×

zbMATH — the first resource for mathematics

Mean-field evolution of fermionic systems. (English) Zbl 1304.82061
Authors’ abstract: The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one-particle density is an orthogonal projection \(\omega_N\) with the appropriate semiclassical structure. Assuming some regularity of the interaction potential, we show that the evolution of such an initial data remains close to a Slater determinant, with reduced one-particle density given by the solution of the Hartree-Fock equation with initial data \(\omega_N\). Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree-Fock dynamics.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
81V70 Many-body theory; quantum Hall effect
35Q40 PDEs in connection with quantum mechanics
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
82B30 Statistical thermodynamics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82D20 Statistical mechanics of solids
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adami, R.; Golse, R.; Teta, A., Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys., 127, 1193-1220, (2007) · Zbl 1118.81021
[2] Ammari, Z.; Nier, F., Mean field propagation of Wigner measures and BBGKY hierarchy for general bosonic states, J. Math. Pures Appl. (9), 95, 585-626, (2011) · Zbl 1251.81062
[3] Bach, V., Error bound for the Hartree-Fock energy of atoms and molecules, Commun. Math. Phys., 147, 527-548, (1992) · Zbl 0771.46038
[4] Bardos, C.; Golse, F.; Gottlieb, A.D.; Mauser, N.J., Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. (9), 82, 665-683, (2003) · Zbl 1029.82022
[5] Ben Arous, G.; Kirkpatrick, K.; Schlein, B., A central limit theorem in many-body quantum dynamics, Commun. math. Phys., 321, 371-417, (2013) · Zbl 1280.81157
[6] Benedikter, N., de Oliveira, G., Schlein. B.: Quantitative derivation of the Gross-Pitaevskii Equation. To appear in Comm. Pure Appl. Math. (2014). arxiv:1208.0373 [math-ph] · Zbl 1320.35318
[7] Chen, L.; Oon Lee, J.; Schlein, B., Rate of convergence towards Hartree dynamics, J. Stat. Phys., 144, 872-903, (2011) · Zbl 1227.82046
[8] Elgart, A.; Erdős, L.; Schlein, B.; Yau, H.-T., Nonlinear Hartree equation as the Mean field limit of weakly coupled fermions, J. Math. Pures Appl. (9), 83, 1241-1273, (2004) · Zbl 1059.81190
[9] Erdős , L.; Yau, H.-T., Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys., 5, 1169-1205, (2001) · Zbl 1014.81063
[10] Erdős, L.; Schlein, B.; Yau, H.-T., Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Inv. Math., 167, 515-614, (2006) · Zbl 1123.35066
[11] Erdős , L.; Schlein, B.; Yau, H.-T., Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. Math. (2), 172, 291-370, (2010) · Zbl 1204.82028
[12] Erdős, L.; Schlein, B.; Yau, H.-T., Rigorous derivation of the Gross-Pitaevskii equation, Phys. Rev. Lett., 98, 040404, (2007)
[13] Erdős, L.; Schlein, B.; Yau, H.-T., Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22, 1099-1156, (2009) · Zbl 1207.82031
[14] Frank, R.L.; Hainzl, C.; Seiringer, R.; Solovej, J.P., Microscopic derivation of Ginzburg-Landau theory, J. Amer. Math. Soc., 25, 667-713, (2012) · Zbl 1251.35156
[15] Fröhlich, J.; Graffi, S.; Schwarz, S., Mean-field- and classical limit of many-body Schrödinger dynamics for bosons, Commun. Math. Phys., 271, 681-697, (2007) · Zbl 1172.82011
[16] Fröhlich, J.; Knowles, A., A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction, J. Stat. Phys., 145, 23-50, (2011) · Zbl 1269.82042
[17] Fröhlich, J.; Knowles, A.; Schwarz, S., On the Mean-field limit of bosons with Coulomb two-body interaction, Commun. Math. Phys., 288, 1023-1059, (2009) · Zbl 1177.82016
