April  2019, 39(4): 1891-1921. doi: 10.3934/dcds.2019080

Asymptotic expansion of the mean-field approximation

1. 

CMLS, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France

2. 

International Research Center on the Mathematics and Mechanics of Complex Systems, MeMoCS, University of L'Aquila, Italy

Received  January 2018 Revised  October 2018 Published  January 2019

We consider the $ N $-body quantum evolution of a particle system in the mean-field approximation. We show that the $ j $th order marginals $ F^N_j(t) $, for factorized initial data $ F(0)^{\otimes N} $, are explicitly expressed, modulo $ N^{-\infty} $, out of the solution $ F(t) $ of the corresponding non-linear mean-field equation and the solution of its linearization around $ F(t) $. The result is valid for all times $ t $, uniformly in $ j = O(N^{\frac12-\alpha}) $ for any $ \alpha>0 $. We establish and estimate the full asymptotic expansion in integer powers of $ \frac1N $ of $ F^N_j(t) $, $ j = O(\sqrt N) $, whose computation at order $ n $ involves a finite number of operations depending on $ j $ and $ n $ but not on $ N $. Our results are also valid for more general models including Kac models. As a by-product we get that the rate of convergence to the mean-field limit in $ \frac1N $ is optimal in the sense that the first correction to the mean-field limit does not vanish.

Citation: Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
References:
[1]

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

[2]

H. van BeijerenO. E. LandfordJ. L. Lebowitz and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Stat. Phys., 22 (1980), 237-257.  doi: 10.1007/BF01008050.  Google Scholar

[3]

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

[4]

C. BoldrighiniA. De Masi and A. Pellegrinotti, Non equilibrium fluctuations in particle systems modelling Reaction-Diffusion equations, Stochastic Processes and Appl., 42 (1992), 1-30.  doi: 10.1016/0304-4149(92)90023-J.  Google Scholar

[5]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Commun. Math. Phys., 56 (1977), 101-113.  doi: 10.1007/BF01611497.  Google Scholar

[6]

S. Caprino and M. Pulvirenti, A cluster expansion approach to a one-dimensional Boltzmann equation: a validity result, Comm. Math. Phys., 166 (1995), 603-631.  doi: 10.1007/BF02099889.  Google Scholar

[7]

S. CaprinoA. De MasiE. Presutti and M. Pulvirenti, A derivation of the Broadwell equation, Comm. Math. Phys., 135 (1991), 443-465.  doi: 10.1007/BF02104115.  Google Scholar

[8]

S. CaprinoM. Pulvirenti and W. Wagner, A particle systems approximating stationary solutions to the Boltzmann equation, SIAM J. Math. Anal., 29 (1998), 913-934.  doi: 10.1137/S0036141096309988.  Google Scholar

[9]

C. Cercignani, The Grad limit for a system of soft spheres, Comm. Pure Appl. Math., 36 (1983), 479-494.  doi: 10.1002/cpa.3160360406.  Google Scholar

[10]

A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamical Limits, Lecture Notes in Mathematics 1501, Springer-Verlag, 1991. doi: 10.1007/BFb0086457.  Google Scholar

[11]

A. De MasiE. OrlandiE. Presutti and L. Triolo, Glauber evolution with Kac potentials. Ⅰ.Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity, 7 (1994), 633-696.  doi: 10.1088/0951-7715/7/3/001.  Google Scholar

[12]

A. De MasiE. OrlandiE. Presutti and L. Triolo, Glauber evolution with Kac potentials. Ⅱ. Fluctuations, Nonlinearity, 9 (1996), 27-51.  doi: 10.1088/0951-7715/9/1/002.  Google Scholar

[13]

A. De MasiE. PresuttiD. Tsagkarogiannis and M. E. Vares, Truncated correlations in the stirring process with births and deaths, Electronic Journal of Probability, 17 (2012), 1-35.  doi: 10.1214/EJP.v17-1734.  Google Scholar

[14]

F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Rational Mech. Anal., 223 (2017), 57-94.  doi: 10.1007/s00205-016-1031-x.  Google Scholar

[15]

C. Graham and S. Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Annals of Probability, 25 (1997), 115-132.  doi: 10.1214/aop/1024404281.  Google Scholar

