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Ukrainian Mathematical Journal

, Volume 58, Issue 9, pp 1329–1346 | Cite as

Initial-value problem for the Bogolyubov hierarchy for quantum systems of particles

  • V. I. Herasymenko
  • V. O. Shtyk
Article

Abstract

We construct cumulant (semi-invariant) representations for a solution of the initial-value problem for the Bogolyubov hierarchy for quantum systems of particles. In the space of sequences of trace-class operators, we prove a theorem on the existence and uniqueness of a solution. We study the equivalence problem for various representations of a solution in the case of the Maxwell-Boltzmann statistics.

Keywords

Quantum System Recurrence Relation Evolution Operator Cluster Expansion BBGKY Hierarchy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti, “Some consideration on the derivation of the nonlinear quantum Boltzmann equation,” J. Statist. Phys., 116, No. 1/4, 381–410 (2004).CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti, “A short review on the derivation of the nonlinear quantum Boltzmann equations,” in: Lect. Notes, Workshop “Mathematical Methods in Quantum Mechanics,” Bressanove (Italy), 2005.Google Scholar
  3. 3.
    C. Bardos, F. Golse, and N. J. Mauser, “Weak coupling limit of the N-particle Schrödinger equation,” Math. Anal. Appl., 2, No. 7, 275–293 (2000).MathSciNetGoogle Scholar
  4. 4.
    C. Bardos, F. Golse, A. Gottlieb, and N. Mauser, “Mean field dynamics of fermions and the time-dependent Hartree-Fock equation,” J. Math. Pures Appl., 82, 665–683 (2003).MathSciNetzbMATHGoogle Scholar
  5. 5.
    F. Golse, “The mean-field limit for the dynamics of large particle systems,” J. Equat. Deriv. Part., No. 9 (2003).Google Scholar
  6. 6.
    F. Castella, “From the von Neumann equation to the quantum Boltzmann equation in a deterministic framework,” J. Statist. Phys., 104, No. 1/2, 387–447 (2001).CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    L. Erdös, “Derivation of macroscopic kinetic equations from microscopic quantum mechanics,” in: Lect. Notes, School Math. Georgiatech. (2001).Google Scholar
  8. 8.
    L. Erdös and H.-T. Yau, “Derivation of the nonlinear Schrödinger equation from a many body Coulomb system,” Adv. Theor. Math. Phys., 5, 1169–1205 (2001).MathSciNetzbMATHGoogle Scholar
  9. 9.
    L. Erdös, M. Salmhofer, and H.-T. Yau, “On the quantum Boltzmann equation,” J. Statist. Phys., 116, No. 116, 367–380 (2004).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Arnold, “Self-consistence relaxation-time models in quantum mechanics,” Commun. Part. Different. Equat., 21, No. 3, 473–506 (1996).zbMATHGoogle Scholar
  11. 11.
    H. Spohn, “Quantum kinetic equations,” in: M. Fannes, C. Maes, and A. Verbeure (editors), On Three Levels (Micro-, Meso-and Macro-Approaches in Physics), Plenum, New York (1994), pp 1–10.Google Scholar
  12. 12.
    N. N. Bogolyubov, Lectures on Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems [in Ukrainian], Radyans’ka Shkola, Kyiv (1949).Google Scholar
  13. 13.
    D. Ya. Petrina, Mathematical Foundations of Quantum Statistical Mechanics. Continuous Systems, Kluwer, Dordrecht (1995).Google Scholar
  14. 14.
    D. Ya. Petrina, “On solutions of the Bogolyubov kinetic equations. Quantum statistics,” Teoret. Mat. Fiz., 13, No. 3, 391–405 (1972).MathSciNetGoogle Scholar
  15. 15.
    D. Ya. Petrina and A. K. Vidybida, “Cauchy problem for the Bogolyubov kinetic equations,” Tr. Mat. Inst. Akad. Nauk SSSR, 86, 370–378 (1975).MathSciNetGoogle Scholar
  16. 16.
    J. Ginibre, “Some applications of functional integrations in statistical mechanics,” in: C. de Witt and R. Stord (editors), Statistical Mechanics and Quantum Field Theory, Gordon and Breach, New York (1971), pp. 329–427.Google Scholar
  17. 17.
    D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin, New York (1969).zbMATHGoogle Scholar
  18. 18.
    F. A. Berezin and M. A. Shubin, Schrödinger Equation [in Russian], Moscow University, Moscow (1983).zbMATHGoogle Scholar
  19. 19.
    T. Kato, Perturbation Theory for Linear Operators, Springer, New York (1966).zbMATHGoogle Scholar
  20. 20.
    R. Dautray and J. L. Lions, Evolution Problems, Springer, Berlin (2000).Google Scholar
  21. 21.
    V. I. Herasymenko and T. V. Ryabukha, “Cumulant representation of solutions of the BBGKY hierarchy of equations,” Ukr. Mat. Zh., 54, No. 10, 1313–1328 (2002).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. I. Herasymenko
    • 1
  • V. O. Shtyk
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyiv

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