tight. Any such limit process is a BM on the carpet F .

If X is a limit process and Tt its semigroup, define the Dirichlet energy on L2(F ) by

EBB(u) = supt>0

1

t〈u − Ttu, u〉 with natural domain.

It is strong local regular conservative, and is self-similar:

EBB(u) =

mF∑i=1

ρF · EBB(u ◦ φi ).

The Laplacian is the corresponding infinitesimal generator.

(Barlow, Bass, Kumagai & Teplyaev 2008) Up to constant multiples, ∃! DF on GSC.


Heat kernel estimates on GSCs

Theorem (Barlow, Bass, Kusuoka, Zhou, · · · )

pt(x , y) � C1t−dH/dW exp

(−C2

(|x − y |dW

t

) 1dW−1

).

Here dH = log mF/ log lF (Hausdorff), dW = log(ρFmF )/ log lF (walk).

dS = 2dH

dW= 2

log mF

log(mFρF )is the spectral dimension of the carpet.

For any carpet, dW > 2 and 1 ≤ dS < dH, indicative of sub-Gaussian diffusion.

Theorem (Hambly ’08, Kajino ’08)

There exists a (log ρF )-periodic function G, bounded away from 0 and ∞, such that theheat kernel trace

K(t) :=

∫F

pt(x , x)µ(dx) = t−dH/dW [G(− log t) + o(1)] as t ↓ 0.


A better estimate of heat kernel trace

Consider GSCs with Dirichlet b.c. on exterior boundary, and Neumann b.c. on theinterior boundaries.

Theorem (Kajino ’09, in prep ’11)

For any GSC F ⊂ Rd , there exist continuous, (log ρF )-periodic functions Gk : R→ R fork = 0, 1, · · · , d such that

K(t) =d∑

k=0

t−dk/dWGk(− log t) +O(

exp(−ct

− 1dW−1

))as t ↓ 0,

where dk := dH (F ∩ {x1 = · · · = xk = 0}).

Rem 1. G0 > 0 and G1 < 0. Numerics suggest that G0 is nonconstant.Rem 2. The analogous result for manifolds Md is

K(t) =d∑

k=0

t−(d−k)/2Gk +O(exp(−ct−1)) as t ↓ 0.

Notation. Gk(x) =∑p∈Z

Gk,p exp

(2πpix

log ρF

).


The function G0

(Numerics on level-5 standard Sierpinski carpet)

10−4

10−3

10−2

10−1

0.25

0.255

0.26

0.265

0.27

0.275

0.28

t

td s/2K

(t)


Spectral zeta function ζ∆(s, γ) := Tr 1(−∆+γ)s =

∑∞j=1(λj + γ)−s

(Mellin integral rep)

ζ∆(s, γ) =1

Γ(s)

∫ ∞0

tse−γtK(t)dt

t, Re(s) >

dS

2.

(Poles of ζ∆(s, 0), for GSC ⊂ R2)

Re(s)

Im(s) Using the Mittag-Leffler decomposition, we findthat the simple poles of ζ∆(s) are (at most)

d⋃k=0

⋃p∈Z

{dk

dW+

2πpi

log ρF

}=:

d⋃k=0

⋃p∈Z

{dk,p

2

},

with residues Res

(ζ∆,

dk,p

2

)=

Gk,p

Γ (dk,p/2).

The poles of the spectral zeta fcn encode the dims of the relevant spectral volumes:

On manifolds Md :

{d

2,

(d − 1)

2, · · · , 1

2, 0

}.

On fractals:

{dk,p

2

}k,p

. (Complex dims, a la Lapidus)


Meromorphic continuation of ζ∆

Re(s)

Im(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).


