Modulated free energy and mean field limit
Modulated free energy and mean field limit
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DOI：
10.5802/slsedp.135
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
D. Bresch;P. Jabin;Zhenfu Wang
中科院分区：
文献类型：
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作者：
D. Bresch;P. Jabin;Zhenfu Wang
这是第一作者于2019年11月19日在IH{E}S为Laurent Schwartz研讨会所做的演讲的相应文件。它涉及到我们最近在BrJaWa [4]的平均场理论中引入的调制自由能。这个物理对象可以看作是由S引入的调制势能的组合。Serfaty [见Proc. Int. Cong. Math.（2018）]以及P. -在平均场极限理论中引入的相对熵E. Jabin，Z. Wang [参见Inventiones 2018]。使用Du [8]和Se 1 [20]中的估计，它首次允许在存在粘性的情况下获得Riesz和Coulomb排斥核的平均场极限的收敛速率。本文的主要目的是解释如何通过BrJaWa [4]中宣布的傅里叶变换方法来覆盖更一般的排斥核，首先是在N $\rightarrow$ +$\infty$时$\sigma$N $\rightarrow $0的情况下，然后如果$\sigma$ > 0是固定的。最后，我们对Patlak-Keller-Segel系统的粒子近似作了评论，该系统与吸引核有关，并参考[C.R.
This is the document corresponding to the talk the first author gave at IH{E}S for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in mean-field theory in BrJaWa [4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in mean field limit theory by P.-E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the mean field limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates in Du [8] and Se1 [20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced in BrJaWa [4] first in the case $\sigma$N $\rightarrow$ 0 when N $\rightarrow$ +$\infty$ and then if $\sigma$ > 0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R.