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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

S. Lang, Algebra, Springer, 2002.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

S. Lang, Algebra, Springer, 2002.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[29]

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

[30]

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

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