The Maxwell Stefan problem with intrinsic randomness

Imagine two liquids, like water and alcohol, mixed in a container. The molecules naturally spread out from areas of high concentration to areas of low concentration - a process called diffusion. Then, alcohol molecules tend to diffuse faster than water molecules because they are smaller and less polar, allowing them to move more easily through the solvent. However, the process of diffusion is not limited to individual substances - the molecules also interact with each other, which influences how fast and in what direction they move. Because of these factors, the Maxwell- Stefan framework (which models the diffusion process of a multicomponent mixture of different species) is a useful way to describe how each substance diffuses in the mixture. Mathematically, the Maxwell-Stefan system is a set of coupled, nonlinear partial differential equations that express the fluxes of the species in terms of their concentration gradients and interaction coefficients. This system consists of the cross-diffusion effect and a reaction term. The Maxwell-Stefan system can exhibit unexpected behaviours due to cross- diffusion effects, such as upwind diffusion, where substances move against their concentration gradient. Phase separation can occur, resulting in spatial variations in concentration. This phenomenon often occurs in systems with strong interactions and varying diffusion rates, as seen in many natural and industrial processes. Additionally, when combined with fluid dynamics, the Maxwell-Stefan system has broad applications in fields like chemical and petroleum engineering, the food and beverage industry, and energy systems. The irregular motion of molecules and the randomness of the environment naturally introduce intrinsic noise into the system. Macroscopic equations are derived as the limiting behaviour of microscopic dynamics, where the movement of molecules is viewed as the result of irregular microscopic motion within the fluid. In this transition to macroscopic models, fluctuations around the mean are typically neglected. However, physical systems are rarely isolated and are often influenced by external noise, such as parameter fluctuations or environmental disturbances. Additionally, noise serves as a tool to capture small-scale perturbations. Thus, incorporating stochastic terms into macroscopic models provides a more realistic mathematical framework. In the project, we focus on studying the Maxwell-Stefan system and its variants, including cases with and without a reaction term, as well as those with and without a fluid field. We will introduce a stochastic term to model random fluctuations. We will start with theoretical questions, such as whether solutions to the system exist, ergodicity, and the effects of gradient noise. Next, we develop numerical methods to simulate these systems. Additionally, we aim to explain phenomena such as ion interactions, transport properties, and conductivity, which cannot be described in the deterministic setting.

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