2015, 14(5): 1941-1960. doi: 10.3934/cpaa.2015.14.1941

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

1. 

The College of Information and Technology, Nanjing University of Chinese Medicine, Nanjing 210046, China

Received  November 2014 Revised  March 2015 Published  June 2015

In this paper, we investigate the dynamics of a boson gas with three-body interactions in $T^2$. We prove that when the particle number $N$ tends to infinity, the BBGKY hierarchy of $k$-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized $N$-body initial datum, we show that this $N\rightarrow\infty$ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.
Citation: Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941
References:
[1]

R. Adami, C. Bardos, F. Golse and A. Teta, Towards a rigorous derivation of the cubic nonlinear Schrodinger equation in dimension one,, \emph{Asymptot. Anal.}, 40 (2004), 93.

[2]

R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one,, \emph{J. Stat. Phys.}, 127 (2007), 1193. doi: 10.1007/s10955-006-9271-z.

[3]

E. Bombieri and J. Pila, The number of integral points on arcs and ovals,, \emph{Duke Math. J.}, 59 (1989), 337. doi: 10.1215/S0012-7094-89-05915-2.

[4]

Thomas Chen and Natasa Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies,, \emph{Discr. Contin. Dyn. Syst. A}, 27 (2010), 715. doi: 10.3934/dcds.2010.27.715.

[5]

Thomas Chen and Natasa Pavlović, The quintic NLS as the mean field limit of a Boson gas with three-body interactions,, \emph{J. Funct. Anal.}, 260 (2011), 959. doi: 10.1016/j.jfa.2010.11.003.

[6]

Thomas Chen and Natasa Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies,, \emph{Proc. Amer. Math. Soc.}, 141 (2013), 279. doi: 10.1090/S0002-9939-2012-11308-5.

[7]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms,, \emph{Ann. H. Poincare}, 15 (2014), 543. doi: 10.1007/s00023-013-0248-6.

[8]

T. Chen and K. Taliaferro, Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy,, \emph{Commun. PDE}, 39 (2014), 1658. doi: 10.1080/03605302.2014.917380.

[9]

T. Chen, C. Hainzl, N. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti,, \emph{CPAM}, ().

[10]

X. Chen, Second order corrections to mean field evolution for weakly interacting Bosons in the case of three-body interactions,, \emph{Arch. Rational Mech. Anal.}, 203 (2012), 455. doi: 10.1007/s00205-011-0453-8.

[11]

X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps,, \emph{J. Math. Pures Appl.}, 98 (2012), 450. doi: 10.1016/j.matpur.2012.02.003.

[12]

X. Chen, On the rigorous derivation of the 3d cubic nonlinear Schrödinger equation with a quadratic trap,, \emph{Arch. Rational Mech. Anal.}, 210 (2013), 365. doi: 10.1007/s00205-013-0645-5.

[13]

X. Chen and J. Holmer, On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics,, \emph{Arch. Rational Mech. Anal.}, 210 (2013), 909. doi: 10.1007/s00205-013-0667-z.

[14]

X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation,, 41pp, ().

[15]

X. Chen and J. Holmer, Focusing quantum many-body dynamics II: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation from 3D,, 48pp, ().

[16]

X. Chen and J. Holmer, On the Klainerman-Machedon onjecture of the quantum BBGKY hierarchy with self-interaction,, preprint, ().

[17]

X. Chen and J. Holmer, Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy,, 48pp, ().

[18]

X. Chen and P. Smith, On the unconditional uniqueness of solutions to the infinite radial Chern- Simons-Schrödinger hierarchy,, \emph{Analysis and PDE}, 7 (2014), 1683. doi: 10.2140/apde.2014.7.1683.

[19]

D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrodinger equation in 1D and 2D,, \emph{Discrete and Continuous Dynamical Systems-Series A}, 19 (2007), 37. doi: 10.3934/dcds.2007.19.37.

[20]

A. Elgart, L. Erdos, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons,, \emph{Arch. Rat. Mech. Anal.}, 179 (2006), 265. doi: 10.1007/s00205-005-0388-z.

[21]

A. Elgart and B. Schlein, Mean field dynamics of Boson stars,, \emph{Commun. Pure Appl. Math.}, 60 (2007), 500. doi: 10.1002/cpa.20134.

[22]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems,, \emph{Invent. Math.}, 167 (2007), 515. doi: 10.1007/s00222-006-0022-1.

[23]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate,, \emph{Ann. of Math.}, 172 (2010), 291. doi: 10.4007/annals.2010.172.291.

[24]

L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential,, \emph{J. Amer. Math. Soc.}, 22 (2009), 1099. doi: 10.1090/S0894-0347-09-00635-3.

