Mathematical Sciences: Multiple Scales for Interacting Particle Systems: Mesoscopic and Macroscopic Equations

第1段在非平衡统计力学的背景下，物理系统的演化是通过相互作用的粒子系统在微观水平上建模的。这种过程涉及大量的成分(粒子)，具有确定性或随机性的动力学，受局部相互作用定律支配。看起来，大群的粒子，原则上以无序的方式演化，往往在更大的空间或时间尺度上以一致的模式组织起来。特别是，这些重新标度的、可能是随机的微观模型有望产生描述宏观量演化的确定性非线性偏微分方程组。这里讨论的问题之一是从相互作用的粒子系统严格推导非线性偏微分方程组。更进一步，人们希望对于更精细的粒子系统，局部随机波动被保存在宏观方程中，而微观细节仍然被隐藏。在这个框架下，我们建议研究(I)具有长程相互作用和弱传播界面的相互作用的粒子系统，以及(Ii)随机离散速度模型和流体动力学方程组。我们特别感兴趣的是理解粒子系统收敛到宏观方程的弱解，而不是流体方程的激波的形成，以及传播界面的所有可能的几何奇点。所提出的技术来自于非线性偏微分方程组，以及概率论和随机过程。分析相互作用的粒子系统的一个关键部分在于不同时空重新标度制度的相互作用，每一种制度都有自己的极限确定性偏微分方程式。本研究项目涉及材料科学、生物学和流体力学中出现的非平衡现象的微观和宏观模型的数学分析。提案的第一部分涉及相变材料(例如，改变晶体结构的合金)的输运性质的推导。类似的技术有望在预测生物过程中的非平衡行为方面被证明是有用的。这里使用的数学工具有望提供有关我们的物理系统的新的详细的定量信息，超越了现象学的一步。在项目的第二部分，我们研究了离散随机模型对气体和流体动力学方程的收敛。除了第一部分提出的问题外，我们希望这样的离散模型能够为气体动力学流动的数值计算提供健壮的算法，特别是在激波存在的情况下。

PARAGRAPH 1 In the context of Nonequilibrium Statistical Mechanics, the evolution of physical systems is modelled at a microscopic level by Interacting Particle Systems. Such processes involve a large number of components (particles) with either deterministic or stochastic dynamics, governed by local laws of interaction. It appears that large groups of particles, which in principle evolve in an unorderly manner, tend to organize themselves in a coherent pattern at some larger space or time scale. In particular, these rescaled, possibly stochastic, microscopic models are expected to give rise to deterministic Nonlinear Partial Differential Equations describing the evolution of macroscopic quantities. One of the problems addressed here, is the rigorous derivation of nonlinear partial differential equations from interacting particle systems . Proceeding a step further, one would hope that for finer scales of the particle systems, the local random fluctuations are preserved in the macroscopic equation, while microscopic details still remain hidden. In this framework, we propose to study (i) Interacting Particle Systems with long range interactions and weakly propagating interfaces, and (ii) Stochastic discrete velocity models and fluid dynamics systems of equations. We are particularly interested in understanding the convergence of particle systems to weak solutions of the macroscopic equations, past the formation of shocks for the fluids equations and past all possible geometric singularities of the propagating interfaces. The proposed techniques are drawn from Nonlinear Partial Differential Equations, as well as, Probability Theory and Stochastic Processes. A crucial part in the analysis of the interacting particle systems, lies in the interplay of the different space-time rescaling regimes present, each one with its own limiting deterministic partial differential equation. PARAGRAPH 2 This research project is concerned with the mathematical analysis of microscopic and macroscopic models of non-equilibrium phenomena arising in material science, biology and fluid mechanics. The first part of the proposal is concerned with the derivation of transport properties for materials undergoing phase transitions (for example alloys changing crystallographic structure). Similar techniques will hopefully prove useful in the prediction of nonequilibrium behavior in biological processes. The mathematical tools employed here, are expected to provide new detailed quantitative information about our physical systems, going one step beyond phenomenology. In the second part of the project, we study the convergence of discrete random models, to gas and fluid dynamics equations. In addition to the issues proposed in the first part, here we hope that such discrete models may yield robust algorithms for the numerical computation of gas dynamics flow, particularly in the presence of shock waves.