Mixing and Waves: Part I The Ehrenfest time · Mixing and Waves: Part I The Ehrenfest time Roman Schubert Montreal April 7, 2008. OutlineIntroductionMixing and universalityThe quantum

Outline Introduction Mixing and universality The quantum classical correspondence The Ehrenfest time Related problems Summary

Mixing and Waves: Part IThe Ehrenfest time

Roman Schubert

Montreal April 7, 2008


Introduction

Mixing and universality

The quantum classical correspondence

The Ehrenfest time

Related problems

Summary


Spectrum and dynamics

wave propagation geodesic flow

mixinginvariant states

eigenfunctionseigenvalues

time to

infinity

.

.

semiclassical limit

invariant measures

• semiclassical limit and t →∞ don’t commute

• need semiclassical techniques which are uniform in t


regular and irregular motion

regular motion:

irregular motion:

• neighbouring trajectories diverge typically

• a typical trajectory fills the space with uniform density


mixing and universality

A dynamical system (X , φt , dµ) is mixing if for a, ρ ∈ L2(X )(∫X ρdµ = 1)

limt→∞

∫X

a φtρ dµ =

∫X

a dµ

Interpretation:

• a observable, ρ a state (i.e., a probability density), then∫a φtρdµ expected value of a at time t: the system forgets

where it came from - ”Universality”

• Other manifestations of universality: If mixing is rapid enougha Central Limit Theorem (CLT) holds

1√T

∫ T

0a Φtdt becomes normally distributed

for T →∞ (if∫X a dµ = 0).


Anosov flows

Definitionφt is called Anosov if for all x ∈ X there is a splitting

TxX = E s(x)⊕ Eu(x)⊕ E 0(x)

such that E 0(x) is spanned by the flow direction and there areconstants λ > 0,C such that

• ‖dφtu‖ ≤ Ce−λt‖u‖ for all u ∈ E s(x) and t > 0

• ‖dφtu‖ ≤ Ceλt‖u‖ for all u ∈ Eu(x) and t < 0

Example: geodesic flow on compact manifold M.

• X = S∗M, dµ Liouville measure on S∗M

• φt Hamiltonian flow generated by H(x , ξ) = 12 |ξ|

2g(x)

Geodesic flow is Anosov if all sectional curvatures are negative.


exponential mixing

Theorem (Dolgopyat 98, Liverani 05)

Assume X is contact and φt : X → X is Anosov and volumepreserving, then there exists a γ > 0 such that for alla, ρ ∈ C 1(X ),

∫ρdµ = 1,∫

a φtρ dµ =

∫a dµ+ O(‖a‖C1‖ρ‖C1e−γ|t|)

Localise initial conditions: ρε(x) := 1ε2d−1 ρ0

(d(x ,x0)ε

), then

‖ρε‖C1 ∼ 1/ε2d , so

limε→0

∫a φtρε dµ =

a(φt(x0)

)for t 1

λ ln 1ε∫

a dµ for t 1γ ln 1

ε

transition from local to global behaviour. Later ε ∼√

~,uncertainty relation


The quantum classical correspondence: Observables

Main idea: classify operators by their action on rapidly oscillatingfunctions, e.g., plane waves:

• for an operator A we define its symbol a by

Aei~ 〈x ,ξ〉 = a(~, x , ξ)e

i~ 〈x ,ξ〉

• Symbol classes: a ∈ Sm,kρ if

|∂αx ∂βξ a| ≤ Cα,β~−k~−ρ(|α|+|β|)〈ξ〉m−|β| , where 〈ξ〉 = (1+|ξ|2)1/2

• Operator classes: A ∈ Ψk,mρ if a ∈ Sk,m

ρ

• ~→ 0 semiclassical limit, ρ governs ”classicality” of symboland A

• Example: H = −~2∆ ∈ Ψ0,20 , H = |ξ|2g(x) + O(~).


using Fourier transformation one gets

Aψ(x) =1

(2π~)d

∫∫a(~, x , ξ)e

i~ 〈x−y ,ξ〉ψ(y) dydξ

symbol determines the operator and we will write A = Op[a].

• product formula: Op[a] Op[b] = Op[a]b] where

a]b = e−i~〈∂ξ,∂y 〉a(x , ξ)b(y , η)|x=y ,ξ=η ∼∑α

(−i~)|α|

α!∂αξ a∂αx b

asymptotic expansion in ~ if ρ < 1/2: cannot localize onscales smaller that

√~, Heisenberg ∆q∆p ≥ ~/2.

• a]b − b]a ∼ i~a, b+ · · · Commutator → Poissonbracket

• ‖Op[a]‖L2 ≤ C∑|α|≤2d+1 ~|α|/2|∂αa|.


Operators and symbols on manifolds

• use covering by local charts to define Op[a], different choiceof local charts defines different Op′[a]. In generalOp[a] = Op′[a] + O(~) for a ∈ S0,0

0 .

