E ective free dynamics of a tracer-particle coupled to a Fermi gas … · 2016. 1. 27. · E ective free dynamics of a tracer-particle coupled to a Fermi gas in the high-density limit

Effective free dynamics of a tracer-particlecoupled to a Fermi gas in the high-density limit

David Mitrouskas

Department Mathematik der LMU München

joint work with Maximilian Jeblick, Sören Petrat and Peter Pickl

Basque Center for Applied Mathematics, September 29

motivation

Phys. Rev. Letters 32, 23 (1974)

1 E. Fermi and E. Teller, Phys. Rev. 72, 399 (1947)

2 M. Gryzinski, Phys. Rev. 111, 900 (1958)

table of contents

1 Microscopic Model

2 Main result and effective model

3 Sketch of the Proof

4 Explanation as a mean field type result

5 Conclusion

table of contents

1 Microscopic Model

5 Conclusion

microscopic model

• N+1 particle wave function on a d-dim. torus of length Λ

Ψ ∈ L2(Td(N+1), ddx1...ddxN ddy)

• non-relativistic Schrödinger equation

i∂tΨt = HNΨt ⇒ Ψt = e−iHN tΨ0

• microscopic Hamiltonian

HN = −∆y −N∑k=1

∆xk + g

N∑k=1

v(xk − y)

- v ∈ C∞0 (Td)- g = 1: not in the weak-coupling regime

microscopic model

HN = −∆y −N∑k=1

∆xk + g

N∑k=1

v(xk − y)

microscopic model

HN = −∆y −N∑k=1

∆xk + g

N∑k=1

v(xk − y)

microscopic model

HN = −∆y −N∑k=1

∆xk + g

N∑k=1

v(xk − y)

microscopic model

• initial condition Ψ0 = φ0 ⊗ Ω0

- φ0 ∈ H2(Td, ddy),∣∣∣∣φ0∣∣∣∣L2(Td) = 1

- ground state of the ideal Fermi gas

Ω0(x1, ..., xN ) =1√N !

∑σ∈SN

(−1)σN∏i=1

ϕσ(i)(xi)

ϕk(x) = Λ− d

2 exp(ipkx), pk = 2πΛ−1zk with N smallest zk ∈ Zd

• Fermi momentum and average density : kF = Cdρ1d

table of contents

1 Microscopic Model

5 Conclusion

main result and effective model

• main result (similar for d = 1)

Let d = 2 and � > 0. Denote average dens. by ρ = N/Λ2.Then there exists a positive constant C� s.t. for all t > 0

limN,Λ→∞ρ=const.

∣∣∣∣e−iHN tΨ0 − e−iHmf tΨ0∣∣∣∣L2(Td(N+1)) ≤ C�t 32 ρ− 18 +�• effective Hamiltonian

Hmf = −∆y −N∑k=1

∆xk + ρv̂(0)

- V → E|Ω0|2 [V ](y) = 〈Ω0, V Ω0〉L2(TdN )(y) = ρv̂(0)

- physically equivalent to free dynamics

Hmf = −∆y −N∑k=1

∆xk + ρv̂(0)

Hmf = −∆y −N∑k=1

∆xk + ρv̂(0)

Hmf = −∆y −N∑k=1

∆xk + ρv̂(0)

table of contents

1 Microscopic Model

5 Conclusion

sketch of the proof

• general strategy

- appropriate expansion of e−iHN tΨ0 around e−iHmf tΨ0

- control of fluctuations by means of oscillatory integrals

- proof relies highly on the anti-symmtry of Ω0

- explicit propagation estimates: rate of convergence

sketch of the proof

• repetitive application of Duhamel’s formula

e−iHN tΨ0 =e−iHmf tΨ0 − i

∫ t0

e−iHN (t−s)(V − ρv̂(0))e−iHmf sΨ0 ds

• ∣∣∣∣e−iHN tΨ0 − e−iHmf tΨ0∣∣∣∣ ≤√∣∣∣∣ΨA(t)∣∣∣∣+ ∣∣∣∣ΨB(t)∣∣∣∣- first order contribution

