# American Institute of Mathematical Sciences

February  2019, 39(2): 747-769. doi: 10.3934/dcds.2019031

## Free energy in a mean field of Brownian particles

 Department of Mathematics, University of Tennessee at Knoxville, 227 Ayres Hall, 1403 Circle Drive, Knoxville TN 37996-1320, USA

* Corresponding author

Received  November 2017 Published  November 2018

Fund Project: X. Chen's research is partially supported by the Simons Foundation #244767. T. Phan's research is partially supported by the Simons Foundation, #354889

We compute the limit of the free energy
 ${1\over Nt_N}\log \mathbb{E}\exp\bigg\{{1\over N}\sum\limits_{1\le j of the mean field generated by the independent Brownian particles $ \{B_j(s)\}$interacting through the non-negative definite function $\gamma(\cdot)$. Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan's large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree's theory on the asymptotics for the ground state energy of bosonic quantum system. Citation: Xia Chen, Tuoc Phan. Free energy in a mean field of Brownian particles. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 747-769. doi: 10.3934/dcds.2019031 ##### References:  [1] A. de Acosta, On large deviations of sums of independent random vectors, Probability in Banach Spaces, Lecture Notes in Math., 1153 (1985), 1-14, Springer, Berlin. doi: 10.1007/BFb0074942. Google Scholar [2] R. Bass, X. Chen and J. Rosen, Large deviations for Riesz potentials of additive processes, Annales de l'Institut Henri Poincare, 45 (2009), 626-666. doi: 10.1214/08-AIHP181. Google Scholar [3] X. Chen, Random Walk Intersections: Large Deviations and Related Topics, Mathematical Surveys and Monographs, 157, American Mathematical Society, RI, 2010. doi: 10.1090/surv/157. Google Scholar [4] X. Chen, Spatial assymptotics for the parabolic Anderson models with generalized time-space Gaussian noise, Ann. Probab., 44 (2016), 1535-1598. doi: 10.1214/15-AOP1006. Google Scholar [5] X. Chen, Y. Z. Hu, J. Song and F. Xing, Exponential asymptotics for time-space Hamiltonians, Annales de l'Institut Henri Poincare, 51 (2015), 1529-1561. doi: 10.1214/13-AIHP588. Google Scholar [6] D. Conus, Moments for the parabolic Anderson model: On a result by Hu and Nualart, Stoch. Anal., 7 (2013), 125-152. Google Scholar [7] D. Conus, M. Joseph and D. Khoshnevisan, On the chaotic character of the stochastic heat equation, before the onset of intermittency, Ann. Probab., 41 (2013), 2225-2260. doi: 10.1214/11-AOP717. Google Scholar [8] D. Conus, M. Joseph, D. Khoshnevisan and S.-Y. Shiu, On the chaotic character of the stochastic heat equation, Ⅱ, Probab. Theor. Rel. Fields, 156 (2013), 483-533. doi: 10.1007/s00440-012-0434-3. Google Scholar [9] R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E's, Electron. J. Probab., 4 (1999), 1-29. doi: 10.1214/EJP.v4-43. Google Scholar [10] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, (2nd ed.) Springer, New York, 1998. doi: 10.1007/978-1-4612-5320-4. Google Scholar [11] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time Ⅲ, Comm. Pure Appl. Math., 29 (1976), 389-461. doi: 10.1002/cpa.3160290405. Google Scholar [12] Y. Z. Hu, J. Huang, K. Le, D. Nualart and S. Tindel, Stochastic heat equation with rough dependence in space, Ann. Probab., 45 (2017), 4561-4616. doi: 10.1214/16-AOP1172. Google Scholar [13] Y. Hu and D. Nualart, Stochastic heat equation driven by fractional noise and local time, Probab. Theor. Rel. Fields, 143 (2009), 285-328. doi: 10.1007/s00440-007-0127-5. Google Scholar [14] W. Hunziker and I. M. Sigal, The quantum N-body problem, J. Math. Phys., 41 (2000), 3448-3510. doi: 10.1063/1.533319. Google Scholar [15] K. Kirkpatrick, B. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004. Google Scholar [16] M. Lewin, P. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose system, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010. Google Scholar show all references ##### References:  [1] A. de Acosta, On large deviations of sums of independent random vectors, Probability in Banach Spaces, Lecture Notes in Math., 1153 (1985), 1-14, Springer, Berlin. doi: 10.1007/BFb0074942. Google Scholar [2] R. Bass, X. Chen and J. Rosen, Large deviations for Riesz potentials of additive processes, Annales de l'Institut Henri Poincare, 45 (2009), 626-666. doi: 10.1214/08-AIHP181. Google Scholar [3] X. Chen, Random Walk Intersections: Large Deviations and Related Topics, Mathematical Surveys and Monographs, 157, American Mathematical Society, RI, 2010. doi: 10.1090/surv/157. Google Scholar [4] X. Chen, Spatial assymptotics for the parabolic Anderson models with generalized time-space Gaussian noise, Ann. Probab., 44 (2016), 1535-1598. doi: 10.1214/15-AOP1006. Google Scholar [5] X. Chen, Y. Z. Hu, J. Song and F. Xing, Exponential asymptotics for time-space Hamiltonians, Annales de l'Institut Henri Poincare, 51 (2015), 1529-1561. doi: 10.1214/13-AIHP588. Google Scholar [6] D. Conus, Moments for the parabolic Anderson model: On a result by Hu and Nualart, Stoch. Anal., 7 (2013), 125-152. Google Scholar [7] D. Conus, M. Joseph and D. Khoshnevisan, On the chaotic character of the stochastic heat equation, before the onset of intermittency, Ann. Probab., 41 (2013), 2225-2260. doi: 10.1214/11-AOP717. Google Scholar [8] D. Conus, M. Joseph, D. Khoshnevisan and S.-Y. Shiu, On the chaotic character of the stochastic heat equation, Ⅱ, Probab. Theor. Rel. Fields, 156 (2013), 483-533. doi: 10.1007/s00440-012-0434-3. Google Scholar [9] R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E's, Electron. J. Probab., 4 (1999), 1-29. doi: 10.1214/EJP.v4-43. Google Scholar [10] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, (2nd ed.) Springer, New York, 1998. doi: 10.1007/978-1-4612-5320-4. Google Scholar [11] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time Ⅲ, Comm. Pure Appl. Math., 29 (1976), 389-461. doi: 10.1002/cpa.3160290405. Google Scholar [12] Y. Z. Hu, J. Huang, K. Le, D. Nualart and S. Tindel, Stochastic heat equation with rough dependence in space, Ann. Probab., 45 (2017), 4561-4616. doi: 10.1214/16-AOP1172. Google Scholar [13] Y. Hu and D. Nualart, Stochastic heat equation driven by fractional noise and local time, Probab. Theor. Rel. Fields, 143 (2009), 285-328. doi: 10.1007/s00440-007-0127-5. Google Scholar [14] W. Hunziker and I. M. Sigal, The quantum N-body problem, J. Math. Phys., 41 (2000), 3448-3510. doi: 10.1063/1.533319. Google Scholar [15] K. Kirkpatrick, B. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004. Google Scholar [16] M. Lewin, P. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose system, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010. Google Scholar  [1] Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 851-879. doi: 10.3934/dcdsb.2018210 [2] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173 [3] Chjan C. Lim. 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