• use global phase function: φ : M × T ∗M → R, homogeneousof degree one in ξ, non-degenerate anddξφ(x , y , ξ) = O(x − y), and define

Op[a]ψ(x) =1

(2π~)d

∫∫a(~, x , ξ)e

i~φ(x ,y ,ξ)ψ(y) dydξ

such functions can be constructed, e.g., using exponentialmap. Can define symbols on T ∗M using covariant derivatives.(Widom 80, Safarov 97)

• Results on calculus of operators carry over from Rn. (Onnon-compact manifolds operators have to be properlysupported)


Quantum time evolution

• unitary operator U(t) = e−i~ tH , solution to

i~∂tU(t) = HU(t) , U(t = 0) = I

• time evolution of states ψ → U(t)ψ

• expectation values of observables A

〈U(t)ψ,AU(t)ψ〉 = 〈ψ,U∗(t)AU(t)ψ〉

• time evolution of observables A→ A(t) = U∗(t)AU(t)solution to

i~dA(t)

dt= [A(t), H] , A(t = 0) = A .


The correspondence principle: Egorov’s theoremAnsatz A(t) = Op[a(t)], a ∈ S0,0

ρ . commutator [A(t), H] hassymbol

a(t)]H − H]a(t) = i~a(t),H+ O(~2−2ρ)

so in leading order ∂ta(t) = a(t),H which is solved bya(t) = a φt .

Theorem (Bambusi, Graffi and Paul (99); Bouzouina andRobert (02))

There exists a k > 1 such that for all a ∈ S0,00 (T ∗M)

‖U(t)∗Op[a]U(t)− Op[a φt ]‖L2 = O(~‖a φt‖C k ) .

Remarks:

• Correspondence principle: for ~→ 0 we find quantum →classical.

• if φt is Anosov, then ‖a φ−t‖C k = O(ekλ|t|)


The correspondence principle: Egorov’s theoremAnsatz A(t) = Op[a(t)], a ∈ S0,0

ρ . commutator [A(t), H] hassymbol

a(t)]H − H]a(t) = i~a(t),H+ O(~2−2ρ)

so in leading order ∂ta(t) = a(t),H which is solved bya(t) = a φt .

Theorem (Bambusi, Graffi and Paul (99); Bouzouina andRobert (02))

There exists a k > 1 such that for all a ∈ S0,00 (T ∗M)

‖U(t)∗Op[a]U(t)− Op[a φt ]‖L2 = O(~‖a φt‖C k ) .

Remarks:

• Correspondence principle: for ~→ 0 we find quantum →classical.

• if φt is Anosov, then ‖a φ−t‖C k = O(ekλ|t|)


Ehrenfest time ISince

|∂a φt | |∂a|eλt

we have for a ∈ S0,00 that

a φt ∈ S0,0ρ for eλt ~−ρ

Ehrenfest time

T ∼ ρ

λln

1

~classical and quantum fluctuations are of the same order

pseudo-differential operator calculus breaks down1978-1979 Berman Zaslavsky, Balasy Berry Tabor Voros: log breaking time,

Ehrenfest time, limit of validity of semiclassics?

But for larger times?

• oscillations on smaler scale than Heisenberg → averaging →mixing → universality?

• Averaged Egorov, e.g., Tr B(U(t)∗Op[a]U(t)−Op[a φt ]) =?


Ehrenfest time ISince

|∂a φt | |∂a|eλt

we have for a ∈ S0,00 that

a φt ∈ S0,0ρ for eλt ~−ρ

Ehrenfest time

T ∼ ρ

λln

1

~classical and quantum fluctuations are of the same order

pseudo-differential operator calculus breaks down1978-1979 Berman Zaslavsky, Balasy Berry Tabor Voros: log breaking time,

Ehrenfest time, limit of validity of semiclassics?

But for larger times?

• oscillations on smaler scale than Heisenberg → averaging →mixing → universality?

• Averaged Egorov, e.g., Tr B(U(t)∗Op[a]U(t)−Op[a φt ]) =?


Van-Vleck formula

Let U(t)ψ(x) =∫

K (t, x , y)ψ(y) dythen

K (t, x , y) =∑γx,y (t)

[Aγ(t)+Oγ,t(~)

]e

i~ Sγ(t) y

x

where the sum is over all geodesics γx ,y from x to y in time t (insome energy window), and Sγ is the action along γ.

• If φt is Anosov then Aγ(t) ∼ te−λγt/2, Oγ,t(~) ~e−λγt/2

(1/√

of unstable Jacobian), so the remainder is, roughly,bounded by ∑

γx,y (t)

~te−λγt/2 ∼ ~ePt

with pressure P = P(−H/2) (H-SRB potential, unstableJacobian).


Van-Vleck formula

Let U(t)ψ(x) =∫

K (t, x , y)ψ(y) dythen

K (t, x , y) =∑γx,y (t)

[Aγ(t)+Oγ,t(~)

]e

i~ Sγ(t) y

x

where the sum is over all geodesics γx ,y from x to y in time t (insome energy window), and Sγ is the action along γ.

