Observation functional An exact generalization of DFT

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Observation functional An exact generalization
of DFT
Philippe CHOMAZ - GANIL

• States, observables, observations
• Variational principles
• Generalized mean-Field
• Hartree-Fock
• Hierarchies and fluctuations
• Exact generalized density functional
• Exact generalized Kohn-Sham Eq.

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A) States, Observables and Observations

• States
• Observables
• Observation

Many-body wave function Hilbert or Fock space
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A) States, Observables and Observations

• States
• Observables
• Observation

Many-body wave function Hilbert or Fock space
Density matrix Liouville space
Scalar product in matrix space
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B) Variational principles

• Static
• Dynamics

Schrödinger equation
Extremum of the action I
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B) Variational principles

• Static
• Dynamics

Zero Temperature minimum energy E
Finite T minimum free energy
Entropy
Liouville equation
Schrödinger equation
Balian and Vénéroni double principle
Extremum of the action I
Observables backward from t1 Density forward from
t0
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C) Generalized mean-field

• Coherent states
• Generalized density
• Extremum action

Group transformation
Lie Algebra
Group parameters
Mean-field ltgt Ehrenfest
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C) Generalized mean-field

• Coherent states
• Generalized density
• Extremum action

Trial observables
Group transformation
Lie Algebra
Group parameters
Mean-field ltgt Ehrenfest
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D) Hartree Fock

• Lie algebra
• Observation
• Trial states
• Hamiltonian
• Independent particles
• Mean Field

One-body observables
One-body density
Independent particle state
Thouless theorem (Slaters)
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E) Hierarchies and fluctuations

• Exact dynamics
• Hierarchy
• Projections ltAgt
• Minimum entropy
• Correlation
• MF Langevin
• Mean-Field

Close the Lie Algebra including A an H
Coupled equations
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F) Exact generalized Density functional

• Exact State
• Exact Observations
• Exact E functional
• Min in a subspace
• Constrained energy
• ltgt external field

Or
Generalized density
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G) Exact Generalized Kohn-Sham Eq.

• Exact E functional
• For a set of observations
• Exact ground state E
• gt exact densities ?
• Variation
• Equivalent to mean-field Eq.
• with Lie algebra including Al , Al , Am

Generalized density
Exact E and ? in an external field U?zlAl
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G) Exact Generalized Kohn-Sham Eq.

• Remarks
• Exact for E
• and all observations ltAl gt ?l ?included in E?
• Easy to go from a set of Al to a reduced set Al
• gt E?min?cst E?

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H) Density functional theory LDA

• The only information needed is the energy
• gt functionals of r
• Local density approximation
• Energy density functional
• Local densities
• matter , kinetic , current
• Mean field

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H) LDA Skyrme case

• Standard case few densities
• Matter isoscalar isovector
• kinetic isoscalar isovector
• Spin isoscalar isovector
• Energy functional
• Mean-field q(n,p)

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H) LDA Skyrme case

• Standard case few densities
• Matter isoscalar isovector
• kinetic isoscalar isovector
• Spin isoscalar isovector
• Energy functional
• Mean-field q(n,p)

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