交互作用粒子系统的分析与计算研究进展

Analysis and Computation for Interacting Particle Systems

作为一类基本的统计物理微观模型,交互作用粒子系统(多体系统)在从传统的物理、化学和生物到社会科学、数据科学等领域都发挥着重要的作用。由于初始数据的不确定性、交互作用的随机性以及环境噪声的影响,大部分的交互作用粒子系统都带有随机性。简要介绍了交互作用粒子系统的研究进展。在理论分析方面,介绍利用相对熵和调整自由能等工具处理具有奇异交互作用的交互作用粒子系统的平均场极限的新研究进展,同时也介绍交互作用粒子系统能量函数景观关于信噪比的精准相变关键值,以及斯蒂弗尔流形上粒子在梯度流的全局收敛性与信噪比的非随机充分条件等的新研究结果。在数值计算方面,介绍基于随机思想设计复杂度低且并行效率高的模拟和抽样算法方面的研究进展。这为我国在相关方面的数学研究和应用提供有力支撑。

As a class of fundamental microscopic models in statistical physics,interacting particle systems(or manybody systems)play important roles in the fields ranging from physics,chemistry,biology to social sciences and data sciences etc.Due to the randomness in the initial data and the interaction,and noise from the environment,most interacting particle systems are stochastic.This paper gives a brief introduction to the research progress on interacting particle systems.For the theoretical analysis,how the tools of relative entropy is reviewed.The modulated free energy etc.could be used for the rigorous justification of the mean field limit with singular kernels.In addition,this paper reviews the recent works that characterize the sharp critical SNR for the phase transition of the energy landscape of particle systems,and sufficient conditions for the SNR under which the energy landscape associated with particles on Stiefel manifolds only has a unique local(global)minimizer.For the scientific computation,this paper reviews several new simulation and sampling algorithms based on random strategies,which enjoy low complexity and high parallel efficiency.These results will provide strong support for the relevant applications.