[18] Ginibre J., Velo G.: The classical field limit of scattering theory for nonrelativistic many-boson systems. I and II. Commun. Math. Phys. 66(1), 37-76 (1979) 68(1):45-68 (1979) · Zbl 0443.35067
[19] Graf, G.M.; Solovej, J.P., A correlation estimate with applications to quantum systems with Coulomb interactions, Rev. Math. Phys., 6, 977-997, (1994) · Zbl 0843.47041
[20] Graffi, S.; Martinez, A.; Pulvirenti, M., Mean-field approximation of quantum systems and classical limit, Math. Models Methods Appl. Sci., 13, 59-73, (2003) · Zbl 1049.81022
[21] Grillakis, M.; Machedon, M.; Margetis, D., Second-order corrections to Mean field evolution of weakly interacting bosons. I, Commun. Math. Phys., 294, 273-301, (2010) · Zbl 1208.82030
[22] Grillakis, M.; Machedon, M.; Margetis, D., Second-order corrections to Mean field evolution of weakly interacting bosons. II, Adv. Math., 228, 1788-1815, (2011) · Zbl 1226.82033
[23] Grillakis, M.; Machedon, M., Pair excitations and the Mean field approximation of interacting bosons. I, Commun. Math. Phys., 324, 601-636, (2013) · Zbl 1277.82034
[24] Grech, P.; Seiringer, R., The excitation spectrum for weakly interacting bosons in a trap, Commun. Math. Phys., 322, 559-591, (2013) · Zbl 1273.82007
[25] Hainzl, C.; Hamza, E.; Seiringer, R.; Solovej, J.P., The BCS functional for general pair interactions, Commun. Math. Phys., 281, 349-367, (2008) · Zbl 1161.82027
[26] Hainzl, C.; Lenzmann, E.; Lewin, M.; Schlein, B., On blowup for time-dependent generalized Hartree-Fock equations, Ann. Henri Poincare, 11, 1023, (2010) · Zbl 1209.85009
[27] Hainzl, C.; Seiringer, R., Critical temperature and energy gap in the BCS equation, Phys. Rev. B, 77, 184517, (2008)
[28] Hainzl, C.; Seiringer, R., Low density limit of BCS theory and Bose-Einstein condensation of fermion pairs, Lett. Math. Phys., 100, 119-138, (2012) · Zbl 1253.82116
[29] Hainzl, C.; Schlein, B., Dynamics of Bose-Einstein condensates of fermion pairs in the low density limit of BCS theory, J. Funct. Anal., 265, 399-423, (2013) · Zbl 1282.35367
[30] Hepp, K., The classical limit for quantum mechanical correlation functions, Commun. Math. Phys., 35, 265-277, (1974)
[31] Kirkpatrick, K.; Schlein, B.; Staffilani, G., Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133, 91-130, (2011) · Zbl 1208.81080
[32] Knowles, A.; Pickl, P., Mean-field dynamics: singular potentials and rate of convergence, Commun. Math. Phys., 298, 101-138, (2010) · Zbl 1213.81180
[33] Lewin, M., Thành Nam, P., Serfaty, S., Solovej, J.P.: Bogoliubov spectrum of interacting Bose gases. To appear in Comm. Pure Appl. Math. (2013). arxiv:1211.2778 [math-ph] · Zbl 1318.82030
[34] Narnhofer, H.; Sewell, G.L., Vlasov hydrodynamics of a quantum mechanical model, Commun. Math. Phys., 79, 9-24, (1981)
[35] Pickl, P., A simple derivation of Mean field limits for quantum systems, Lett. Math. Phys., 97, 151-164, (2011) · Zbl 1242.81150
[36] Rodnianski, I.; Schlein, B., Quantum fluctuations and rate of convergence towards Mean field dynamics, Commun. Math. Phys., 291, 31-61, (2009) · Zbl 1186.82051
[37] Ruijsenaars, S.N.M., On Bogoliubov transformations. II. the general case, Ann. Phys., 116, 105-134, (1978)
[38] Seiringer, R.: (2011) The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306(2), 565-578 · Zbl 1226.82039
[39] Solovej, J.P.: Many Body Quantum Mechanics. Lecture Notes. Summer (2007). Available at http://www.mathematik.uni-muenchen.de/ sorensen/Lehre/SoSe2013/MQM2/skript.pdf · Zbl 0492.35067
[40] Spohn, H., Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys., 52, 569-615, (1980)
[41] Spohn, H., On the Vlasov hierarchy, Math. Methods Appl. Sci., 3, 445-455, (1981) · Zbl 0492.35067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.