[25]

P. Gressman, V. Sohinger and G. Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy,, \emph{J. Funct. Anal.}, 266 (2014), 4705. doi: 10.1016/j.jfa.2014.02.006.

[26]

M. G. Grillakis and M. Machedon, Pair excitations and the mean .eld approximation of interacting Bosons, I,, \emph{Commun. Math. Phys.}, 324 (2013), 601. doi: 10.1007/s00220-013-1818-7.

[27]

M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. I,, \emph{Commun. Math. Phys.}, 294 (2010), 273. doi: 10.1007/s00220-009-0933-y.

[28]

M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. II,, \emph{Adv. Math.}, 228 (2011), 1788. doi: 10.1016/j.aim.2011.06.028.

[29]

Y. Hong, K. Taliaferro and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity,, 26pp, ().

[30]

Kay Kirkpatrick, Benjamin Schlein and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics,, \emph{American Journal of Mathematics}, 133 (2011), 91. doi: 10.1353/ajm.2011.0004.

[31]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy,, \emph{Comm. Math. Phys.}, 279 (2008), 169. doi: 10.1007/s00220-008-0426-4.

[32]

M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems,, \emph{Adv. Math.}, 254 (2014), 570. doi: 10.1016/j.aim.2013.12.010.

[33]

E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases,, \emph{Phys. Rev. Lett.}, 88 (2002), 1.

[34]

E. H. Lieb and R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas,, \emph{Comm. Math. Phys.}, 224 (2001), 17. doi: 10.1007/s002200100533.

[35]

E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation,, \textbf{34} (2005), 34 (2005).

[36]

I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics,, \emph{Comm. Math. Phys.}, 291 (2009), 31. doi: 10.1007/s00220-009-0867-4.

[37]

L. Pitaevskii, Vortex lines in an imperfect Bose-gas,, \emph{Sov. phys. JETP}, 13 (1961), 451.

[38]

P. Pickl, A simple derivation of mean field limits for quantum systems,, \emph{Lett. Math. Phys.}, 97 (2011), 151. doi: 10.1007/s11005-011-0470-4.

[39]

H. Spohn, Kinetic equations from Hamiltonian dynamics,, \emph{Rev. Mod. Phys.}, 52 (1980), 569. doi: 10.1103/RevModPhys.52.569.

[40]

V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbbT^3$ from the dynamics of many-body quantum systems,, \emph{Ann. I.H.Poincar\'e-AN}., (). doi: 10.1016/j.anihpc.2014.09.005.

show all references

References:
[1]

R. Adami, C. Bardos, F. Golse and A. Teta, Towards a rigorous derivation of the cubic nonlinear Schrodinger equation in dimension one,, \emph{Asymptot. Anal.}, 40 (2004), 93.

[2]

R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one,, \emph{J. Stat. Phys.}, 127 (2007), 1193. doi: 10.1007/s10955-006-9271-z.

[3]

E. Bombieri and J. Pila, The number of integral points on arcs and ovals,, \emph{Duke Math. J.}, 59 (1989), 337. doi: 10.1215/S0012-7094-89-05915-2.

[4]

Thomas Chen and Natasa Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies,, \emph{Discr. Contin. Dyn. Syst. A}, 27 (2010), 715. doi: 10.3934/dcds.2010.27.715.

[5]

Thomas Chen and Natasa Pavlović, The quintic NLS as the mean field limit of a Boson gas with three-body interactions,, \emph{J. Funct. Anal.}, 260 (2011), 959. doi: 10.1016/j.jfa.2010.11.003.

[6]

Thomas Chen and Natasa Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies,, \emph{Proc. Amer. Math. Soc.}, 141 (2013), 279. doi: 10.1090/S0002-9939-2012-11308-5.

[7]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms,, \emph{Ann. H. Poincare}, 15 (2014), 543. doi: 10.1007/s00023-013-0248-6.

[8]

T. Chen and K. Taliaferro, Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy,, \emph{Commun. PDE}, 39 (2014), 1658. doi: 10.1080/03605302.2014.917380.

[9]

T. Chen, C. Hainzl, N. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti,, \emph{CPAM}, ().

[10]

X. Chen, Second order corrections to mean field evolution for weakly interacting Bosons in the case of three-body interactions,, \emph{Arch. Rational Mech. Anal.}, 203 (2012), 455. doi: 10.1007/s00205-011-0453-8.

[11]

X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps,, \emph{J. Math. Pures Appl.}, 98 (2012), 450. doi: 10.1016/j.matpur.2012.02.003.

