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Between Green's Functions and Transport Equations

PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS III Kiel August 22 – 25, 2005. Between Green's Functions and Transport Equations. B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička , Acad. Sci. of CR, Praha.

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Between Green's Functions and Transport Equations

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  1. PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS III Kiel August 22 – 25, 2005 Between Green's Functions and Transport Equations B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  2. PROGRESS IN NON-EQUILIBRIUM GREEN'S FUNCTIONS III Kiel August 22 – 25, 2005 Between Green's Functions and Transport Equations: Reconstruction Theorems and the Role of InitialConditions B. Velický, Charles University and Acad. Sci. of CR, Praha A. Kalvová, Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  3. Topical Problems in Statistical Physics TU Chemnitz, November 30, 2005 Between Green's Functions and Transport Equations: Correlated Initial Condition for Restart Process Kalvová, Acad. Sci. of CR, Praha B. Velický, Charles University and Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  4. Topical Problems in Statistical Physics TU Chemnitz, November 30, 2005 Between Green's Functions and Transport Equations: Correlated Initial Condition for Restart Process Time Partitioning for NGF Kalvová, Acad. Sci. of CR, Praha B. Velický, Charles University and Acad. Sci. of CR, Praha V. Špička, Acad. Sci. of CR, Praha

  5. Prologue Between GF and Transport Equations …

  6. (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance many-body Hamiltonian many-body density matrix additive operator Between GF and Transport Equations …

  7. (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Between GF and Transport Equations …

  8. a closed equation for (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Quantum Transport Equation generalized collision term Between GF and Transport Equations …

  9. a closed equation for (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Quantum Transport Equation interaction term Between GF and Transport Equations …

  10. a closed equation for • existence, construction of • incorporation of the initial condition QUESTIONS (Non-linear) quantum transport non-equilibrium problem Many-body system Initial state External disturbance Response many-body Hamiltonian many-body density matrix additive operator one-particle density matrix Quantum Transport Equation interaction term Between GF and Transport Equations …

  11. DIRECT use a NGF solver This talk: orthodox study of quantum transport using NGF TWO PATHS INDIRECT use NGF to construct a Quantum Transport Equation Between GF and Transport Equations …

  12. Lecture on NGF This talk: orthodox study of quantum transport using NGF DIRECT TWO PATHS use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation Between GF and Transport Equations …

  13. This talk: orthodox study of quantum transport using NGF DIRECT TWO PATHS use a NGF solver Lecture on NGF…continuation Between GF and Transport Equations …

  14. This talk: orthodox study of quantum transport using NGF Real timeNGF choices DIRECT TWO PATHS use a NGF solver Lecture on NGF…continuation Between GF and Transport Equations …

  15. This talk: orthodox study of quantum transport using NGF DIRECT TWO PATHS use a NGF solver INDIRECT use NGF to construct a Quantum Transport Equation

  16. Standard approach based on GKBA Real timeNGF our choice  GKBE Between GF and Transport Equations …

  17. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form Between GF and Transport Equations …

  18. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz widely used: KBA (for steady transport), GKBA (transients, optics) Between GF and Transport Equations …

  19. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz GKBA Lipavsky, Spicka, Velicky, Vinogradov, Horing Haug + Frankfurt team, Rostock school, Jauho, … Between GF and Transport Equations …

  20. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz GKBA Resulting Quantum Transport Equation Between GF and Transport Equations …

  21. Standard approach based on GKBA Real timeNGF our choice  GKBE  Specific physical approximation -- self-consistent form  Elimination of by an Ansatz GKBA Resulting Quantum Transport Equation • Famous examples: • Levinson eq. • (hot electrons) • Optical quantum • Bloch eq. • (optical transients) Between GF and Transport Equations …

  22. Act I • reconstruction Between GF and Transport Equations …

  23. Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Between GF and Transport Equations …

  24. Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Reminiscences: BBGKY, Hohenberg-Kohn Theorem Between GF and Transport Equations …

  25. Exact formulation -- Reconstruction Problem GENERAL QUESTION: conditions under which a many-body interacting system can be described in terms of its single-time single-particle characteristics Reminiscences: BBGKY, Hohenberg-Kohn Theorem Here: time evolution of the system Between GF and Transport Equations …

  26. … in fact: express , a double-time correlation function, by its time diagonal  Eliminate by an Ansatz GKBA Exact formulation -- Reconstruction Problem New look on the NGF procedure: Any Ansatz is but an approximate solution… ¿Does an answer exist, exact at least in principle? Between GF and Transport Equations …

  27. Reconstruction Problem – Historical Overview INVERSION SCHEMES Between GF and Transport Equations …

  28. Reconstruction Problem – Historical Overview INVERSION SCHEMES Between GF and Transport Equations …

  29. Parallels G E N E R A L S C H E M E LABEL Bogolyubov Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for Between GF and Transport Equations …