Photon gas in GSC: H =√−∆, µ = 0

Proposition

The grand canonical partition function Ξ for a non-interacting, spinless photon gas in abounded domain F is given by

log Ξ =1

πi

∫ σ+i∞

σ−i∞β−2tΓ(2t)ζR(2t + 1)ζ∆(t)dt,

where σ ∈ R, σ > dS/2, ζR is the Riemann zeta function, and ζ∆ is the spectral zetafunction 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 idealspinless photon gas confined in F is given by

log Ξ(L) = 2d∑

k=0

∑p∈Z

(L

β

)dk,p Γ (dk,p)

Γ (dk,p/2)ζR (dk,p + 1) Gk,p +

β

2Lζ∆

(−1

2

).

We interpret ECas := 12Lζ∆(−1/2) as the Casimir energy, the energy of

(zero-temperature) vacuum fluctuations.


Blackbody radiation in GSC

Recall that the appropriate thermodynamic volume is the spectral volume(Akkermans, Dunne & Teplyaev ’10)

VS(ΛL) = (4π)dS/2Γ

(dS

2

)Res

(ζ∆,

dS

2

)LdS .

Let E(L) := 1VS(ΛL)

ωβ,0(H) be the energy density of the photon gas in a GSC of side L.

Proposition

For large L,

E(L) =1

βdS+1

dS

π(dS+1)/2Γ

(dS + 1

2

)ζR(dS + 1)

+1

βdS+1

8

G0,0(4π)dS/2

∞∑p=1

Re

{e4πpi

log(L/β)log R

Γ(d0,p)

Γ(d0,p/2)ζR(d0,p + 1)G0,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) = dS/2.

Equation of state: P = 1dSE .


Massive Bose gas in GSC: H = −∆, µ ≤ inf Spec(H)

Proposition

The grand canonical partition function Ξ for a non-interacting, spinless massive Bose gasin a bounded domain F is given by

log Ξ(β, µ) =1

2πi

∫ σ+i∞

σ−i∞

(L2

β

)t

Γ(t)ζR(t + 1)ζ∆ (t,−µ)dt.

Consider the unbounded carpet F∞ =⋃∞

n=0 lnFF . We exhaust F∞ by taking an increasingfamily of carpets {Λn}n = {lnFF}n.

Lemma

As Λn →∞, the density of Bose gas in a GSC at (β, µ) is

ρΛn (β, µ) =1

(4πβ)dS/2G0,0

∞∑m=1

emβµG0

(− log

(mβ

(lF )2n

))m−dS/2 + o(1).

In particular, the upper critical density ρc(β) := lim supΛn→∞ ρΛn (β, 0) <∞ iff dS > 2.

Caveat: Numerics suggests that the infinite-volume limit does not exist!


Bose-Einstein condensation (BEC) in GSCAlternatively one could write

ρL(β, µ) =1

VS(ΛL)

∞∑m=1

emβµK

(mβ

L2

).

∑∞m=1 K(mt) <∞⇐⇒ BM is transient.

What is the significance of finite ρc(β)?

Theorem (C. ’11)

Assume dS > 2. For each ρtot > 0, choose µn to be the unique root of ρΛn (β, µn) = ρtot.If ρtot > ρc(β), then limΛn→∞ µn = 0 and the lower bound on the density in the groundstate

ρ0(β) := lim infΛn→∞

1

VS(Λn)

1

eβ(E0(Λn)−µn) − 1= ρtot − ρc(β).

Interpretation: Any Bose gas in excess of ρc(β) must condense in the lowesteigenstate of the Laplacian. This is the fractal version of Bose-Einsteincondensation, a genuine quantum phase transition. (In Euclidean setting: theory 1924,

experimentally realized in R3 1995, Nobel Physics Prize 2001)


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 dS > 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).


Probing BEC in non-integer dimensions: Menger sponges

MS(3,1) MS(4,2) MS(5,3) MS(6,4)Rigorous bnds ondS [Barlow & Bass ’99]

2.21 ∼ 2.60 2.00 ∼ 2.26 1.89 ∼ 2.07 1.82 ∼ 1.95

Numerical dS [C.] 2.51... - 2.01... -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?)


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 & EuropeanScience Foundation


joe p. chenteplyaev/cornell4/...outline 1 the set-up for quantum statistical mechanics 2 what we...

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