Free energy in a mean field of Brownian particles
Free energy in a mean field of Brownian particles
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DOI：
10.3934/dcds.2019031
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Xia Chen;T. Phan
中科院分区：
文献类型：
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作者：
Xia Chen;T. Phan
我们计算了独立布朗粒子通过非负定函数相互作用产生的平均场的自由能极限。我们的主要定理与具有时间白噪声、空间白噪声或有色噪声的抛物型Anderson模型的高阶矩渐近有关。我们的方法与著名的Donsker-Varadhan关于I.I.D.的大偏差原理建立了一个新的联系。无限维空间中的随机变量。作为主要定理的应用，我们对Hartree关于玻色量子系统基态能量渐近性的理论进行了概率处理。
We compute the limit of the free energy \begin{document}${1\over Nt_N}\log \mathbb{E}\exp\bigg\{{1\over N}\sum\limits_{1\le j of the mean field generated by the independent Brownian particles \begin{document}$ \{B_j(s)\}$\end{document} interacting through the non-negative definite function \begin{document}$\gamma(\cdot)$\end{document} . 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.