×

Tracer particles coupled to an interacting boson gas. (English) Zbl 1485.81069

Summary: In this work, we investigate the mean-field limit of a model consisting of \(m\geqslant 1\) tracer particles, coupled to an interacting boson field. We assume the mass of the tracer particles and the expected number of bosons to be of the same order of magnitude \(N\geqslant 1\) and we investigate the \(N\to\infty\) limit. In particular, we show that the limiting system can be effectively described by a pair of variables \((\mathbf{X}_t,\varphi_t)\in\mathbb{R}^{3m}\times H^1(\mathbb{R}^3)\) that solve a mean-field equation. Our methods are based on proving estimates for the number of bosonic particles in a suitable fluctuation state \(\Omega_{N,t}\). The main difficulty of the problem comes from the fact that the interaction with the tracer particles can create or destroy bosons for states close to the vacuum.

MSC:

81V73 Bosonic systems in quantum theory
82D05 Statistical mechanics of gases
81T70 Quantization in field theory; cohomological methods
81T28 Thermal quantum field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Cazenave, T.; Haraux, A., An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications, vol. 13 (1998), Oxford University Press · Zbl 0926.35049
[2] Chen, T.; Hainzl, C.; Pavlović, N.; Seiringer, R., Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, Commun. Pure Appl. Math., 68, 10, 1845-1884 (2015) · Zbl 1326.35332
[3] Chen, X.; Holmer, J., On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc., 18, 6, 1161-1200 (2016) · Zbl 1342.35322
[4] Chen, X.; Holmer, J., Correlation structures, many-body scattering processes, and the derivation of the Gross-Pitaevskii hierarchy, Int. Math. Res. Not., 2016: 10, 3051-3110 (2016) · Zbl 1404.35407
[5] Chen, T.; Pavlović, N., On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discrete Contin. Dyn. Syst., 27, 2, 715-739 (2010) · Zbl 1190.35207
[6] Chen, T.; Pavlović, N., Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in \(d = 3\) based on spacetime norms, Ann. Henri Poincaré, 15, 3, 543-588 (2014) · Zbl 1338.35406
[7] Chen, T.; Soffer, A., Mean field dynamics of a quantum tracer particle interacting with a boson gas, J. Funct. Anal., 276, 3, 971-1006 (2019) · Zbl 1414.82026
[8] Deckert, D.-A.; Fröhlich, J.; Pickl, P.; Pizzo, A., Effective dynamics of a tracer particle interacting with an ideal Bose gas, Commun. Math. Phys., 328, 2, 597-624 (2014) · Zbl 1391.81074
[9] Erdös, L.; Schlein, B.; Yau, H.-T., Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Commun. Pure Appl. Math., 59, 12, 1659-1741 (2006) · Zbl 1122.82018
[10] Erdös, L.; Schlein, B.; Yau, H.-T., Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167, 515-614 (2007) · Zbl 1123.35066
[11] Erdös, L.; Schlein, B.; Yau, H.-T., Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Am. Math. Soc., 22, 4, 1099-1156 (2009) · Zbl 1207.82031
[12] Erdös, L.; Schlein, B.; Yau, H.-T., Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensates, Ann. Math. (2), 172, 1, 291-370 (2010) · Zbl 1204.82028
[13] Fröhlich, J.; Gang, Z., Ballistic motion of a tracer particle coupled to a Bose gas, Adv. Math., 259, 252-268 (2014) · Zbl 1311.82032
[14] Fröhlich, J.; Gang, Z., Emission of Cherenkov radiation as a mechanism for Hamiltonian friction, Adv. Math., 264, 183-235 (2014) · Zbl 1305.82043
[15] Fröhlich, J.; Gang, Z.; Soffer, A., Friction in a model of Hamiltonian dynamics, Commun. Math. Phys., 315, 2, 401-444 (2012) · Zbl 1263.82033
[16] Ginibre, J.; Velo, G., The classical field limit of scattering theory for nonrelativistic many-boson systems. I, Commun. Math. Phys., 66, 37-76 (1979) · Zbl 0443.35067
[17] Ginibre, J.; Velo, G., The classical field limit of scattering theory for non-relativistic many-boson systems. II, Commun. Math. Phys., 68, 45-68 (1979) · Zbl 0443.35068
[18] Gressman, P.; Sohinger, V.; Staffilani, G., On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal., 266, 7, 4705-4764 (2014) · Zbl 1297.35215
[19] Grillakis, M.; Machedon, M.; Margetis, A., Second-order corrections to mean field evolution for weakly interacting bosons. I, Commun. Math. Phys., 294, 1, 273-301 (2010) · Zbl 1208.82030
[20] Grillakis, M.; Machedon, M., Pair excitations and the mean field approximation of interacting bosons, I, Commun. Math. Phys., 324, 2, 601-636 (2013) · Zbl 1277.82034
[21] Grillakis, M.; Machedon, M., Pair excitations and the mean field approximation of interacting bosons, II, Commun. Partial Differ. Equ., 42, 1, 24-67 (2017) · Zbl 1371.35232
[22] Hepp, K., The classical limit for quantum mechanical correlation functions, Commun. Math. Phys., 35, 265-277 (1974)
[23] Hott, M., Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev trace norms, Rev. Math. Phys., 33, 9, Article 2150029 pp. (2022) · Zbl 1476.35214
[24] Kato, T., Linear evolution equations of “hyperbolic” type, II, J. Math. Soc. Jpn., 25, 4, 648-666 (1973) · Zbl 0262.34048
[25] Kirkpatrick, K.; Schlein, B.; Staffilani, G., Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics, Am. J. Math., 133, 1, 91-130 (2011) · Zbl 1208.81080
[26] Klainerman, S.; Machedon, M., On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Commun. Math. Phys., 279, 1, 169-185 (2008) · Zbl 1147.37034
[27] Lampart, J.; Pickl, P., Dynamics of a tracer particle interacting with excitations of a Bose-Einstein condensate · Zbl 1496.82015
[28] Lenzmann, E., Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10, 1, 43-64 (2007) · Zbl 1171.35474
[29] Lewin, M.; Nam, P. T.; Rougerie, N., Derivation of Hartree’s theory for generic mean-field Bose systems, Adv. Math., 254, 570-621 (2014) · Zbl 1316.81095
[30] Lewin, M.; Nam, P. T.; Schlein, B., Fluctuations around Hartree states in the mean-field regime, Am. J. Math., 137, 6, 1613-1650 (2015) · Zbl 1329.81430
[31] Pickl, P., A simple derivation of mean field limits for quantum systems, Lett. Math. Phys., 97, 2, 151-164 (2011) · Zbl 1242.81150
[32] Reed, M.; Simon, B., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Methods of Modern Mathematical Physics (1975), Elsevier Science · Zbl 0308.47002
[33] Rodnianski, I.; Schlein, B., Quantum fluctuations and rate of convergence towards mean field dynamics, Commun. Math. Phys., 291, 1, 31-61 (2009) · Zbl 1186.82051
[34] Spohn, H., Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52, 3, 569-615 (1980)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.