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June  2022, 42(6): 3005-3037. doi: 10.3934/dcds.2022006

The mean-field limit of the Lieb-Liniger model

MIT, Department of Mathematics, 77 Massachusetts Ave, Cambridge, MA 02139-4307, USA

*Corresponding author: Matthew Rosenzweig

Received  July 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

Fund Project: The author acknowledges financial support from the University of Texas at Austin and the Simons Collaboration on Wave Turbulence

We consider the well-known Lieb-Liniger (LL) model for $ N $ bosons interacting pairwise on the line via the $ \delta $ potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the $ \delta $ potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the $ N $-body wave function in a single particle variable. By further exploiting the $ L^2 $-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of $ N $-body initial states.

Citation: Matthew Rosenzweig. The mean-field limit of the Lieb-Liniger model. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3005-3037. doi: 10.3934/dcds.2022006
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show all references

References:
[1]

R. AdamiC. BardosF. Golse and A. Teta, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal., 40 (2004), 93-108. 

[2]

R. AdamiF. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys., 127 (2007), 1193-1220.  doi: 10.1007/s10955-006-9271-z.

[3]

Z. Ammari and S. Breteaux, Propagation of chaos for many-boson systems in one dimension with a point pair-interaction, Asymptot. Anal., 76 (2012), 123-170.  doi: 10.3233/ASY-2011-1064.

[4]

C. BardosF. Golse and N. J. Mauser, Weak coupling limit of the $N$-particle Schrödinger equation, Methods Appl. Anal., 7 (2000), 275-293.  doi: 10.4310/MAA.2000.v7.n2.a2.

[5]

N. Benedikter, M. Porta and B. Schlein, Effective Evolution Equations from Quantum Dynamics, SpringerBriefs in Mathematical Physics, 7. Springer, Cham, 2016. doi: 10.1007/978-3-319-24898-1.

[6]

H. Bethe, Zur theorie der metalle, Zeitschrift Für Physik, 71 (1931), 205-226.  doi: 10.1007/BF01341708.

[7]

L. Boßmann, Derivation of the 1d nonlinear Schrödinger equation from the 3d quantum many-body dynamics of strongly confined bosons, J. Math. Phys., 60 (2019), 031902, 30. doi: 10.1063/1.5075514.

[8]

L. BoßmannN. PavlovićP. Pickl and A. Soffer, Higher order corrections to the mean-field description of the dynamics of interacting Bosons, J. Stat. Phys., 178 (2020), 1362-1396.  doi: 10.1007/s10955-020-02500-8.

[9]

L. Boßmann and S. Teufel, Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons, Ann. Henri Poincaré, 20 (2019), 1003-1049.  doi: 10.1007/s00023-018-0738-7.

[10]

C. Brennecke and B. Schlein, Gross-Pitaevskii dynamics for Bose-Einstein condensates, Anal. PDE, 12 (2019), 1513-1596.  doi: 10.2140/apde.2019.12.1513.

[11]

R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), 937-964.  doi: 10.4310/CMS.2011.v9.n4.a1.

[12]

T. ChenC. HainzlN. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, Comm. Pure Appl. Math., 68 (2015), 1845-1884.  doi: 10.1002/cpa.21552.

[13]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588.  doi: 10.1007/s00023-013-0248-6.

[14]

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), 631-676.  doi: 10.1007/s00205-016-0970-6.

[15]

X. Chen and J. Holmer, The derivation of the $\Bbb T^3$ energy-critical NLS from quantum many-body dynamics, Invent. Math., 217 (2019), 433-547.  doi: 10.1007/s00222-019-00868-3.

[16]

J. Chong, Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing NLS in $\mathbb{R}^3$, J. Math. Phys., 62 (2021), Paper No. 042106, 38 pp. doi: 10.1063/1.5099113.

[17]

S. DettmerD. HellwegP. RyyttyJ. J. ArltW. ErtmerK. SengstockD. S. PetrovG. V. ShlyapnikovH. KreutzmannL. Santos and M. Lewenstein, Observation of phase fluctuations in elongated bose-einstein condensates, Phys. Rev. Lett., 87 (2001), 160406. 

[18]

V. DunjkoV. Lorent and M. Olshanii, Bosons in cigar-shaped traps: Thomas-fermi regime, tonks-girardeau regime, and in between, Phys. Rev. Lett., 86 (2001), 5413-5416. 

[19]

L. ErdösB. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math., 59 (2006), 1659-1741.  doi: 10.1002/cpa.20123.

[20]

L. ErdösB. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614.  doi: 10.1007/s00222-006-0022-1.

[21]

L. ErdösB. 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.  doi: 10.1090/S0894-0347-09-00635-3.

[22]

L. ErdösB. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370.  doi: 10.4007/annals.2010.172.291.

[23]

L. Erdös and H.-T. Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys., 5 (2001), 1169-1205.  doi: 10.4310/ATMP.2001.v5.n6.a6.

