IST Austria Mathematical Many Body Theory and its Applications Bilbao, June 17, 2016
Derivation = show that solution of microscopic eq. is close to solution of effective eq. (with fewer degrees of freedom) Motivation: microscopic eq.s usually too complicated to solve exactly for large particle number: ask for typical behavior (what do most particles do?) Examples: Boltzmann eq. from Newton s eq. to the Vlasov eq.... from Schrödinger s eq. to the Note: static or dynamic setting Hartree eq. Gross-Pitaevskii eq. BCS eq. Hartree-Fock eq....
Microscopic Dynamics Non-relativistic quantum mechanics: Here: N particles at time t R described by wave function ψ t L 2 (R 3N ) connection to experiment: ψ t (x 1,..., x N ) 2 probability density microscopic law of motion: Schrödinger eq. i t ψ t = Hψ t, with self-adjoint Hamiltonian H determines ψ t from initial conditions ψ 0 : ψ t = e iht ψ 0 fermions: ψ t antisymmetric Hamiltonian: H = N H 0 j + i<j if ψ 0 is antisymmetric, then so is ψ t v (N) ( x i x j )
Effective Dynamics Approximation: most simple antisymmetric state choose orthonormal ϕ t 1,..., ϕt N L2 (R 3 ) effective N-particle state: N ϕt j = φ t HF, with φ t HF(x 1,..., x N ) = (N!) 1/2 σ S N ( 1) σ N ϕ t σ(j) (x j). Here: the states ϕ t 1,..., ϕt N solve the Hartree-Fock eq.s ( ) N N ( ) i t ϕ t j = H 0 ϕ t j + v (N) ϕ t k 2 ϕ t j v (N) ϕ t k ϕt j ϕ t k k=1 } {{ } direct term k=1 } {{ } exchange term ϕ 0 1,..., ϕ0 N orthonormal ϕt 1,..., ϕt N orthonormal direct term = mean-field interaction without exchange term: fermionic Hartree eq.s in many scaling limits: exchange term subleading important in physics and chemistry: electrons (atoms, molecules, solid states), protons/neutrons (nuclei, neutron star)
Goal If we choose an initial condition ψ 0 N ϕ0 j, then also e iht ψ 0 N ϕt j in an appropriate scaling limit. Remarks: e iht N ϕ0 j = N ϕt j in general only without interaction! interaction correlations (entanglement) bosons: ψ t N ϕt (x j ) Hartree limit or NLS limit More exactly: convergence in the macroscopic sense! let γ ψ be the reduced one-particle density matrix: γ ψ (x, y) = dx 2... dx N ψ(x, x 2,..., x N )ψ(y, x 2,..., x N ) φ HF = N ϕ j, i.e., γ φhf (x, y) = N 1 N ϕ j(x)ϕ j (y) let tr be the trace norm Goal: show that tr γ ψ 0 γ φ 0 HF 0 = allows to control one-particle observables γ ψ t γ φ t tr HF 0
Scaling Limits For a rigorous derivation we need a mean-field scaling limit. So far two limits have been considered: high density limit leading order is described by the classical Vlasov eq. Hartree(-Fock) eq.s are a correction large volume limit leading order is described by free quantum dynamics with certain phase factors Hartree(-Fock) eq.s are a correction
Scaling I: Correction to Vlasov Schrödinger eq. with ψ 0 in O(1) volume (density O(N)): Remarks: in 1/3 t ψ t = N 2/3 N xj ψ t + N 1 i<j kinetic and interaction energy of same order mean-field limit (many particles interact weakly) v(x i x j )ψ t semi-classical limit (small parameter ε N 1/3 ) closeness to Vlasov eq. (for semiclassical initial data) note: exchange term subleading
Scaling I: Correction to Vlasov in 1/3 t ψ t = N 2/3 N xj ψ t + N 1 i<j v(x i x j )ψ t Overview of results: Closeness to Hartree(-Fock) eq.s: Elgart, Erdös, Schlein, Yau (2004) small times, analytic v Benedikter, Porta, Schlein (2013) all times, v in particular bounded, explicit error estimates SP, Pickl (2014) similar result, result can be stated without reference to Fock-space construction Closeness to Vlasov eq. (for Wigner transform of ψ): Narnhofer, Sewell (1981), Spohn (1981) regular v, no error estimates note: also several results on closeness of Hartree to Vlasov