[16]

K. Hepp and E. H. Lieb, Phase transitions in reservoir-driven open systems with applications to lasers and superconductors, Helv. Phys, Acta, 46 (1973), 573. Google Scholar

[17]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, 3 (1956), 171-197.  Google Scholar

[18]

M. Kac, Probability and Related Topics in Physical Sciences, Interscience, London-New York, 1959.  Google Scholar

[19]

A. Knowles and P. Pickl, Mean-field dynamics: Singular potentials and rate of convergence, Com. Math.Physics, 298 (2010), 101-138.  doi: 10.1007/s00220-010-1010-2.  Google Scholar

[20]

M Lachowicz and M Pulvirenti, A stochastic system of particles modelling the Euler equation, Arch. Ration. Mech. Anal., 109 (1990), 81-93.  doi: 10.1007/BF00377981.  Google Scholar

[21]

S. Lang, Algebra, Springer, 2002. Google Scholar

[22]

M. LewinP. T. NamS. Serfaty and J. P. Solovej, Bogoliubov spectrum of interacting Bose gases, Commun. Pur. Appl. Math., 68 (2015), 413-471.  doi: 10.1002/cpa.21519.  Google Scholar

[23]

M. LewinP. T. Nam and B. Schlein, Fluctuations around Hartree states in the mean-field regime, Am. J. Math., 137 (2015), 1613-1650.  doi: 10.1353/ajm.2015.0040.  Google Scholar

[24]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Inventiones Mathematicae, 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.  Google Scholar

[25]

D. Mitrouskas, S. Petrat and P. Pickl, Bogoliubov corrections and trace norm convergence for the Hartree dynamics, preprint. Google Scholar

[26]

T. Paul, M. Pulvirenti and S. Simonella, On the size of kinetic chaos for mean field models, to appear in ARMA. Google Scholar

[27]

M. Pulvirenti and S. Simonella, The Boltzmann Grad limit of a hard sphere system: Analysis of the correlation error, Inventiones Mathematicae, 207 (2017), 1135-1237.  doi: 10.1007/s00222-016-0682-4.  Google Scholar

[28]

B. Schlein, Derivation of effective evolution equations from microscopic quantum dynamics, Evolution Equations, 511–572, Clay Math. Proc., 17, Amer. Math. Soc., Providence, RI, 2013.  Google Scholar

[29]

H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.  Google Scholar

[30]

H. Spohn, Fuctuations around the Boltzmann equation, J. Stat.l Physics, 26 (1981), 285-305.  doi: 10.1007/BF01013172.  Google Scholar

show all references

References:
[1]

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

[2]

H. van BeijerenO. E. LandfordJ. L. Lebowitz and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Stat. Phys., 22 (1980), 237-257.  doi: 10.1007/BF01008050.  Google Scholar

[3]

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

[4]

C. BoldrighiniA. De Masi and A. Pellegrinotti, Non equilibrium fluctuations in particle systems modelling Reaction-Diffusion equations, Stochastic Processes and Appl., 42 (1992), 1-30.  doi: 10.1016/0304-4149(92)90023-J.  Google Scholar

[5]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Commun. Math. Phys., 56 (1977), 101-113.  doi: 10.1007/BF01611497.  Google Scholar

[6]

S. Caprino and M. Pulvirenti, A cluster expansion approach to a one-dimensional Boltzmann equation: a validity result, Comm. Math. Phys., 166 (1995), 603-631.  doi: 10.1007/BF02099889.  Google Scholar

[7]

S. CaprinoA. De MasiE. Presutti and M. Pulvirenti, A derivation of the Broadwell equation, Comm. Math. Phys., 135 (1991), 443-465.  doi: 10.1007/BF02104115.  Google Scholar

[8]

S. CaprinoM. Pulvirenti and W. Wagner, A particle systems approximating stationary solutions to the Boltzmann equation, SIAM J. Math. Anal., 29 (1998), 913-934.  doi: 10.1137/S0036141096309988.  Google Scholar

[9]

C. Cercignani, The Grad limit for a system of soft spheres, Comm. Pure Appl. Math., 36 (1983), 479-494.  doi: 10.1002/cpa.3160360406.  Google Scholar