[24]

J. EsteveJ. B. TrebbiaT. SchummA. AspectC. I. Westbrook and I. Bouchoule, Observations of density fluctuations in an elongated bose gas: Ideal gas and quasicondensate regimes, Prl, 96 (2006), 130403. 

[25]

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, English edition, Classics in Mathematics, Springer, Berlin, 2007.

[26]

J. Fröhlich, T.-P. Tsai and H.-T. Yau, On a classical limit of quantum theory and the non-linear Hartree equation, GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal., Special Volume, Part Ⅰ, (2000), 57–78. doi: 10.1007/978-3-0346-0422-2_3.

[27]

M. Gaudin, The Bethe Wavefunction, Cambridge University Press, New York, 2014, Translated from the 1983 French original by Jean-Sébastien Caux. doi: 10.1017/CBO9781107053885.

[28]

J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. Ⅰ, Comm. Math. Phys., 66 (1979), 37-76.  doi: 10.1007/BF01197745.

[29]

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Table 1.  Notation
Symbol Definition
$ A\lesssim B, \ A\sim B $ There are absolute constants $ C_1, C_2>0 $ such that $ A\leq C_1 B $ or $ C_2B\leq A\leq C_1 B $; subscript $ p $ (e.g. $ A\lesssim_p B $) denotes dependence on parameter $ p $
$ \underline{x}_{k}, \ \underline{x}_{i;i+k} $ $ (x_1, \ldots, x_k), \ (x_i, \ldots, x_{i+k}) $, where $ x_j\in {\mathbb{R}} $ for $ j\in\left\{ {1, \ldots, k} \right\} $ or $ j\in\left\{ {i, \ldots, i+k} \right\} $
$ d \underline{x}_{k}, \ d \underline{x}_{i;i+k} $ $ dx_1 \cdots dx_k, \ dx_i \cdots dx_{i+k} $
$ {\mathbb{N}}, \ {\mathbb{N}}_0 $ natural numbers, natural numbers inclusive of zero
$ \left\langle {\cdot}|{\cdot} \right\rangle $ $ L^2( {\mathbb{R}}^N) $ inner product with physicist's convention: $\left\langle {f}|{g}\right\rangle := \int_{ {\mathbb{R}}^N}d \underline{x}_N \overline{f( \underline{x}_N)}g( \underline{x}_N) $
$ \left\langle \cdot , \cdot \right\rangle $ duality pairing
$ \left\langle {{\cdot}} \right| \ \left| {{\cdot}} \right\rangle $ Dirac's bra-ket notation: see footnote 2
$ A_{i_1\cdots i_k}^{(k)} $ subscript denotes that the operator on $ L^2( {\mathbb{R}}^N) $ acts in the variables $ (x_{i_1}, \ldots, x_{i_k}) $
$ \varphi^{\otimes k} $ $ k $-fold tensor product of $ \varphi $ with itself realized as $ \varphi^{\otimes k}( \underline{x}_k) = \prod_{i=1}^k \varphi(x_i), \ \underline{x}_k\in {\mathbb{R}}^k $
$ {\rm{Tr}}_{1, \ldots, N} $ trace on $ L^2( {\mathbb{R}}^N) $
$ {\rm{Tr}}_{k+1, \ldots, N} $ partial trace on $ L^2( {\mathbb{R}}^N) $ over $ x_{k+1}, \ldots, x_N $ coordinates
$ {\bf{1}}, \ {\bf{1}}_N $ identity operator on $ L^2( {\mathbb{R}}) $ and on $ L^2( {\mathbb{R}}^N) $
$ H_{N}, \ H_{N, \varepsilon} $ LL Hamiltonian and regularized LL Hamiltonian: see (1.2) and (1.15)
$ p_j, \ q_j $ projectors $ {\bf{1}}^{\otimes j-1}\otimes p \otimes {\bf{1}}^{N-j}, \ {\bf{1}}^{\otimes j-1} \otimes q \otimes {\bf{1}}^{N-j} $: see (3.2)
$ P_k $ projector onto subspace of $ k $ particles not in the state $ \varphi(t) $: see (3.4)
$ \widehat{f}, \ \widehat{f}^{-1} $ operator $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $ defined by $ \widehat{f} := \sum_{k=0}^N f(k)P_k $, for $ f: {\mathbb{Z}}\rightarrow {\mathbb{C}} $: see (3.6)
$ n_N, m_N $ $ \widehat{n_N}, \widehat{m_N} $ functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see 3.1
