**Benjamin Schlein**^{[a]}

*Derivation of effective evolution equations from many-body quantum mechanics
*

**Pages:** (* in press *)

**Received:** 31 December 2016

**Accepted in revised form:** 23 May 2017

**Mathematics Subject Classification (2010):** 82C10.

**Keywords:** Quantum dynamics, Bose-Einstein condensation, Gross-Pitaevskii dynamics, Hartree-Fock equation.

**Author address:**

[a]: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

**Abstract:**
In these notes, based on a mini-course held at the summer school ''Methods and Models of Kinetic Theory''
that took place in Porto Ercole in June 2016, we review some of the recent developments in the derivation
of effective evolution equations starting from many-body quantum mechanics.
We discuss the derivation of the Hartree equation in the bosonic mean-field limit,
of the Gross-Pitaevskii equation describing the dynamics of initially trapped Bose-Einstein
condensates and of the Hartree-Fock equation for fermions in a joint mean-field and semiclassical limit.

**References**

[1]
R. Adami, F. Golse and A. Teta, *Rigorous derivation of the cubic NLS in dimension one*,
J. Stat. Phys. 127 (2007), 1193–1220.
MR2331036

[2]
Z. Ammari and F. Nier,
*Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states*,
J. Math. Pures Appl. 95 (2011), no. 6, 585–626.
MR2802894

[3]
A. Athanassoulis, T. Paul, F. Pezzotti and M. Pulvirenti,
*Strong semiclassical approximation of Wigner functions for the Hartree dynamics*,
Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011), no. 4, 525–552.
MR2904998

[4]
V. Bach, S. Breteaux, S. Petrat, P. Pickl and T. Tzaneteas,
*Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction*,
J. Math. Pures Appl. 105 (2016), no. 1, 1–30.
MR3427937

[5]
C. Bardos, F. Golse, A. D. Gottlieb and N. J. Mauser,
*Mean field dynamics of fermions and the time-dependent Hartree-Fock equation*,
J. Math. Pures Appl. (9) 82 (2003), no. 6, 665–683.
MR1996777

[6]
C. Bardos, F. Golse and N. J. Mauser,
*Weak coupling limit of the \(N\)-particle Schrödinger equation*,
Methods Appl. Anal. 7 (2000), no. 2, 275–293.
MR1869286

[7]
N. Benedikter, G. de Oliveira and B. Schlein,
*Quantitative derivation of the Gross-Pitaevskii equation*, Comm. Pure Appl. Math. 68 (2015), no. 8, 1399–1482.
MR3366749

[8]
N. Benedikter, V. Jakšić, M. Porta, C. Saffirio and B. Schlein,
*Mean-field evolution of fermionic mixed states*, Comm. Pure Appl. Math. 69 (2016), n. 12, 2250–2303.
MR3570479

[9]
N. Benedikter, M. Porta, C. Saffirio and B. Schlein,
*From the Hartree dynamics to the Vlasov equation*,
Arch. Ration. Mech. Anal. 221 (2016), no. 1, 273–334.
MR3483896

[10]
N. Benedikter, M. Porta and B. Schlein,
*Mean-field evolution of fermionic systems*, Comm. Math. Phys 331 (2014), 1087–1131.
MR3248060

[11]
N. Benedikter, M. Porta and B. Schlein,
*Mean-field dynamics of fermions with relativistic dispersion*, J. Math. Phys. 55 (2014), 021901, 10 pp.
MR3202863

[12]
X. Chen and J. Holmer,
*Focusing quantum many-body dynamics: the rigorous derivation of the \(1D\) focusing cubic nonlinear Schrödinger equation*,
Arch. Ration. Mech. Anal. 221 (2016), no. 2, 631–676.
MR3488534

[13]
L. Chen, J. O. Lee and B. Schlein, *Rate of convergence towards Hartree dynamics*,
J. Stat. Phys. 144 (2011), no. 4, 872–903.
MR2826623

[14]
T. Chen, C. Hainzl, N. Pavlović and R. Seiringer,
*Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti*,
Comm. Pure Appl. Math. 68 (2015), no. 10, 1845–1884.
MR3385343

[15]
T. Chen and N. Pavlović,
*The quintic NLS as the mean-field limit of a boson gas with three-body interactions*,
J. Funct. Anal. 260 (2011), no. 4, 959–997.
MR2747009

[16]
A. Elgart, L. Erdős, B. Schlein and H.-T. Yau,
*Nonlinear Hartree equation as the mean field limit of weakly coupled fermions*,
J. Math. Pures Appl. (9) 83 (2004), no. 10, 1241–1273.
MR2092307

[17]
A. Elgart and B. Schlein,
*Mean field dynamics for boson stars*, Comm. Pure Appl. Math. 60 (2007), no. 4, 500–545.
MR2290709

