| EPSRC Reference: |
EP/W018616/1 |
| Title: |
Diffusion in random media: Quantifying the large-scale effects |
| Principal Investigator: |
Giunti, Dr A |
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| Department: |
Mathematics |
| Organisation: |
Imperial College London |
| Scheme: |
EPSRC Fellowship |
| Starts: |
01 September 2022 |
Ends: |
31 August 2027 |
Value (£): |
1,136,617
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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.
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.imperial.ac.uk |