Derivation of the time dependent Hartree (Fock) equation...2.Derivation of Hartree (Fock) equations...

Derivation of the time dependent Hartree (Fock)equation

Peter Pickl

Mathematical Institute

LMU

13. April 2016

Peter Pickl Mathematical Institute LMU

Derivation of the time dependent Hartree (Fock) equation

Dictionary

1. �Derivation� means prove of validity

2. �Hartree (Fock)�: time dependent Hartree (Fock)

Dictionary

Overview

1. Derivation of Hartree equations for Bosons

2. Derivation of Hartree (Fock) equations for Fermions

3. Special case: A tracer particle in the fermi sea

Overview

Mean �eld for the bosons: The Hartree equation

H =N∑j=1

−∆j +N∑j=1

At(xj) + (N − 1)−1∑k

H =N∑j=1

−∆j +N∑j=1

At(xj) + (N − 1)−1∑k

H =N∑j=1

−∆j +N∑j=1

At(xj) + (N − 1)−1∑k

H =N∑j=1

−∆j +N∑j=1

At(xj) + (N − 1)−1∑k

H =N∑j=1

−∆j +N∑j=1

At(xj) + (N − 1)−1∑k

H =N∑j=1

−∆j +N∑j=1

At(xj) + (N − 1)−1∑k

I Assuming Ψt ≈∏N

j=1 φt(xj) and �nding φt : easy.

I Proving Ψt ≈∏N

j=1 φt(xj): hard.In particular error propagation

I Example: Gross-Pitaevskii for dilute gases: NOT a mean �eldsituation!

Mean �eld for �particle 1�

V

x1x x x x x x x x x x23 45 67 89 1011

W (x1) = (N − 1)−1∑N

j=2 V (x1 − xj) for �xed, |φ0|2- distributedx2, . . . , xN .Law of large numbers: |φ0|2 close to the empirical density ρ0.W (x1) ≈ V ? |φ0|2(x1) (�Mean �eld�).

E�ective Dynamics: Hartree equation

idtφt =(−∆ + At + V ? |φt |2

)φt .

V

x1x x x x x x x x x x23 45 67 89 1011

W (x1) = (N − 1)−1∑N

idtφt =(−∆ + At + V ? |φt |2

)φt .

V

x1x x x x x x x x x x23 45 67 89 1011

W (x1) = (N − 1)−1∑N

idtφt =(−∆ + At + V ? |φt |2

)φt .

V

x1x x x x x x x x x x23 45 67 89 1011

W (x1) = (N − 1)−1∑N

idtφt =(−∆ + At + V ? |φt |2

)φt .

Grönwall argument

V

x1x x x x x x x x x x23 45 67 89 1011 x12,13,14

Let αt be a measure for the dirt in the condensate:

dtαt ≤ C (αt + O(1))

Grönwall: αt stays small if α0 was small (αt ≤ eCtα0 + O(1))

Grönwall argument

V

x1x x x x x x x x x x23 45 67 89 1011 x12,13,14

Grönwall argument

V

x1x x x x x x x x x x23 45 67 89 1011 x12,13,14

Usual approach

Control reduced one particle density matrix µΨt1

dtµΨt1

depends on µΨt2

dtµΨt2

depends on µΨt3

...Technically di�cult (e.g. Erdös, Schlein, Yau (2009)

Usual approach

dtµΨt1

depends on µΨt2

dtµΨt2

depends on µΨt3

Usual approach

dtµΨt1

depends on µΨt2

dtµΨt2

depends on µΨt3

Usual approach

dtµΨt1

depends on µΨt2

dtµΨt2

depends on µΨt3

Usual approach

dtµΨt1

depends on µΨt2

dtµΨt2

depends on µΨt3

Usual approach

dtµΨt1

depends on µΨt2

dtµΨt2

depends on µΨt3

Introduction of a counting measure:

For every j = 1, . . . ,N and φ ∈ L2(R3) let pφj be the projector given by

pφj = |φ〉〈φ|j .

Let qφj = 1− pφj .

According to Ψ �expected relative number� of particles not in the state φ

α(Ψ, φ) = N−1N∑j=1

〈Ψ, qφj Ψ〉 = 〈Ψ, qφk Ψ〉 = ‖q

φk Ψ‖

2 .

α(Ψ, φ) = 0 ⇔ Ψ =∏N

j=1 φ(xj).

α(Ψ, φ) = N−1N∑j=1

〈Ψ, qφj Ψ〉 = 〈Ψ, qφk Ψ〉 = ‖q

φk Ψ‖

2 .