[18]
L. Erdős, B. Schlein and H.-T. Yau,
*Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems*,
Invent. Math. 167 (2007), no. 3, 515–614.
MR2276262

[19]
L. Erdős, B. Schlein and H.-T. Yau,
*Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate*,
Ann. of Math. (2) 172 (2010), no. 1, 291–370.
MR2680421

[20]
L. Erdős, B. Schlein and H.-T. Yau,
*Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential*,
J. Amer. Math. Soc. 22 (2009), 1099–1156.
MR2525781

[21]
L. Erdős and H.-T. Yau,
*Derivation of the nonlinear Schrödinger equation from a many-body Coulomb system*,
Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205.
MR1926667

[22]
J. Fröhlich and A. Knowles,
*A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction*,
J. Stat. Phys. 145 (2011), no. 1, 23–50.
MR2841931

[23]
J. Fröhlich, A. Knowles and A. Pizzo,
*Atomism and quantization*, J. Phys. A 40 (2007), no. 12, 3033–3045.
MR2313859

[24]
J. Fröhlich, A. Knowles and S. Schwarz,
*On the mean-field limit of bosons with Coulomb two-body interaction*,
Comm. Math. Phys. 288 (2009), no. 3, 1023–1059.
MR2504864

[25]
F. Golse, C. Mouhot and T. Paul,
*On the mean field and classical limits of quantum mechanics*, Comm. Math. Phys. 343 (2016), 165–205.
MR3475664

[26]
P. Grech and R. Seiringer, *The excitation spectrum for weakly interacting bosons in a trap*,
Comm. Math. Phys. 322 (2013), no. 2, 559–591.
MR3077925

[27]
J. Ginibre and G. Velo,
*The classical field limit of scattering theory for nonrelativistic many-boson systems, I and II*,
Comm. Math. Phys. 66 (1979), no. 1, 37–76 and 68 (1979), no. 1, 45–68.
MR0530915,
MR0539736

[28]
K. Hepp,
*The classical limit for quantum mechanical correlation functions*,
Comm. Math. Phys. 35 (1974), 265–277.
MR0332046

[29]
K. Kirkpatrick, B. Schlein and G. Staffilani,
*Derivation of the two dimensional nonlinear Schrödinger equation from many-body quantum dynamics*,
Amer. J. Math. 133 (2011), no. 1, 91–130.
MR2752936

[30]
S. Klainerman and M. Machedon,
*On the uniqueness of solutions to the Gross-Pitaevskii hierarchy*,
Comm. Math. Phys. 279 (2008), no. 1, 169–185.
MR2377632

[31]
A. Knowles and P. Pickl,
*Mean-field dynamics: singular potentials and rate of convergence*,
Comm. Math. Phys. 298 (2010), no. 1, 101–138.
MR2657816

[32]
E. H. Lieb and R. Seiringer,
*Proof of Bose-Einstein condensation for dilute trapped gases*,
Phys. Rev. Lett. 88 (2002), 170409.
DOI: https://doi.org/10.1103/PhysRevLett.88.170409

[33]
E. H. Lieb, R. Seiringer and J. Yngvason,
*Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional*,
Phys. Rev. A 61 (2000), 043602.
DOI: https://doi.org/10.1103/PhysRevA.61.043602

[34]
P.-L. Lions and T. Paul,
*Sur les mesures de Wigner* (French), Rev. Mat. Iberoamericana 9 (1993), 553–618.
MR1251718

[35]
H. Narnhofer and G. L. Sewell,
*Vlasov hydrodynamics of a quantum mechanical model*,
Comm. Math. Phys. 79 (1981), no. 1, 9–24.
MR0609224

[36]
S. Petrat and P. Pickl,
*A new method and a new scaling for deriving fermionic mean-field dynamics*,
Math. Phys. Anal. Geom. 19 (2016), no. 1, Art. 3, 51 pp.
MR3461406

[37]
P. Pickl,
*Derivation of the time dependent Gross-Pitaevskii equation with external fields*,
Rev. Math. Phys. 27 (2015), no. 1, 1550003, 45 pp.
MR3317556

[38]
M. Porta, S. Rademacher, C. Saffirio and B. Schlein,
*Mean-field evolution of fermions with Coulomb interaction*,
J. Stat. Phys. 166 (2017), no. 6, 1345–1364.
MR3612230

[39]
I. Rodnianski and B. Schlein,
*Quantum fluctuations and rate of convergence towards mean field dynamics*,
Comm. Math. Phys. 291 (2009), no. 1, 31–61.
MR2530155

[40]
H. Spohn,
*Kinetic equations from Hamiltonian dynamics: Markovian limits*,
Rev. Modern Phys. 52 (1980), no. 3, 569–615.
MR0578142

[41]
H. Spohn,
*On the Vlasov hierarchy*,
Math. Methods Appl. Sci. 3 (1981), no. 4, 445–455.
MR0657065

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