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