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

Heisenberg picture of quantum kinetic evolution in mean-field limit

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.
Citation: Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & 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).   Google Scholar

[2]

C. Cercignani, "Mathematical Methods in Kinetic Theory,", Springer-Verlag, (1990).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[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.  doi: 10.1016/S0021-7824(03)00023-0.  Google Scholar

[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.  doi: 10.1007/s00220-007-0207-5.  Google Scholar

[9]

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

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.  doi: 10.1088/0305-4470/37/42/002.  Google Scholar

[23]

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

[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., 43 (4852).  doi: 10.1088/1751-8113/43/48/485203.  Google Scholar

[25]

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

show all references

References:
[1]

C. Cercignani, "The Boltzmann Equation And Its Applications,", Springer-Verlag, (1987).   Google Scholar

[2]

C. Cercignani, "Mathematical Methods in Kinetic Theory,", Springer-Verlag, (1990).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[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.  doi: 10.1016/S0021-7824(03)00023-0.  Google Scholar

[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.  doi: 10.1007/s00220-007-0207-5.  Google Scholar

[9]

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

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.  doi: 10.1088/0305-4470/37/42/002.  Google Scholar

[23]

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

[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., 43 (4852).  doi: 10.1088/1751-8113/43/48/485203.  Google Scholar

[25]

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

[1]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443

[2]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[3]

Jin-Cheng Jiang, Chi-Kun Lin, Shuanglin Shao. On one dimensional quantum Zakharov system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5445-5475. doi: 10.3934/dcds.2016040

[4]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492

[5]

Harald Markum, Rainer Pullirsch. Classical and quantum chaos in fundamental field theories. Conference Publications, 2003, 2003 (Special) : 596-603. doi: 10.3934/proc.2003.2003.596

[6]

Thibaut Allemand. Derivation of a two-fluids model for a Bose gas from a quantum kinetic system. Kinetic & Related Models, 2009, 2 (2) : 379-402. doi: 10.3934/krm.2009.2.379

[7]

Jingwei Hu, Shi Jin. On kinetic flux vector splitting schemes for quantum Euler equations. Kinetic & Related Models, 2011, 4 (2) : 517-530. doi: 10.3934/krm.2011.4.517

[8]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845

[9]

Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic & Related Models, 2011, 4 (4) : 1049-1062. doi: 10.3934/krm.2011.4.1049

[10]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165

[11]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the quantum Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 273-294. doi: 10.3934/cpaa.2017013

[12]

Paolo Antonelli, Pierangelo Marcati. Quantum hydrodynamics with nonlinear interactions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 1-13. doi: 10.3934/dcdss.2016.9.1

[13]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119

[14]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1

[15]

Dmitry Jakobson. On quantum limits on flat tori. Electronic Research Announcements, 1995, 1: 80-86.

[16]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013

[17]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086

[18]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037

[19]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015

[20]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]