250043 VU Kinetic Theory Applied to Biology (2021S)

Emergent phenomena are ubiquitous in nature: it corresponds to the appearance of large-scale structure from underlying microscopic dynamics. At the microscopic level particles or agents interact following some rules, but the macroscopic structures are not encoded directly in these rules and, therefore, it is a challenge to explain how the macroscopic or observable dynamics emerge from the microscopic ones. Examples of emergence are collective dynamics (flocks of birds, school of fish, pedestrians…), network formation (capillary formation, leaf venation, formation of gullies…), opinion dynamics, tumor growth, tissue development… Understanding emergence in science is key to explaining why observable phenomena take place. The mathematical tools to studying emergence come from kinetic theory, which originally was developed to study problems in Mathematical Physics in the field of gas dynamics. The application of these tools to explore questions coming from biology poses many new interesting challenges at the level of the modeling and mathematical analysis.

This course will be a short introduction to classical and modern techniques in kinetic theory to derive continuum equations from discrete equations.

The topics covered in this course include:
1. What is emergence and how does kinetic theory contributes to its study?
2. Discrete models or agent-based models (ordinary differential equations).
2. Mean-field limits: from discrete models to transport equations.
3. Transport equations: the case of the linear Boltzmann equation, existence of solutions.
3. Hydrodynamic limits: from transport equations to macroscopic models.
a. Boltzmann and Vlasov equations.
b. Hilbert expansion method.
c. Generalised Collision Invariant.

The course will be a combination of theory and exercises done during the class. Articles will be distributed and worked in class in which the theory learned will be applied. The active participation of students during class is expected.

Appropriate breaks will be taken during the class.