• If φt is Anosov then Aγ(t) ∼ te−λγt/2, Oγ,t(~) ~e−λγt/2

(1/√

of unstable Jacobian), so the remainder is, roughly,bounded by ∑

γx,y (t)

~te−λγt/2 ∼ ~ePt

with pressure P = P(−H/2) (H-SRB potential, unstableJacobian).


Ehrenfest time II

• remainder term small for

t T ∼ 1

Pln

1

~

• What for larger times?

• note that ∑γx,y (t)

|Aγ |2 <∞

• so CLT for long orbits might lead to CLT for time evolution

K (t, x , y) ∼∑γx,y (t)

Aγ(t)ei~ Sγ(t)

1979 Balasz Berry - random wave conjecture: equidistribution and universal

fluctuations.


Tomsovic Heller 1991Numerical experiments: Semiclassics remains accurate fort 1/

√~. waves become equidistributed, fluctuations satisfy CLT


Remarks: Ehrenfest time

• Can control accuracy of semiclassical approximations up toEhrenfest time scales

t TE ∼ ln1

~

two sources of breakdown:• positive Liapunov exponents, inducing rapid oscillations on

scales shorter than√

~, standart semicassical techniques breakdown

• exponential proliferation of orbits, one would have toincorporate the cancellation from rapidly oscillatingphase-factors to go beyond

• Beyond Ehrenfest time: expect universality from exponentialmixing and CLT

• effective averaging from uncertainty principle• generic long orbits become dense and behave universal.


Remarks: Ehrenfest time

• Can control accuracy of semiclassical approximations up toEhrenfest time scales

t TE ∼ ln1

~

two sources of breakdown:• positive Liapunov exponents, inducing rapid oscillations on

scales shorter than√

~, standart semicassical techniques breakdown

• exponential proliferation of orbits, one would have toincorporate the cancellation from rapidly oscillatingphase-factors to go beyond

• Beyond Ehrenfest time: expect universality from exponentialmixing and CLT

• effective averaging from uncertainty principle• generic long orbits become dense and behave universal.


Related problems: equidistribution

Theorem (RS 05)

Let M is compact and of neg. curvature, take ψ = f ei~ϕ with

supp f small, and assume

Λ := (x ,dϕ(x)) , x ∈ supp f ⊂ S∗M ,

and Λ is transversal to the stable foliation (i.e. for all x ∈ Λ,TxΛ ∩ E s(x) = 0). Then there exists constants Γ, γ > 0 suchthat for all a ∈ S0,0

0

〈U(t)ψ,Op[a]U(t)ψ〉 = ‖ψ‖L2

∫S∗M

a dµ+ O(~eΓ|t|) + O(e−γ|t|)

Problems:

• larger times

• coherent states


Localized states

TheoremLet δ > 0, a0 ∈ C∞(M) and set

a(~, x) =1

~dδ2

a0

(x − x0

~δ)

ψ0(x) = a(~, x)ei~ϕ(x) .

Assume φt is Anosov, Λϕ ⊂ S∗M, and Λϕ is transversal to thestable foliation. Then there exists constants Γ, λ ≥ γ > 0 suchthat for all f ∈ C∞0 (T ∗M) and if ‖ψ‖L2 = 1

lim~→0〈U(t)ψ,Op[f ]U(t)ψ〉 =

f(φ−t(x0,dϕ(x0))

)if t << δ

λ ln 1~∫

S∗M f dµ if δγ ln 1

~ << t << 1−δΓ ln 1

~


Related problems: rate of quantum ergodicity

−∆ψn = λ2nψn

• N(λ) := |λn ≤ λ|• on manifolds of negative curvature (Zelditch 94)

1

N(λ)

∑λn≤λ

∣∣∣∣〈ψn,Op[a]ψn〉 −∫

S∗Madµ

∣∣∣∣2 1/ lnλ

main tool: avaraging Op[a]→ 1T

∫ T0 U

∗(t) Op[a]U(t) dt, RHSis 1/T ,

• Conjecture: RHS= O(λ−1) (on surfaces)


Related problems: Weyls law

• for surfaces

N(λ) = cλ2 + R(λ)

• upper bound in negative curvature: Berard (77)R(λ) = O(λ/ lnλ)

• lower bound: Jacobson, Polterovich, Toth (07)R(λ) = Ω((log λ)P(−H/2)/h−ε)

• Conjecture: Randol (81): R(λ) = O(λ1/2)

• Estimates on L∞ norms:

• Berard (77): ‖ψn‖L∞ = O(λ/ lnλ)• Conjecture (Iwaniec Sarnak 95): ‖ψn‖L∞ = Oε(λ

ε) for allε > 0.


Summary: time scales

• Ehrenfest time

TE ∼1

λln

1

~

-exponential proliferation oforbits,-small-scale oscillations

• Heisenberg time, time scaleto resolve spectrum:

TH ∼1

~d−1

h

t

T

T

H

known region

E

1

terra incognita

Mixing and Waves: Part I The Ehrenfest time · Mixing and Waves: Part I The Ehrenfest time Roman Schubert Montreal April 7, 2008. OutlineIntroductionMixing and universalityThe quantum

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