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

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粒子系统在微观水平上模拟的。这样的过程包含了大量具有确定性或随机动力学的组分（粒子），受局部相互作用规律的支配。看来，原则上以无序方式进化的大群粒子，倾向于在更大的空间或时间尺度上以连贯的模式组织起来。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。这里讨论的问题之一，是相互作用粒子系统非线性偏微分方程的严格推导。再进一步，人们希望在粒子系统的更细尺度上，局部随机波动保留在宏观方程中，而微观细节仍然隐藏。在此框架下，我们建议研究(i)具有远距离相互作用和弱传播界面的相互作用粒子系统，以及（ii）随机离散速度模型和方程流体动力学系统。我们特别感兴趣的是理解粒子系统对宏观方程弱解的收敛性，通过流体方程的激波形成以及通过传播界面的所有可能的几何奇点。提出的技术是从非线性偏微分方程，以及概率论和随机过程中得出的。在相互作用粒子系统的分析中，一个至关重要的部分在于存在的不同时空重标度制度的相互作用，每个制度都有自己的极限确定性偏微分方程。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。该提案的第一部分涉及相变材料（例如改变晶体结构的合金）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证明是有用的。这里使用的数学工具，有望提供关于我们的物理系统的新的详细的定量信息，超越现象学。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性。除了第一部分中提出的问题之外，在这里我们希望这种离散模型可以为气体动力学流动的数值计算提供鲁棒算法，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.