  30. Parallels G E N E R A L S C H E M E LABEL Bogolyubov Postulate/Conjecture: typical systems are controlled by a hierarchy of times separating the initial, kinetic, and hydrodynamic stages. A closed transport equation holds for Between GF and Transport Equations …

  31. Parallels G E N E R A L S C H E M E LABEL TDDFT Runge – Gross Theorem: Let be local. Then, for a fixed initial state , the functional relation is bijective and can be inverted. NOTES: U must be sufficiently smooth. no enters the theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional. Between GF and Transport Equations …

  32. Parallels G E N E R A L S C H E M E LABEL TDDFT Runge – Gross Theorem: Let be local. Then, for a fixed initial state , the functional relation is bijective and can be inverted. NOTES: U must be sufficiently smooth. no enters the theorem. This is an existence theorem, systematic implementation based on the use of the closed time path generating functional. Between GF and Transport Equations …

  33. Parallels G E N E R A L S C H E M E LABEL Schwinger Closed Time Contour Generating Functional (Schwinger): Used to invert the relation EXAMPLES OF USE: Fukuda et al. … Inversion technique based on Legendre transformation  Quantum kinetic eq. Leuwen et al. … TDDFT context Between GF and Transport Equations …

  34. Parallels G E N E R A L S C H E M E LABEL Schwinger Closed Time Contour Generating Functional (Schwinger): Used to invert the relation EXAMPLES OF USE: Fukuda et al. … Inversion technique based on Legendre transformation  Quantum kinetic eq. Leuwen et al. … TDDFT context Between GF and Transport Equations …

  35. Parallels: Lessons for the Reconstruction Problem G E N E R A L S C H E M E • „Bogolyubov“: importance of the time hierarchy REQUIREMENTCharacteristic times should emerge in a constructive manner during the reconstruction procedure. • „TDDFT“ :analogue of the Runge - Gross Theorem REQUIREMENTConsider a general non-local disturbance U in order to obtain the full 1-DM  as its dual. • „Schwinger“: explicit reconstruction procedure REQUIREMENTA general operational method for the reconstruction (rather than inversion in the narrow sense). Its success in a particular case becomes the proof of the Reconstruction theorem at the same time. LABEL NGF Reconstruction Theorem

  36. Reconstruction Problem – Summary INVERSION SCHEMES Between GF and Transport Equations …

  37. Reconstruction Problem – Summary INVERSION SCHEMES Between GF and Transport Equations …

  38. Reconstruction theorem:Reconstruction equations Keldysh IC:simple initial state permits to concentrate on the other issues DYSON EQUATIONS Two well known “reconstruction equations” easily follow: RECONSTRUCTION EQUATIONS LSV, Vinogradov … application! Between GF and Transport Equations …

  39. Reconstruction theorem:Reconstruction equations Keldysh IC:simple initial state permits to concentrate on the other issues DYSON EQUATIONS Two well known “reconstruction equations” easily follow: RECONSTRUCTION EQUATIONS • Source terms … the Ansatz • For t=t' … tautology  … input Between GF and Transport Equations …

  40. GKB EQ. DYSON EQ. RECONSTRUCTION EQ. Reconstruction theorem:Coupled equations Between GF and Transport Equations …

  41. Reconstruction theorem: operational description NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation   reconstruction equations Dyson eq. Between GF and Transport Equations …

  42. Reconstruction theorem: formal statement NGF RECONSTRUCTION THEOREM determination of the full NGF restructured as a DUAL PROCESS quantum transport equation   reconstruction equations Dyson eq. "THEOREM" The one-particle density matrix and the full NGF of a process are in a bijective relationship, Between GF and Transport Equations …

  43. Act II • reconstruction • and initial conditions • NGF view Between GF and Transport Equations …

  44. General initial state For an arbitrary initial state at start from the NGF Problem of determination of G extensively studied Fujita  Hall  Danielewicz  … Wagner  Morozov&Röpke … KlimontovichKremp … Bonitz&Semkat … Take over the relevant result for : The self-energy depends on the initial state (initial correlations) has singular components Between GF and Transport Equations …

  45. General initial state For an arbitrary initial state at start from the NGF Problem of determination of G extensively studied Fujita  Hall  Danielewicz  … Wagner  Morozov&Röpke … KlimontovichKremp … Bonitz&Semkat … Take over the relevant result for : The self-energy depends on the initial state (initial correlations) has singular components Morawetz Between GF and Transport Equations …

  46. General initial state: Structure of Structure of Between GF and Transport Equations …

  47. General initial state: Structure of Structure of Danielewicz notation Between GF and Transport Equations …

  48. General initial state: Structure of Structure of Danielewicz notation Between GF and Transport Equations …

  49. GKB EQ. DYSON EQ. RECONSTRUCTION EQ. General initial state: A try at the reconstruction

  50. GKB EQ. DYSON EQ. RECONSTRUCTION EQ. General initial state: A try at the reconstruction To progress further, narrow down the selection of the initial states

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