Scaling I: Correction to Vlasov Theorem (SP, Pickl (2014); before: Benedikter, Porta, Schlein (2013)) Let ψ t be solution to Schrödinger eq., let ϕ t 1,..., ϕt N be solutions to Hartree or Hartree-Fock eq.s. Assume v L 1 (R 3 ) and for the Fourier transform ˆv, d 3 k (1 + k 2 ) ˆv(k) <. Assume for the orthonormal initial conditions ϕ 0 1,..., ϕ0 N, sup (1 + k ) 1 [ γ φ 0 k R 3 HF, e ikx] tr CN 1/3, [ γ φ 0 HF, ] tr C. Remark: Then ( γ ψ t γ φ t tr HF e ect γψ 1/2 0 γ φ 0 HF + N 1/2). rate can be improved to N 5/6 under stronger conditions on initial state for closeness to Vlasov eq.: convergence rate N 1/3 tr
Scaling I: Correction to Vlasov Desirable: Coulomb interaction v(x) = x 1 physically relevant for electrons for Coulomb interaction the scaling limit follows from a rescaling of space and time variables in 1/3 t ψ t = N 2/3 N xj ψ t + N 1 i<j t = N t, x i = N 1/3 x i unscaled Schrödinger equation 1 x i x j ψt important application to fermionic stars with gravitational interaction N i t ψ t = xj + m 2 ψ t κ 1 x i x j ψt i<j for N > cκ 3/2, gravitational collapse in mean-field description for t N 1/3 Also desirable: a mean-field limit that is not semi-classical, but quantum (e.g., interference effects in the leading order)
Scaling II: Correction to Free Quantum Dynamics Initial data ψ 0 in O(N) volume (density O(1)) need long-range interaction! Schrödinger eq.: Remarks: i t ψ t = N xj ψ t + N βs x i x j s ψ t kinetic and interaction energy of same order for β s = 1 s/3 i<j mean-field limit (many particles interact weakly) but average force per particle is small, O(N 1/3 ) closeness to free dynamics (with t, x dependent phase) note: exchange term subleading
Scaling II: Correction to Free Quantum Dynamics i t ψ t = N xj ψ t + N β i<j v(x i x j )ψ t Overview of results for closeness to Hartree(-Fock) eq.s: β = 1 (subleading interaction energy): Bardos, Golse, Gottlieb, Mauser (2003) bounded v Fröhlich, Knowles (2011) v(x) = x 1 Note: β = β s, v(x) = x s : main results, see next slides for t N 1/3, we recover the previous scaling limit by rescaling x N 1/3 x
Scaling II: Correction to Free Quantum Dynamics Theorem (SP, Pickl (in preparation)) Let ψ t be solution to Schrödinger eq. with v (N) (x) = ±N 2/3 x 1. Let ϕ t 1,..., ϕt N be solutions to i tϕ t j = ϕ t j and set ϕ t j = e iθ(x,t) ϕ t j, with θ(x, t) = t 0 (v (N) ϕ s j 2 )(x)ds. Assume for the orthonormal initial conditions ϕ 0 1,..., ϕ0 N, Then Remark: N 4 ϕ 0 2 j CN. ( γ ψ t γ ϕ t tr γψ j C(t) 0 γ ϕ 0 j 1/2 tr convergence rate N 1/3 expected to be optimal + N 1/3 ).
Scaling II: Correction to Free Quantum Dynamics Let ψ t be solution to Schrödinger eq., let ϕ t 1,..., ϕt N be solutions to Hartree or Hartree-Fock eq.s. Assume N ϕ 0 2 j KN. Theorem (SP, Pickl (2014)) For v (N) (x) = ±N βs x s and 0 < s < 3/5: ( γ ψ t γ φ t tr HF e Ct γψ 1/2 0 γ φ 0 HF Theorem (Bach, Breteaux, SP, Pickl, Tzaneteas (2015)) For v (N) (x) = ±N 2/3 x 1 : ( γ ψ t γ φ t tr HF e Ct N 1/3 γψ 1/2 0 γ φ 0 HF tr tr + N 1/2 ). + N 1/6 ). Remarks: stronger conditions on convergence of initial states generalization: if ϕ t j 2 KN, v(x) = λ x 1, then Ct 30λN 2/3 K 1/2 t note: correction to free dynamics for K N 1/3
Scaling II: Correction to Free Quantum Dynamics Theorem (SP, Pickl (in preparation)) Assume additionally that N 4 ϕ 0 2 j CN. Then for v (N) (x) = ±N 2/3 x 1 and some δ > 0 tbd: ( ) γ ψ t γ φ t tr γψ HF C(t) 0 γ φ 0 1/2 + N 1/2 δ. Remarks: extra assumption necessary to avoid clustering of particles Hartree(-Fock) is a subleading correction here open problem: assume semiclassical initial conditions and extend to t N 1/3 HF tr
The Method Two key steps: introduce a certain functional α(ψ, ϕ 1,..., ϕ N ) to measure closeness of ψ to N ϕ j (Pickl (2011)) bring t α(ψ t, ϕ t 1,..., ϕt N ) into a nice form and do a Gronwall estimate
The Method: α-functional Definition Define the projectors N p m = q m = 1 N p m, ϕ t j (x m ) ϕ t j (x m ) = 1 1 }{{} m 1 times P (N,k) = q 1... q k p k+1... p N = ( q 1... q k p k+1... p N )sym. For any f : {0,..., N} [0, 1], define α f (ψ, ϕ 1,..., ϕ N ) = N k=0 Nγ φhf 1 1, }{{} N m times f (k) ψ, P (N,k) ψ.