α(Ψ, φ) = 0 ⇔ Ψ =∏N

j=1 φ(xj).

α(Ψ, φ) = N−1N∑j=1

〈Ψ, qφj Ψ〉 = 〈Ψ, qφk Ψ〉 = ‖q

φk Ψ‖

2 .

α(Ψ, φ) = 0 ⇔ Ψ =∏N

j=1 φ(xj).

α(Ψ, φ) = N−1N∑j=1

〈Ψ, qφj Ψ〉 = 〈Ψ, qφk Ψ〉 = ‖q

φk Ψ‖

2 .

α(Ψ, φ) = 0 ⇔ Ψ =∏N

j=1 φ(xj).

α(Ψ, φ) = N−1N∑j=1

〈Ψ, qφj Ψ〉 = 〈Ψ, qφk Ψ〉 = ‖q

φk Ψ‖

2 .

α(Ψ, φ) = 0 ⇔ Ψ =∏N

j=1 φ(xj).

Example:

Ψ(x1, . . . , xN) =(χ(x1, . . . xk)

∏Nj=k+1 φ(xj)

)sym

with χ ∈ L2(R3k), pjχ = 0 for all 1 ≤ j ≤ k

⇒ α(Ψ, φ) = k/N.

Lemma: α(Ψt , φt)→ 0 is equivalent to convergence of µt to |φt〉〈φt | inoperator norm.

Example:

Ψ(x1, . . . , xN) =(χ(x1, . . . xk)

∏Nj=k+1 φ(xj)

)sym

⇒ α(Ψ, φ) = k/N.

Example:

Ψ(x1, . . . , xN) =(χ(x1, . . . xk)

∏Nj=k+1 φ(xj)

)sym

⇒ α(Ψ, φ) = k/N.

Deriving the mean �eld equation:

Goal: show that |dtαt | < C (αt + o(1))⇒ αt � 1 by GrönwallLet

hj := −∆j + A + V ? |φt |2(xj) .

Sincedtq

φtj = −i [hj , q

φtj ]

dtα(Ψt , φt) = i〈Ψt , [H − h1, qφt1 ]Ψt〉 .

hj := −∆j + A + V ? |φt |2(xj) .

Sincedtq

φtj = −i [hj , q

φtj ]

hj := −∆j + A + V ? |φt |2(xj) .

Sincedtq

φtj = −i [hj , q

φtj ]

hj := −∆j + A + V ? |φt |2(xj) .

Sincedtq

φtj = −i [hj , q

φtj ]

hj := −∆j + A + V ? |φt |2(xj) .

Sincedtq

φtj = −i [hj , q

φtj ]

dtα(Ψt , φt) = i〈Ψt , [−∆1 + (N − 1)−1N∑j=2

V (x1 − xj)− h1, qφt1 ]Ψt〉

= i〈Ψt , [V (x1 − x2)− V ? |φt |2(x1), qφt1 ]Ψt〉

= i(〈Ψt ,

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

Ψt〉 − c.c.)

= i(〈Ψt , (pφt1 + q

φt1

)(pφt2

+ qφt2

)(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

(pφt2

+ qφt2

)Ψt〉− c.c.

).

All terms 〈Ψt ,A12Ψt〉 where A12 is either selfadjoint or invariant underadjunction plus simultaneous exchange of the variables x1 and x2 cancelout.

dtα(Ψt , φt) = i〈Ψt , [−∆1 + (N − 1)−1N∑j=2

V (x1 − xj)− h1, qφt1 ]Ψt〉

= i(〈Ψt ,

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

Ψt〉 − c.c.)

= i(〈Ψt , (pφt1 + q

φt1

)(pφt2

+ qφt2

)(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

(pφt2

+ qφt2

)Ψt〉− c.c.

).

dtα(Ψt , φt) = i〈Ψt , [−∆1 + (N − 1)−1N∑j=2

V (x1 − xj)− h1, qφt1 ]Ψt〉

= i(〈Ψt ,

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

Ψt〉 − c.c.)

= i(〈Ψt , (pφt1 + q

φt1

)(pφt2

+ qφt2

)(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

(pφt2

+ qφt2

)Ψt〉− c.c.

).

dtα(Ψt , φt) = i〈Ψt , [−∆1 + (N − 1)−1N∑j=2

V (x1 − xj)− h1, qφt1 ]Ψt〉

= i(〈Ψt ,

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

Ψt〉 − c.c.)

= i(〈Ψt , (pφt1 + q

φt1

)(pφt2

+ qφt2

)(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

(pφt2

+ qφt2

)Ψt〉− c.c.

