e ective free dynamics of a tracer-particle coupled to a fermi gas … · 2016. 1. 27. · e ective...
TRANSCRIPT
-
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
2 Main result and effective model
3 Sketch of the Proof
4 Explanation as a mean field type result
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
• 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
• 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
• 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
• 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
2 Main result and effective model
3 Sketch of the Proof
4 Explanation as a mean field type result
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
-
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
-
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
-
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
-
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
-
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
• 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
• 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
• 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
limN,Λ→∞ρ=const.
∣∣∣∣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
• sufficient control
limN,Λ→∞ρ=const.
∣∣∣∣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
• sufficient control
limN,Λ→∞ρ=const.
∣∣∣∣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
limN,Λ→∞ρ=const.
∣∣∣∣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
limN,Λ→∞ρ=const.
∣∣∣∣ΨB(t)∣∣∣∣2 ≤ t3(Cρ− 14 + 7a4 + Caρ− 1a)
-
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
-
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
-
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
-
explanation as a mean field type result
• 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)]
-
explanation as a mean field type result
-
explanation as a mean field type result
-
explanation as a mean field type result
-
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
-
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
• 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
• 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
• 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
-
Thank you for your attention
Microscopic ModelMain result and effective modelSketch of the ProofExplanation as a mean field type resultConclusion