Diffusion in random media: Quantifying the large-scale effects

在自然界中，许多材料尽管具有相当复杂的微观结构，但在宏观水平上呈现出相对均匀的外观。这种现象的一个例子出现在气体的研究中：在分子水平上描述气体需要考虑每个分子及其与其他分子的相互作用。然而，在比分子尺度大得多的尺度上，气体的某些性质（例如密度或温度）可以用一个根据方程演化的单一量来表示。另一个例子是研究污染物通过地下水流的迁移，这是多孔介质中迁移的一个典型例子。要研究污染物如何在土壤中扩散，就必须了解输送污染物的水的行为，而要全面描述水的动力学，就必须了解作为障碍物的每一粒土壤（多孔介质）的几何形状。然而，在比土壤颗粒大得多的尺度上，流体的运动由一个更简单的方程来模拟。在后者中，土壤的影响减少到一个单一的系数（有效渗透率），出现在其中一项。前面的例子有一个共同的特点，即从微观到宏观描述时，复杂性大大降低。使这种减少更加惊人的是，微观水平上可用的信息，如分子相互作用或多孔介质的几何形状，通常是纯统计的。从数学的角度来看，这产生了微观层面的现实建模需要是随机的。在这个框架中，在宏观尺度上获得的复杂性的降低转化为随机的，可能是高维的，系统被确定性方程所取代。这种近似的严格的数学证明，以及对基本机制的充分理解，长期以来一直是涉及概率，数学物理和分析领域的密集研究活动的中心。该研究项目的重点是回答以前的问题，在多孔介质中的气体动力学和输运的相关模型，其中一个严格的理论仍然缺乏。这是追求在随机介质中的扩散过程的数学框架，更确切地说，在一个相互作用的粒子系统和随机环境中的随机行走的缩放限制。主要目标是发展一个定量的理论，允许明确量化的近似误差时，取代宏观的微观描述。到目前为止，在纯数学文献中考虑的大多数设置中，这种定量信息是完全缺失的。为实现这一目标而设想的战略是基于一种新的方法，这种方法依赖于分析和概率技术的结合。特别是，它利用了随机系数的椭圆算子的定量均匀化的最新发展。后者适用于复合材料中的扩散环境，并为本项目中考虑的环境提供了一个有前途的灵感来源。本研究项目将持续五年，将包括两名博士后研究助理和一名帝国理工学院资助的博士生。它也将受益于与J. C. J.J. L. Vel\'azquez（波恩大学），他们是国际领先的概率和偏微分方程分析专家。该项目还设想在世界各地的大学和国际会议上开展密集的传播活动，并组织一个旨在促进粒子系统领域跨学科专家之间互动的研讨会。

In nature, a lot of materials present a relatively uniform aspect at a macroscopic level, in spite of a fairly complicated microscopic structure. An example of this phenomenon arises, for instance, in the study of gases: Describing the gas at the molecular level requires to take into account each molecule and its interaction with the others. Nevertheless, at scales that are much larger than the molecular one, some properties of the gas (e.g its density or its temperature) may be represented by a single quantity that evolves according to an equation. Another case is the study of pollutants transported by groundwater flows, that is a prototypical example for transport in a porous medium. To investigate how the pollutant spreads through the soil, it is necessary to know the behaviour of the water that transports it. A full description of the dynamics of the water, in turn, requires to know the geometry of every grain of soil (the porous medium) that acts as an obstacle. However, at scales that are much larger than the ones of the grains of soil, the motion of the fluid is modelled by a simpler equation. In the latter, the effect of the soil reduces to a single coefficient (the effective permeability) that appears in one of the terms. The previous examples share as common feature a drastic reduction in complexity that takes place when passing from their microscopic to their macroscopic description. What makes this reduction even more striking is that the information available for the microscopic level, such as for the molecular interactions or for the geometry of the porous medium, is typically purely statistical. From a mathematical point of view, this yields that a realistic modelling of the microscopic level needs to be stochastic. In this framework, the reduction of complexity obtained at the macroscopic scales translates into stochastic, and possibly high-dimensional, systems being replaced by deterministic equations. A rigorous mathematical justification of this approximation, and a full understanding of the underlying mechanisms, have long been at the centre of an intense research activity that involves the fields of probability, mathematical physics and analysis. This research project focusses on answering the previous questions for relevant models of gas dynamics and transport in porous media where a rigorous theory is still missing. This is pursued in the mathematical framework of diffusion processes in random media and, more precisely, in the one of scaling limits for interacting particle systems and random walks in random environments. The main goal is to develop a quantitative theory that allows to explicitly quantify the approximation error produced when replacing the microscopic description with the macroscopic one. This quantitative information is so far totally missing in most of the settings considered in the pure mathematical literature. The strategy envisioned to achieve this is based on a new approach that relies on a combination of analytic and probabilistic techniques. In particular, it leverages on the recent developments in the quantitative homogenization of elliptic operators with random coefficients. The latter applies to contexts of diffusion in composite materials and provides a promising source of inspiration for the settings considered in this project.This research project will be pursued on a five-year span and will include two postdoctoral research assistants and one Imperial College funded PhD student. It will also benefit from the collaboration with J.-C. Mourrat (NYU Courant) and J.J. L. Vel\'azquez (Uni of Bonn), who are international leading experts in probability and analysis of PDEs. This project also envisages an intense dissemination activity in universities and international conferences around the world, together with the organisation of a workshop designed to promote the interaction between interdisciplinary specialists working in particle systems.