2.2018 - Hydrodynamical limits for different types of active particles models
Advisor: Julien Barré
At variance with molecules in ordinary fluids, ”active particles” can pro-
pel themselves using some internal energy reservoir, or taking energy from
their environment. This type of particles can model a wealth of different
systems, often biological in nature, from bacterial colonies to herds of large
mammals.
For ordinary or ”active” fluids, there are three levels of description: i) mi-
croscopic, or particle level; ii) mesoscopic, or kinetic: the system is described
by a density in space and velocity; iii) macroscopic, or hydrodynamic: the
system is described by a few macroscopic fields, such as the density in space,
and the local mean velocity. These three levels of descriptions are related by
scaling limits, where N, the number of particles, is large. These limits have
been well understood for a long time for ordinary fluids, at least formally.
Following [1], a large body of physics literature addressed the issue of
deriving hydrodynamical equations for active fluids, starting from a kinetic
description. These works are similar in spririt to a bifurcation computa-
tion, relying on a small parameter measuring the distance from an instabil-
ity threshold. The kinetic description is often of Boltzmann type, assuming
binary collisions.
Independently, mathematicians, starting from a kinetic description of
Fokker-Planck type (adequate for mean-field like interactions), have stud-
ied the same mesoscopic to macroscopic limit with different tools (Hilbert or
Chapman-Enskog expansions, collisional invariants) which precisely assume
that the instability threshold is far away [2].
Unsurprisingly, the hydrodynamical equations obtained in these two cases
are different. Is it possible to connect them through a further limiting pro-
cedure? Is it possible to use collisional invariants methods in the context of
Boltzmann-like kinetic models such as those in [1]? The goal of the intern-
ship is to address at least one of these questions on a well chosen model,
More precisely, after a careful reading of [1] and [2], the student will
pick up a few points among the ones suggested below, according to his/her
preferences:
• Obtain hydrodynamical equations with both bifurcation and collisional
invariant methods on the same well chosen model.
• Perform a formal limit procedure to connect the two sets of hydrody-
namical equations, if possible.
• Adapt the collisional invariant methods to a Boltzmann-like kinetic
equation for active particles.
• Run numerical simulations at the particle level to check some analytical
results and explore new regimes.
• Study the inclusion of finite N fluctuations in the preceding descrip-
tions.
This internship may be followed by a PhD thesis.
References
[1] Bertin, E., Droz, M., et Gr ́egoire, G. (2006). Boltzmann and hydrody-
namic description for self-propelled particles. Physical Review E, 74(2),
022101.
[2] Degond, P., et Motsch, S. (2008). Continuum limit of self-driven parti-
cles with orientation interaction. Mathematical Models and Methods in
Applied Sciences, 18, 1193-1215.
