Diffusion in random media: Quantifying the large-scale effects

在自然界中，许多材料虽然具有相当复杂的微观结构，但在宏观层面上却呈现出相对均匀的一面。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到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，气体的某些特性（例如密度或温度）可以用一个根据方程演变的单个量来表示。另一个例子是地下水对污染物运移的研究，这是多孔介质中运移的一个典型例子。为了研究污染物是如何在土壤中扩散的，有必要了解运输污染物的水的行为。反过来，对水动力学的全面描述需要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，流体的运动可以用一个更简单的方程来模拟。在后一种情况下，土壤的影响减少到出现在其中一项中的单个系数（有效渗透性）。前面的例子有一个共同的特点，那就是当从微观描述过渡到宏观描述时，复杂性急剧降低。更令人惊讶的是，微观层面的信息，如分子相互作用或多孔介质的几何结构，通常都是纯粹的统计信息。从数学的角度来看，这意味着微观层面的现实模型必须是随机的。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，被确定性方程所取代的系统。长期以来，一项涉及概率、数学物理和分析领域的激烈研究活动的中心，一直是对这种近似的严格数学证明，以及对其潜在机制的充分理解。本研究项目的重点是回答多孔介质中气体动力学和输运的相关模型，这是一个严格的理论仍然缺乏的问题。这是在随机介质中扩散过程的数学框架中进行的，更准确地说，是在随机环境中相互作用的粒子系统和随机游动的尺度限制之一中进行的。主要目标是发展一种定量理论，允许明确量化用宏观描述代替微观描述时产生的近似误差。到目前为止，这种定量信息在大多数纯数学文献中完全缺失。实现这一目标的战略设想是基于一种依赖于分析和概率技术相结合的新方法。特别地，它利用了具有随机系数的椭圆算子的定量均匀化的最新进展。后者适用于复合材料的扩散环境，并为本项目中考虑的设置提供了有希望的灵感来源。该研究项目为期五年，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的合作。Mourrat （NYU Courant）和j.j.l. Vel\'azquez（波恩大学），他们是pde概率和分析方面的国际领先专家。该项目还设想在世界各地的大学和国际会议上进行密集的传播活动，同时组织一个研讨会，旨在促进粒子系统跨学科专家之间的互动。

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.