Variational Modelling: Energies, gradient flows, and large deviations

TL;DR

This paper introduces Variational Modelling as a rational approach for gradient-flow systems, emphasizing diffusion processes with entropic and Wasserstein terms, supported by detailed methodology and applications.

Contribution

It provides a comprehensive explanation of Variational Modelling methodology, especially for diffusion, integrating entropic and Wasserstein terms, with detailed motivation and application guidance.

Findings

Detailed methodology for Variational Modelling

Application to diffusion with entropic and Wasserstein terms

Guidance on using the methodology in new situations

Abstract

These are lecture notes for various Summer and Winter schools that I have given. The notes describe the methodology called Variational Modelling, and focus on the application to the modelling of gradient-flow systems. I describe the methodology itself in great detail, and explain why this is a rational modelling route. A central example is diffusion, in combination with various other processes, and a large part of the notes are devoted to this phenomenon. In the Variational Modelling methodology, diffusion is commonly modelled by including entropic terms in the driving functional and Wasserstein-type terms in the dissipation. I explain how to understand these objects, motivate them from more basic models, and how to use them in new situations.