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Heisenberg picture of quantum kinetic evolution in mean-field limit

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  • We develop a rigorous formalism for the description of the evolution of observables of quantum systems of particles in the mean-field scaling limit. The corresponding asymptotics of a solution of the initial-value problem of the dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution of marginal observables and the evolution of quantum states described in terms of a one-particle marginal density operator are established. Such approach gives the alternative description of the kinetic evolution of quantum many-particle systems.
    Mathematics Subject Classification: Primary: 35Q40; Secondary: 47d06.


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  • [1]

    C. Cercignani, "The Boltzmann Equation And Its Applications," Springer-Verlag, 1987.


    C. Cercignani, "Mathematical Methods in Kinetic Theory," Springer-Verlag, 1990.


    C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, 1994.


    C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, "Many-Particle Dynamics and Kinetic Equations," Kluwer Acad. Publ., 1997.


    R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, Journal of Statistical Physics, 127 (2007), 1193-1220.doi: 10.1007/s10955-006-9271-z.


    A. Arnold, Mathematical properties of quantum evolution equation, Lecture Notes in Mathematics, 1946 (2008), 45-109.doi: 10.1007/978-3-540-79574-2.


    C. Bardos, F. Golse, A. D. Gottlieb and N. J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, Journal de Mathématiques Pures et Appliqués, 82 (2003), 665-683.doi: 10.1016/S0021-7824(03)00023-0.


    J. Fröhlich, S. Graffi and S. Schwarz, Mean-field and classical limit of many-body Schrödinger dynamics for bosons, Communications in Mathematical Physics, 271 (2007), 681-697.doi: 10.1007/s00220-007-0207-5.


    A. Michelangeli, Strengthened convergence of marginals to the cubic nonlinear Schrödinger equation, Kinetic and Related Models, 3 (2010), 457–-471.doi: 10.3934/krm.2010.3.457.


    F. Pezzotti and M. Pulvirenti, Mean-field limit and semiclassical expansion of quantum particle system, Annales Henri Poincaré, 10 (2009), 145-187.doi: 10.1007/s00023-009-0404-1.


    D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, A short review on the derivation of the nonlinear quantum Boltzmann equations, Communications in Mathematical Sciences, 5 (2007), 55-71.


    L. Erdös, M. Salmhofer and H.-T. Yau, On the quantum Boltzmann equation, Journal of Statistical Physics, 116 (2004), 367-380.doi: 10.1023/B:JOSS.0000037224.56191.ed.


    G. Borgioli and V. I. Gerasimenko, Initial-value problem of the quantum dual BBGKY hierarchy, Nuovo Cimento, 33 C (2010), 71-78.doi: 10.1393/ncc/i2010-10564-6.


    G. Borgioli and V. I. Gerasimenko, The dual BBGKY hierarchy for the evolution of observables, Riv. Mat. Univ. Parma, 4 (2001), 251-267.doi: 10.1393/ncc/i2010-10564-6.


    M. M. Bogolyubov, "Lectures on Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems," (Ukrainian) Rad. Shkola, 1949.


    H. Spohn, Kinetic equations from Hamiltonian dynamics, Reviews of Modern Physics, 52 (1980), 569-615.doi: 10.1103/RevModPhys.52.569.


    R. Dautray and J. L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," 5, Springer-Verlag, 1992.


    D. Ya. Petrina, "Mathematical Foundations of Quantum Statistical Mechanics. Continuous Systems," Kluwer Acad. Publ., 1995.


    V. I. Gerasimenko and V. O. Shtyk, Evolution of correlations of quantum many-particle systems, J. Stat. Mech. Theory Exp., 3 (2008), P03007.doi: 10.1088/1742-5468/2008/03/P03007.


    O. Bratelli and D. W. Robinson, "Operator Algebras and Quantum Statistical Mechanics," 2, Springer, 1997.


    V. I. Gerasimenko, Groups of operators for evolution equations of quantum many-particle systems, Operator Theory: Adv. and Appl., 191 (2009), 341-355.


    V. I. Gerasimenko, T. V. Ryabukha and M. O. Stashenko, On the structure of expansions for the BBGKY hierarchy solutions, J. Phys. A: Math. Gen., 37 (2004), 9861-9872.doi: 10.1088/0305-4470/37/42/002.


    J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer, 2006.


    V. I. Gerasimenko and Zh. A. Tsvir, A description of the evolution of quantum states by means of the kinetic equation, J. Phys. A: Math. Theor., v. 43, 485203 (19pp), 2010.doi: 10.1088/1751-8113/43/48/485203.


    V. I. Gerasimenko and D. O. Polishchuk, Dynamics of correlations of Bose and Fermi particles, Math. Meth. Appl. Sci., 33 (2011), 76-93.doi: 10.1002/mma.1336.

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