The Method: α-functional α f = N k=0 f (k) ψ, P (N,k) ψ f is called weight function specifies sense of closeness freedom in choosing f gives great technical advantage We use here: f (k) = n(k) := k N α n = ψ, q 1 ψ { k f (k) = m (γ) (k) := N, for k N γ (0 < γ 1) γ 1, otherwise Relation to trace norm: γ ψ γ φhf tr 8α n 2 γ ψ γ φhf tr γ ψ γ φhf tr 8α m (γ) 2 N 1 γ γ ψ γ φhf tr
The Method: t α(t) Strategy: bring t α f (ψ t, ϕ t 1,..., ϕt N ) into nice form prove bound t α f (t) C ( α f (t) + N δ), δ > 0 by Gronwall: α f (t) e Ct α f (0) + (e Ct 1)N δ Most simple case: time derivative of α n (t): t α n (t) = t ψ t, q1ψ t t = i ψ t, [ H H1 mf, q1 t ] ψ t = i ψ t, [ (N 1)v (N) 12 V (N) 1, q1] t ψ t, where V (N) 1 = direct term or V (N) 1 = direct + exchange term. Note: only difference of Schrödinger interaction and effective interaction enters!
The Method: t α(t) Insert identities: p 1 + q 1 = 1 = p 2 + q 2 : ( ) t α n (t) = 2 Im ψ t, q1 t (N 1)p2v t (N) 12 pt 2 (v (N) ρ t N) 1 exch p1ψ t t + 2 Im ψ t, q1q t 2 t (N 1)v (N) 12 pt 1p2ψ t t + 2 Im ψ t, q1q t 2 t (N 1)v (N) 12 pt 1q2ψ t t. need to control 3 types of transitions! ( recall α(t) = ψ t, q t 1 ψt = q t 1 ψt 2 ) Note: for general α f (t) one finds same formal expression with ψ t N N f (k) P (N,k) ψ t k=0 n (k) = N 1 { N m (γ) γ, for k N (k) = γ 0, otherwise
Sketch of Proof for Scaling I i t ψ t = N 1/3 N xj ψ t + N 2/3 i<j v(x i x j )ψ t Use Fourier decomposition v 12 = d 3 k ˆv(k)e ik(x1 x2). Then: t α n (t) = 2N 2/3 Im d 3 k ˆv(k) [ ( )( ψ t, q1e t ikx1 p1 t (N 1)p2e t ikx2 p2 t tr(e ikx p t ) + p1e t ikx1 p1 )ψ t t ( )( + (N 1) ψ t, q1e t ikx1 p1 t q2e t ikx2 p2 )ψ t t ( )( ] + (N 1) ψ t, q1e t ikx1 p1 t q2e t ikx2 q2 )ψ t t. note: ψ, A 1 ψ N 1 A tr for antisymmetric ψ use q t e ikx p t tr [e ikx, p t ] tr C(t)N 2/3 (assumption) first term: gain q 2 by diagonalization
Sketch of Proof for Scaling II with Coulomb Interaction i t ψ t = N xj ψ t + N 2/3 i<j v(x i x j )ψ t Use Fefferman-de la Llave decomposition: (let X r,z (x) = 1 x z r ) x 1 x 2 1 = d 3 z R 3 0 dr πr 5 X r,z(x 1 )X r,z (x 2 ) = dµ(ω) X ω 1 X ω 2 Then (dγ(a) := N A j): t α n (t) = 2N 2/3 Im dµ(ω) ( N ψ 1 t, dγ q t X ω p t)( ( dγ p t X ω p t) tr(x ω p t ) )ψ t + 2(N 1)N 2/3 Im ψ t, q t 1q t 2v 12 p t 1p t 2ψ t + 2(N 1)N 2/3 Im ψ t, q t 1q t 2v 12 p t 1q t 2ψ t.
Sketch of Proof for Scaling II with Coulomb Interaction First term: N 5/3 ( dµ(ω) ψ t, dγ q t X ω p t)( ( dγ p t X ω p t) tr(x ω p t ) )ψ t ) use dγ (p t X ω p t tr(x ω p t ) to gain q t neglecting q t : N 5/3 = N 5/3 dµ(ω) p t X ω tr p t X ω tr d 3 x d 3 yρ t N(x)ρ t N(y) x y 1 N 5/3 N v ρ t N C by Hölder and Lieb-Thirring inequalities
Sketch of Proof for Scaling II with Coulomb Interaction Third term: N 1/3 Im ψ t, q1q t 2v t 12 p1q t 2ψ t t need to control v 2 ρ t N Hardy: v 2 ρ t N CN, no orthonormality used! need to gain extra N 1/3 by using m (γ) one finds N 2 N m (γ) (k) P (N,k) q 1 q 2 ψ t N γ 1 α m (γ), k=0 i.e., gain for small γ need to show: estimates for other terms are ok with m (γ)
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