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March  2011, 4(1): 385-399. doi: 10.3934/krm.2011.4.385

Heisenberg picture of quantum kinetic evolution in mean-field limit

Gerasimenko Viktor 1,

1. 

Institute of Mathematics of NAS of Ukraine, 3, Tereshchenkivs'ka Str., 01601 Kyiv-4, Ukraine

Received  September 2010 Revised  October 2010 Published  January 2011

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.
Keywords: evolution of observables, dual quantum BBGKY hierarchy, Quantum kinetic equation, quantum many-particle system., mean-field limit.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 47d0.
Citation: Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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.

[20]

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

[21]

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

[22]

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.

[23]

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

[24]

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.

[25]

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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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.

[20]

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

[21]

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

[22]

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.

[23]

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

[24]

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.

[25]

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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