[10]

A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamical Limits, Lecture Notes in Mathematics 1501, Springer-Verlag, 1991. doi: 10.1007/BFb0086457.  Google Scholar

[11]

A. De MasiE. OrlandiE. Presutti and L. Triolo, Glauber evolution with Kac potentials. Ⅰ.Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity, 7 (1994), 633-696.  doi: 10.1088/0951-7715/7/3/001.  Google Scholar

[12]

A. De MasiE. OrlandiE. Presutti and L. Triolo, Glauber evolution with Kac potentials. Ⅱ. Fluctuations, Nonlinearity, 9 (1996), 27-51.  doi: 10.1088/0951-7715/9/1/002.  Google Scholar

[13]

A. De MasiE. PresuttiD. Tsagkarogiannis and M. E. Vares, Truncated correlations in the stirring process with births and deaths, Electronic Journal of Probability, 17 (2012), 1-35.  doi: 10.1214/EJP.v17-1734.  Google Scholar

[14]

F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Rational Mech. Anal., 223 (2017), 57-94.  doi: 10.1007/s00205-016-1031-x.  Google Scholar

[15]

C. Graham and S. Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Annals of Probability, 25 (1997), 115-132.  doi: 10.1214/aop/1024404281.  Google Scholar

[16]

K. Hepp and E. H. Lieb, Phase transitions in reservoir-driven open systems with applications to lasers and superconductors, Helv. Phys, Acta, 46 (1973), 573. Google Scholar

[17]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, 3 (1956), 171-197.  Google Scholar

[18]

M. Kac, Probability and Related Topics in Physical Sciences, Interscience, London-New York, 1959.  Google Scholar

[19]

A. Knowles and P. Pickl, Mean-field dynamics: Singular potentials and rate of convergence, Com. Math.Physics, 298 (2010), 101-138.  doi: 10.1007/s00220-010-1010-2.  Google Scholar

[20]

M Lachowicz and M Pulvirenti, A stochastic system of particles modelling the Euler equation, Arch. Ration. Mech. Anal., 109 (1990), 81-93.  doi: 10.1007/BF00377981.  Google Scholar

[21]

S. Lang, Algebra, Springer, 2002. Google Scholar

[22]

M. LewinP. T. NamS. Serfaty and J. P. Solovej, Bogoliubov spectrum of interacting Bose gases, Commun. Pur. Appl. Math., 68 (2015), 413-471.  doi: 10.1002/cpa.21519.  Google Scholar

[23]

M. LewinP. T. Nam and B. Schlein, Fluctuations around Hartree states in the mean-field regime, Am. J. Math., 137 (2015), 1613-1650.  doi: 10.1353/ajm.2015.0040.  Google Scholar

[24]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Inventiones Mathematicae, 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.  Google Scholar

[25]

D. Mitrouskas, S. Petrat and P. Pickl, Bogoliubov corrections and trace norm convergence for the Hartree dynamics, preprint. Google Scholar

[26]

T. Paul, M. Pulvirenti and S. Simonella, On the size of kinetic chaos for mean field models, to appear in ARMA. Google Scholar

[27]

M. Pulvirenti and S. Simonella, The Boltzmann Grad limit of a hard sphere system: Analysis of the correlation error, Inventiones Mathematicae, 207 (2017), 1135-1237.  doi: 10.1007/s00222-016-0682-4.  Google Scholar

[28]

B. Schlein, Derivation of effective evolution equations from microscopic quantum dynamics, Evolution Equations, 511–572, Clay Math. Proc., 17, Amer. Math. Soc., Providence, RI, 2013.  Google Scholar

[29]

H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.  Google Scholar

[30]

H. Spohn, Fuctuations around the Boltzmann equation, J. Stat.l Physics, 26 (1981), 285-305.  doi: 10.1007/BF01013172.  Google Scholar

[1]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[2]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[3]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167

[4]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[7]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[8]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[9]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[10]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[11]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[12]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[13]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[14]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[15]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[16]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[17]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[18]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[19]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[20]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (82)
  • HTML views (95)
  • Cited by (1)

Other articles
by authors

[Back to Top]