).

dtα(Ψt , φt) = i〈Ψt , [−∆1 + (N − 1)−1N∑j=2

V (x1 − xj)− h1, qφt1 ]Ψt〉

= i(〈Ψt ,

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

Ψt〉 − c.c.)

= i(〈Ψt , (pφt1 + q

φt1

)(pφt2

+ qφt2

)(V (x1 − x2)− V ? |φt |2(x1)

)qφt1

(pφt2

+ qφt2

)Ψt〉− c.c.

).

Only three types remain: pp . . . pq, pp . . . qq and pq . . . qq

I = i〈Ψt , pφt1 pφt2

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

= i〈Ψt , pφt1 pφt2V (x1 − x2)pφt2 q

φt1

Ψt〉 − i〈Ψt , pφt1 pφt2V ? |φt |2(x1)pφt2 q

φt1

Ψt〉

II = i〈Ψt , pφt1 pφt2V (x1 − x2)qφt1 q

φt2

Ψt〉

III = i〈Ψt , pφt1 qφt2

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

I : pφt2V (x1 − x2)pφt2 =

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

I : pφt2V (x1 − x2)pφt2 =|φt(x2)〉〈φt(x2)|V (x1 − x2)|φt(x2)〉〈φt(x2)|

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

=|φt(x2)〉V ? |φt |2(x1)〈φt(x2)| = pφt2 V ? |φt |2

=pφt2V ? |φt |2pφt2

⇒ I = 0.

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

⇒ I = 0.

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

⇒ I = 0.

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

⇒ I = 0.

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1pφt2

Ψt〉

φt1

Ψt〉

φt2

Ψt〉

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

⇒ I = 0.

I = 0

φt2

Ψt〉

III = i〈Ψt , qφt2 pφt1

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

|III | ≤ ‖qφt2

Ψt‖22‖V ‖∞ = Cα(Ψt , φt)

dtα(Ψt , φt) ≤ C (α(Ψt , φt) + N−1)

I = 0

φt2

Ψt〉

III = i〈Ψt , qφt2 pφt1

(V (x1 − x2)− V ? |φt |2(x1)

)qφt1qφt2

Ψt〉 .

|III | ≤ ‖qφt2

Ψt‖22‖V ‖∞ = Cα(Ψt , φt)

dtα(Ψt , φt) ≤ C (α(Ψt , φt) + N−1)

Fermions

I Microscopic system: Ψ0 = ΛNj=1φ

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Fermions

j0

H =∑N

j=1−∆j +∑N

j=1 At(xj) + N−2/3∑

k

Results for Fermions

I Semiclassical situation: N. Benedikter, M. Porta, B. Schlein (2014)

I Quantum situation: S. Petrat, P.P. (2015); V. Bach, S. Breteaux, S.Petrat, P. P., T. Tzaneteas (2015)

I Problem: potential of leading order, force small!

I Goal: Consider coupling N−1/3.

Special situation

I Tracer in �lled Fermi sea: Ψ0 = χ(y)∧N

j=1 φj(xj)

I Interaction with tracer and gas particles: HI =∑N

j=1 V (y , xj)

I Empirics: free evolution of tracer

I Strong contrast to bosonic case Ψ0 = χ(y)(∏N

j=1 φj(xj))sym

Brownian motion.

I Mean �eld works much better in fermionic case!

Special situation

j=1 φj(xj)

j=1 V (y , xj)

j=1 φj(xj))sym

Brownian motion.

Special situation

j=1 φj(xj)

j=1 V (y , xj)

j=1 φj(xj))sym

Brownian motion.

Special situation

j=1 φj(xj)

j=1 V (y , xj)

j=1 φj(xj))sym

Brownian motion.

Special situation

j=1 φj(xj)

j=1 V (y , xj)

j=1 φj(xj))sym

Brownian motion.

Special situation

j=1 φj(xj)

j=1 V (y , xj)

j=1 φj(xj))sym

Brownian motion.

Special situation

j=1 φj(xj)

j=1 V (y , xj)

j=1 φj(xj))sym

Brownian motion.

Special situation

1d: easy: Momentum and energy conservation.

Special situation

Higher dimensions

Estimate of �uctuations

Variance of force at some position y(fermions: purble, bosons: blue)

I Fluctuation of force is much smaller for fermions, still largeI Correlation due to antisymmetry reduces �uctuations.I Fluctuations caused by particles with high momentum

Momentum transfer small.Peter Pickl Mathematical Institute LMU