ΨA(t) =

∫ t0

eiHmf s(V − ρv̂(0))e−iHmf sΨ0 ds

- higher order contributions ΨB(t)

sketch of the proof

• repetitive application of Duhamel’s formula

e−iHN tΨ0 =e−iHmf tΨ0 − i

∫ t0

e−iHN (t−s)(V − ρv̂(0))e−iHmf sΨ0 ds

• ∣∣∣∣e−iHN tΨ0 − e−iHmf tΨ0∣∣∣∣ ≤√∣∣∣∣ΨA(t)∣∣∣∣+ ∣∣∣∣ΨB(t)∣∣∣∣- first order contribution

ΨA(t) =

∫ t0

eiHmf s(V − ρv̂(0))e−iHmf sΨ0 ds

- higher order contributions ΨB(t)

sketch of the proof

• controlling the norm of ΨA- too rough:

∣∣∣∣ΨA(t)∣∣∣∣2 ≤ t2Var[V ] = Ct2ρ 12- better:

∣∣∣∣ΨA(t)∣∣∣∣2 = N∑k=1

∞∑l=N+1

|v̂(pk − pl)|2

Λ2︸︷︷︸=Var[V ]

∣∣∣∣ ∫ t0

ei(Ek−El)shkl(s)φ0︸︷︷︸oscillating integrand

ds∣∣∣∣2

with hkl(s) = e−i∆sei(pk−pl)yei∆s

• fast decay of v̂ (Paley-Wiener ⇒ no large mom. transfer)• oscillation of ei(Ek−El)s for |pk| ≈ kF

(Ek − El) = p2k − p2l = (|pk|+ |pl|)(|pk| − |pl|) ≥ kF · �(kF )

sketch of the proof

∣∣∣∣ΨA(t)∣∣∣∣2 = N∑k=1

∞∑l=N+1

|v̂(pk − pl)|2

Λ2︸︷︷︸=Var[V ]

∣∣∣∣ ∫ t0

ds∣∣∣∣2

sketch of the proof

∣∣∣∣ΨA(t)∣∣∣∣2 = N∑k=1

∞∑l=N+1

|v̂(pk − pl)|2

Λ2︸︷︷︸=Var[V ]

∣∣∣∣ ∫ t0

ds∣∣∣∣2

sketch of the proof

• denote v̂a(pk − pl) = θ(ρa − 2πΛ−1|pk − pl|)v̂(pk − pl)

∣∣∣∣ΨA∣∣∣∣2 ≈ Ct2 [ N∑k=1

∞∑l=N+1

+

{stationary points}

1

k2F · �(kF )2N∑k=1

∞∑l=N+1

]|v̂a(pk − pl)|2

Λ2

• sufficient control

∣∣∣∣ΨA(t)∣∣∣∣2 ≤ t2(Cρ− 14 + 3a2 + Caρ− 1a)

sketch of the proof

• denote v̂a(pk − pl) = θ(ρa − 2πΛ−1|pk − pl|)v̂(pk − pl),

∣∣∣∣ΨA∣∣∣∣2 ≈ Ct2 [ N∑k=1

∞∑l=N+1

+

{stationary points}

1

k2F · �(kF )2N∑k=1

∞∑l=N+1

]|v̂a(pk − pl)|2

Λ2

∣∣∣∣ΨA(t)∣∣∣∣2 ≤ t2(Cρ− 14 + 3a2 + Caρ− 1a)

sketch of the proof

• denote v̂a(pk − pl) = θ(ρa − 2πΛ−1|pk − pl|)v̂(pk − pl),

∣∣∣∣ΨA∣∣∣∣2 ≈ Ct2 [ N∑k=1

∞∑l=N+1

+

{stationary points}

1

k2F · �(kF )2N∑k=1

∞∑l=N+1

]|v̂a(pk − pl)|2

Λ2

∣∣∣∣ΨA(t)∣∣∣∣2 ≤ t2(Cρ− 14 + 3a2 + Caρ− 1a)

sketch of the proof

ΨB =

∫ t0

∫ s10

eiHNs2(V -ρv̂(0))eiHmf (s1-s2)(V -ρv̂(0))e-iHmf s1Ψ0 ds2 ds1

• controlling the norm of ΨB- full time-evolution e−iHN t (need to reexpand)

