Research themes

2. Mathematical background: Multiscale analysis and kinetic theory

The idea of multiscale analysis is to derive mathematically a particular physical model from another one that contains less information than the original one. The two models or equations are at different scales and deriving one model from the other requires averaging and a limiting process. This means that a model at atomistic scale explains how particular physical phenomena may arise at observable scale.

The kinetic equation has a solution that depends on space, time and velocity. The macroscopic equation depends only on space and time. The latter will be derived by averaging over the velocities the solution to the kinetic equation, and by performing a limiting process.

The limiting process is based on rescaling space and time. In physical terms, this means the following. The micro time scale is the typical time a particle takes to change its velocities. For observable changes to happen in the bulk of the particles, we need to speed up time and consider macro time scales. In the same manner, we also make a zoom out in space, to focus on the bulk of particles instead of the individual particles.

This rescaling in time and space has to be done properly so that it stands out interesting phenomena: if we speed up time too much, the particles may escape to infinity and we will see nothing, i.e., in the limit we will get zero. If we do not speed up time fast enough, no changes will occur on the bulk of particles and no phenomena will arise.

Summarising, in the limiting process information is lost and at the same time, the dynamics of the bulk of particles, that were only implicit in the kinetic equation, stand out. Notice that due to the loss of information during the limiting process, it is possible that different kinetic models lead to the same macroscopic equation.

3. The study of emergent phenomena using kinetic theory

I specialise in the modelling of emergent phenomena using kinetic theory. Emergent phenomena consists of macroscopic/observable structures that emerge from the local interaction of many individuals, particles or agents.

As an example, I am a daily supervisor of Sophie Hetch (PhD student of Prf. Pierre Degond) on a project in collaboration with the Francis Crick Institute where the goal is to understand the morphogenesis of organs from the cellular dynamics. Specifically, the aim is to test mechanical feedbacks at cellular level by which the so-called imaginal disc of the Drosophila fly may stop growing after reaching its mature state.

Kinetic theory, originally developed to study physical phenomena such as gas dynamics, provides the perfect mathematical framework to investigate emergent phenomena for the following reasons:

• From cells to tissues: Kinetic theory deals with the different scales of a given system. It links the microscopic description of a system, where the dynamics are given by the interaction of individuals agents (cells, molecules,…), with its macroscopic description, where large scale structures are observed (tissues, organs,…). As an analogy, in a paper that is in preparation, we study bird swarming starting from the dynamics of the individual birds (agents), and from this we derive the macroscopic dynamics (flocking).

• Holistic view: By linking the different scales, in some cases kinetic theory can be used to look for the principles at the microscopic level that explain the phenomena observed. In other words, not only do we want to reproduce experimental data, but also hypothesize and test their causes. In the organ morphogenesis project mentioned above, the goal is to test mathematically the conjecture made by the experimental group of biologists at the Francis Crick Institute (a conjecture which is very difficult to test experimentally).

• Biological phenomena as integrated systems: At the microscopic level we can consider different type of agents and their interactions. For example, in a current master project that I am co-supervising, we study the emergence of capillaries (angiogenesis) by considering the interactions between different agents consisting of oxygen, blood and vessels.

• Computational efficiency: Typically, the systems we are interested in are formed by a large number of agents making the numerical simulations costly, sometimes impossible to carry out. By deriving the corresponding macroscopic dynamics, we can study the patterns of the system by simulating models that are more cost effective.

• Not widely exploited, new edge in research: In contraposition with statistical modelling, image analysis and the fast-growing field of big data analysis, the study of biological systems using kinetic theory is not so well known among the biology community.

• Broad applicability: Many different biological scenarios present common structures and therefore, once a methodology is developed for one particular system, it can be applied to others.

PAPERS:

• Continuum dynamics of the intention field under weakly cohesive social interactions, Pierre Degond, Jian-Guo Liu and Thomas Tardiveau, Math. Models Methods Appl. Sci. accepted, (2016). Arxiv version.

• A new flocking model through body attitude coordination (with Pierre Degond, Imperial College London, and Amic Frouvelle, Universite Paris Dauphine). Math. Models Methods Appl. Sci., accepted, (2016). Arxiv version.

3.1 Self-organised systems.

Self-organised systems consists of groups of individuals where there is not a clear leader (fish schooling, flocks of birds, spermatozoon,…) but that form large-scale structures due to their local interactions. These local interactions are typically given by simple rules, such as, individuals try to align their direction of movement. Classical model for self-organisation are the Vicsek model and the Cucker-Smale model.

From self-organised systems, modelled at the level of the individuals, we try to derive the coarse-grained equations, i.e., the equations that describe the macroscopic patterns that arise from the dynamics of the system. This allows to better understand these macroscopic structures and to perform simulations more efficiently when a high number of individuals are present.

PAPERS:

• A new flocking model through body attitude coordination (with Pierre Degond, Imperial College London, and Amic Frouvelle, Universite Paris Dauphine). Math. Models Methods Appl. Sci., accepted, (2016). Arxiv version.

3.2 Opinion dynamics.

As in self-organised systems, we see opinion dynamics as being ruled by the local interactions of individuals in a large system. Which is the asymptotic evolution of the opinions? Is a consensus reached? How does the evolution depend on the particular rules of interaction?

PAPERS:

• Continuum dynamics of the intention field under weakly cohesive social interactions, Pierre Degond, Jian-Guo Liu and Thomas Tardiveau, Math. Models Methods Appl. Sci. accepted, (2016). Arxiv version.

4. Interacting stochastic particle systems.

Kinetic equations can be approximated using stochastic particle systems. Proving that the particle system approximates the kinetic equation when the number of particles grows to infinity is the key goal here. We use probabilistic tools coming from the theory of Markov Chains, Martingales and Skorohod spaces.

PAPERS:

• Isotropic Wave Turbulence with simplified kernels: existence, uniqueness and mean-field limit for a class of instantaneous coagulation-fragmentation processes, 2015. Arxiv version.

5. Fractional diffusion phenomena arising from microscopic models.

My dissertation focuses on a particular macroscopic model: the fractional diffusion equation, which is used in different scenarios, from the way a contaminant spreads under water to the flying pattern of albatrosses. The fractional diffusion equation is similar to the standard diffusion equation; both cases describe a diffusion process. To illustrate what they model, take for example the way heat changes in a plate full of hot soup, it will be warmer in the centre and colder in the borders but, with time, the centre will cool down and the temperature of the soup will become uniform. We say that the temperature has `spread’ or diffused. The fractional diffusion equation describes a diffusion process where the diffusion happens `faster’ than in the standard diffusion equation (and with particular qualitative properties).

I look to explain how fractional diffusion phenomena arises from different microscopic systems. With Dr. Sabine Hittmeir, we obtain fractional diffusion phenomena for different physical quantities like the total energy and the momentum. With Professor Antoine Mellet, we start from a one-dimensional solid (FPU-beta chain) and study how the temperature changes in it (in the same way that it is studied how the temperature changes in the plate of soup). Finally, I try to use also probabilistic techniques to show how this phenomena emerges from particles systems where its elements take large velocities or change their velocity at anomalous rate.

PAPERS:

• Anomalous transport in FPU-β chains (with Antoine Mellet) at the Journal of Statistical Physics, 2015. Arxiv version, journal version.

• Kinetic derivation of fractional Stokes and Stokes-Fourier systems (with Sabine Hittmeir), Kinetic and Related models, 2015. Arxiv version, journal version.