[12]

X. Chen, On the rigorous derivation of the 3d cubic nonlinear Schrödinger equation with a quadratic trap,, \emph{Arch. Rational Mech. Anal.}, 210 (2013), 365. doi: 10.1007/s00205-013-0645-5.

[13]

X. Chen and J. Holmer, On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics,, \emph{Arch. Rational Mech. Anal.}, 210 (2013), 909. doi: 10.1007/s00205-013-0667-z.

[14]

X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation,, 41pp, ().

[15]

X. Chen and J. Holmer, Focusing quantum many-body dynamics II: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation from 3D,, 48pp, ().

[16]

X. Chen and J. Holmer, On the Klainerman-Machedon onjecture of the quantum BBGKY hierarchy with self-interaction,, preprint, ().

[17]

X. Chen and J. Holmer, Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy,, 48pp, ().

[18]

X. Chen and P. Smith, On the unconditional uniqueness of solutions to the infinite radial Chern- Simons-Schrödinger hierarchy,, \emph{Analysis and PDE}, 7 (2014), 1683. doi: 10.2140/apde.2014.7.1683.

[19]

D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrodinger equation in 1D and 2D,, \emph{Discrete and Continuous Dynamical Systems-Series A}, 19 (2007), 37. doi: 10.3934/dcds.2007.19.37.

[20]

A. Elgart, L. Erdos, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons,, \emph{Arch. Rat. Mech. Anal.}, 179 (2006), 265. doi: 10.1007/s00205-005-0388-z.

[21]

A. Elgart and B. Schlein, Mean field dynamics of Boson stars,, \emph{Commun. Pure Appl. Math.}, 60 (2007), 500. doi: 10.1002/cpa.20134.

[22]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems,, \emph{Invent. Math.}, 167 (2007), 515. doi: 10.1007/s00222-006-0022-1.

[23]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate,, \emph{Ann. of Math.}, 172 (2010), 291. doi: 10.4007/annals.2010.172.291.

[24]

L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential,, \emph{J. Amer. Math. Soc.}, 22 (2009), 1099. doi: 10.1090/S0894-0347-09-00635-3.

[25]

P. Gressman, V. Sohinger and G. Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy,, \emph{J. Funct. Anal.}, 266 (2014), 4705. doi: 10.1016/j.jfa.2014.02.006.

[26]

M. G. Grillakis and M. Machedon, Pair excitations and the mean .eld approximation of interacting Bosons, I,, \emph{Commun. Math. Phys.}, 324 (2013), 601. doi: 10.1007/s00220-013-1818-7.

[27]

M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. I,, \emph{Commun. Math. Phys.}, 294 (2010), 273. doi: 10.1007/s00220-009-0933-y.

[28]

M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. II,, \emph{Adv. Math.}, 228 (2011), 1788. doi: 10.1016/j.aim.2011.06.028.

[29]

Y. Hong, K. Taliaferro and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity,, 26pp, ().

[30]

Kay Kirkpatrick, Benjamin Schlein and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics,, \emph{American Journal of Mathematics}, 133 (2011), 91. doi: 10.1353/ajm.2011.0004.

[31]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy,, \emph{Comm. Math. Phys.}, 279 (2008), 169. doi: 10.1007/s00220-008-0426-4.

[32]

M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems,, \emph{Adv. Math.}, 254 (2014), 570. doi: 10.1016/j.aim.2013.12.010.

[33]

E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases,, \emph{Phys. Rev. Lett.}, 88 (2002), 1.

[34]

E. H. Lieb and R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas,, \emph{Comm. Math. Phys.}, 224 (2001), 17. doi: 10.1007/s002200100533.

[35]

E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation,, \textbf{34} (2005), 34 (2005).

[36]

I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics,, \emph{Comm. Math. Phys.}, 291 (2009), 31. doi: 10.1007/s00220-009-0867-4.

[37]

L. Pitaevskii, Vortex lines in an imperfect Bose-gas,, \emph{Sov. phys. JETP}, 13 (1961), 451.

[38]

P. Pickl, A simple derivation of mean field limits for quantum systems,, \emph{Lett. Math. Phys.}, 97 (2011), 151. doi: 10.1007/s11005-011-0470-4.

[39]

H. Spohn, Kinetic equations from Hamiltonian dynamics,, \emph{Rev. Mod. Phys.}, 52 (1980), 569. doi: 10.1103/RevModPhys.52.569.

[40]

V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbbT^3$ from the dynamics of many-body quantum systems,, \emph{Ann. I.H.Poincar\'e-AN}., (). doi: 10.1016/j.anihpc.2014.09.005.

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