$ \mu, \nu $ $ \widehat{\mu}, \widehat{\nu} $ functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see (3.33) and (3.52)
$ \beta_N $ time-dependent functional of solution $ \varphi $ to (1.8) and $ \Phi_N $ to (1.3): see 3.1
$ \tau_n $ shift operator on $ {\mathbb{C}}^{ {\mathbb{Z}}} $: see (3.16)
$ \underline{\Delta}_k $ Laplacian on $ {\mathbb{R}}^k $: $ \underline{\Delta}_k := \sum_{i=1}^k \Delta_i $
$ \left[ \cdot , \cdot \right]$ commutator bracket: $\left[ A, B \right] := AB-BA $
Symbol Definition
$ A\lesssim B, \ A\sim B $ There are absolute constants $ C_1, C_2>0 $ such that $ A\leq C_1 B $ or $ C_2B\leq A\leq C_1 B $; subscript $ p $ (e.g. $ A\lesssim_p B $) denotes dependence on parameter $ p $
$ \underline{x}_{k}, \ \underline{x}_{i;i+k} $ $ (x_1, \ldots, x_k), \ (x_i, \ldots, x_{i+k}) $, where $ x_j\in {\mathbb{R}} $ for $ j\in\left\{ {1, \ldots, k} \right\} $ or $ j\in\left\{ {i, \ldots, i+k} \right\} $
$ d \underline{x}_{k}, \ d \underline{x}_{i;i+k} $ $ dx_1 \cdots dx_k, \ dx_i \cdots dx_{i+k} $
$ {\mathbb{N}}, \ {\mathbb{N}}_0 $ natural numbers, natural numbers inclusive of zero
$ \left\langle {\cdot}|{\cdot} \right\rangle $ $ L^2( {\mathbb{R}}^N) $ inner product with physicist's convention: $\left\langle {f}|{g}\right\rangle := \int_{ {\mathbb{R}}^N}d \underline{x}_N \overline{f( \underline{x}_N)}g( \underline{x}_N) $
$ \left\langle \cdot , \cdot \right\rangle $ duality pairing
$ \left\langle {{\cdot}} \right| \ \left| {{\cdot}} \right\rangle $ Dirac's bra-ket notation: see footnote 2
$ A_{i_1\cdots i_k}^{(k)} $ subscript denotes that the operator on $ L^2( {\mathbb{R}}^N) $ acts in the variables $ (x_{i_1}, \ldots, x_{i_k}) $
$ \varphi^{\otimes k} $ $ k $-fold tensor product of $ \varphi $ with itself realized as $ \varphi^{\otimes k}( \underline{x}_k) = \prod_{i=1}^k \varphi(x_i), \ \underline{x}_k\in {\mathbb{R}}^k $
$ {\rm{Tr}}_{1, \ldots, N} $ trace on $ L^2( {\mathbb{R}}^N) $
$ {\rm{Tr}}_{k+1, \ldots, N} $ partial trace on $ L^2( {\mathbb{R}}^N) $ over $ x_{k+1}, \ldots, x_N $ coordinates
$ {\bf{1}}, \ {\bf{1}}_N $ identity operator on $ L^2( {\mathbb{R}}) $ and on $ L^2( {\mathbb{R}}^N) $
$ H_{N}, \ H_{N, \varepsilon} $ LL Hamiltonian and regularized LL Hamiltonian: see (1.2) and (1.15)
$ p_j, \ q_j $ projectors $ {\bf{1}}^{\otimes j-1}\otimes p \otimes {\bf{1}}^{N-j}, \ {\bf{1}}^{\otimes j-1} \otimes q \otimes {\bf{1}}^{N-j} $: see (3.2)
$ P_k $ projector onto subspace of $ k $ particles not in the state $ \varphi(t) $: see (3.4)
$ \widehat{f}, \ \widehat{f}^{-1} $ operator $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $ defined by $ \widehat{f} := \sum_{k=0}^N f(k)P_k $, for $ f: {\mathbb{Z}}\rightarrow {\mathbb{C}} $: see (3.6)
$ n_N, m_N $ $ \widehat{n_N}, \widehat{m_N} $ functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see 3.1
$ \mu, \nu $ $ \widehat{\mu}, \widehat{\nu} $ functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see (3.33) and (3.52)
$ \beta_N $ time-dependent functional of solution $ \varphi $ to (1.8) and $ \Phi_N $ to (1.3): see 3.1
$ \tau_n $ shift operator on $ {\mathbb{C}}^{ {\mathbb{Z}}} $: see (3.16)
$ \underline{\Delta}_k $ Laplacian on $ {\mathbb{R}}^k $: $ \underline{\Delta}_k := \sum_{i=1}^k \Delta_i $
$ \left[ \cdot , \cdot \right]$ commutator bracket: $\left[ A, B \right] := AB-BA $
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