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

第1款 在非平衡统计力学的背景下， 物理系统是在微观层次上由相互作用粒子模拟的 系统.这些过程涉及大量的组分（颗粒）， 无论是确定性或随机动力学，由当地的法律， 互动似乎原则上， 以无序的方式进化，倾向于以连贯的方式组织自己， 在更大的空间或时间尺度上的模式。特别是，这些重新调整， 可能是随机的，微观模型预计会产生 确定性非线性偏微分方程描述的演变 宏观量 这里解决的问题之一是严格推导非线性 相互作用粒子系统的偏微分方程。程序 更进一步，人们希望， 粒子系统的更精细尺度，局部随机波动被保留在宏观方程中，而 微观细节仍然被隐藏。在这个框架下，我们提出研究（i）具有长程相互作用和弱相互作用的相互作用粒子系统， 传播界面，和（ii）随机离散速度模型和流体 动力学方程组我们特别有兴趣了解 粒子系统收敛到宏观弱解 方程，过去的冲击形成的流体方程和过去的所有 传播界面可能的几何奇异性。拟议 技术来自非线性偏微分方程，以及， 概率论与随机过程分析的一个关键部分是 相互作用的粒子系统，在于不同粒子之间的相互作用。 空间-时间重新缩放制度存在，每一个都有自己的限制 确定性偏微分方程 第2款 该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。建议的第一部分是关于 相变材料输运性质的推导 过渡（例如合金改变晶体结构）。类似 技术将有望证明在预测非平衡态 生物过程中的行为。这里使用的数学工具是 预计将提供新的详细的定量信息，我们的物理 系统，超越现象学的一步。在项目的第二部分，我们研究离散随机模型的收敛性，气体和流体动力学 方程除了第一部分提出的问题外，我们希望 这种离散模型可以产生用于数值计算的鲁棒算法， 气体动力学流动的计算，特别是在存在冲击波的情况下。

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.