- immediate recollisions require energy renormalization

Hmf = −∆y −N∑k=1

∆xk + ρv̂(0)− Ea(ρ)

with Ea(ρ)/ρ→ 0 for ρ→∞

• sufficient control at third order

∣∣∣∣ΨB(t)∣∣∣∣2 ≤ t3(Cρ− 14 + 7a4 + Caρ− 1a)

sketch of the proof

ΨB =

∫ t0

∫ s10

eiHNs2(V -ρv̂(0))eiHmf (s1-s2)(V -ρv̂(0))e-iHmf s1Ψ0 ds2 ds1

• controlling the norm of ΨB- full time-evolution e−iHN t (need to reexpand)

- immediate recollisions require energy renormalization

Hmf = −∆y −N∑k=1

∆xk + ρv̂(0)− Ea(ρ)

with Ea(ρ)/ρ→ 0 for ρ→∞

• sufficient control at third order

∣∣∣∣ΨB(t)∣∣∣∣2 ≤ t3(Cρ− 14 + 7a4 + Caρ− 1a)

table of contents

1 Microscopic Model

5 Conclusion

explanation as a mean field type result

• fluctuations of V around ρv̂(0) in d dimensions

Var[V ](y) = 〈Ω0, (V − ρv̂(0))2Ω0〉L2(TdN )(y) = Cdρd−1d

- P|Ω0|2(∣∣∑N

k=1 v(xk − y)− ρv̂(0)∣∣) ≈ 1 does not hold

- random forces in the gas are large (strong coupling g = 1)

• Comparison with bosons (symmetric initial Ω0)- fluctuations: VarB [V ] = Cdρ (due to

√N law)

- lifetime of φ0 decreases for increasing density ρ1

1 Emission of Cerenkov radiation as a mechanism for Hamiltonianfriction, J. Fröhlich and Z. Gang, Advances in Mathematics 264(2014) 183–235

• fluctuations of V around ρv̂(0) in d dimensions

Var[V ](y) = 〈Ω0, (V − ρv̂(0))2Ω0〉L2(TdN )(y) = Cdρd−1d

- P|Ω0|2(∣∣∑N

k=1 v(xk − y)− ρv̂(0)∣∣) ≈ 1 does not hold

- random forces in the gas are large (strong coupling g = 1)

• Comparison with bosons (symmetric initial Ω0)- fluctuations: VarB [V ] = Cdρ (due to

√N law)

- lifetime of φ0 decreases for increasing density ρ1

1 Emission of Cerenkov radiation as a mechanism for Hamiltonianfriction, J. Fröhlich and Z. Gang, Advances in Mathematics 264(2014) 183–235

• first glance: mean field theory seems not to apply

• closer look at fluctuations:

- (V − ρv̂(0))Ψ0 = Λ−1∑Nk=1 V (pk)Ψ0 with

V (pk)Ψ0 =

∞∑l=N+1

v̂(pk − pl)Λ

ei(pk−pl)ya∗(pk)a(pl)Ψ0

- {V (pk)}Nk=1 family of uncorrelated random variables

Var[V ] = Λ−2N∑k=1

Var[V (pk)]

table of contents

1 Microscopic Model

5 Conclusion

conclusion

• long lived resonance of φ0 emerges under e−iHN t (ρ→∞)- previously known in the physics literature (ion stopping)

• open question for 3d model- applied method does not yield sufficient estimates

• fermions: mean field regime � weak-coupling regime- compared to bosons: mean field regime = weak-coupling regime

• further research- thermal initial states, ground states in periodic external potential,

long time behaviour of φ0, similar result in the fully interacting case

conclusion

Thank you for your attention

Microscopic ModelMain result and effective modelSketch of the ProofExplanation as a mean field type resultConclusion

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E ective free dynamics of a tracer-particle coupled to a Fermi gas … · 2016. 1. 27. · E ective free dynamics of a tracer-particle coupled to a Fermi gas in the high-density limit