# Introduction to the mathematical modeling of multi-scale phenomena

This module combines class discussions with numerical explorations and visualizations of experiments, to help students discover basic ideas of multiscale modeling.

## Introduction

We start with two important ideas that are illustrated on a variety of examples. First, phenomena that appear similar in nature can occur at completely different scales. Understanding them requires a universal theory, which is in some way scale-free. Second, most systems involve more than one scale, and are often described by models that are developed at a specific scale. Modeling complex systems across scales is a current challenge of multiscale analysis.

## Scales and scalings

This short sections introduces simple concepts of dimensional analysis and scalings. It also illustrates the existence of self-similar solutions to partial differential equations.

## Diffusion

The last section is devoted to the concept of diffusion at the microscopic and macroscopic levels. It includes applications of random walks to foraging behaviors and to bacterial motion.