Joint PhD call of the SFB 65 "Taming Complexity in Partial Differential Systems"

Project Part in the SFB 65

Project Part 14 (Sara Merino-Aceituno & Christian Schmeiser) - link

Type of position

PhD position.

Duration of employment

3 years.

Affiliation

The successful applicant will be employed at the Faculty of Mathematics of the University of Vienna (Vienna, Austria) and will register to the Doctoral Program of the University of Vienna. An association with the Vienna School of Mathematics (joint doctoral school between the University of Vienna and TU Wien) is possible.

Theme

mathematics, partial differential equations, modelling, analysis, kinetic theory.

Collective dynamics are ubiquitous: schools of fish, pedestrian dynamics, herds, ant colonies, opinion dynamics, ... These systems are the paramount example of self-organisation, i.e, systems in which agents self-organise collectively in a spontaneous way, without having a leader coordinating them. Self-organisation arises purely from the local interaction between the individuals. The natural starting point to model these systems corresponds to discrete models that describe the dynamic of each one of the individuals. However, to understand how self-organisation takes place, one needs to consider systems with a large number of individuals, on the one hand, and to look at large time and spatial scales, on the other hand.

The mathematical tools of kinetic theory have been applied to the study of self-organisation in collective dynamics. The tools of kinetic theory allow us, firstly, to study the system when there is a large number of individuals (tending to infinity) through a process called mean-field limit. In the mean-field limit, we derive from the discrete dynamics of the individuals, the time-evolution of the distribution of the individuals (i.e., how proportions of individuals behave on average). The time-evolution of this distribution is given by a transport (or kinetic) equation. From this transport equation, it is possible to determine the dynamics of the system for large time-scales and large-spatial scales. To begin with, one expects these systems to converge to some kind of equilibrium distribution for large times. To completely characterise this equilibrium distribution, we typically need to determine the evolution of some of its moments (like the total mass and the mean). This characterisation of the moments of the equilibrium is obtained when one also looks at long-space scales in an asymptotic process called hydrodynamic limit.

This project focuses on the analysis of models for collective dynamics. We will look at a particular model for collective dynamics based on the classical Vicsek model, that we name kinetic Vicsek-BGK model. In this model particles are assumed to be self-propelled and to move at a constant speed in a given orientation. Particles try to align their orientation with the ones of the neighbours. We investigate the dynamics under the presence of a large number of individuals and the long-time dynamics.
The mathematical goals of the project include the investigation of the mean-field limit (limit when the number of agents grows to infinity) and the study of the convergence or not to local equilibrium for long-time dynamics and the rate of convergence to this equilibrium. On a second stage, the results of this analysis will be applied to the study of pattern formation and to other models in collective dynamics based on apolar alignment (particles try to align in the same direction rather than in the same orientation).
This project will bring one step forward the understanding of the mechanisms behind self-organisation in these widely-used models for collective dynamics.

Degree and master in Mathematics. Solid background on mathematical analysis in general, specially on functional analysis. General knowledge on ordinary differential equations and on partial differential equations.

Some related results exist in the literature for different Vicsek-type models. For studies on the limit when the number of particles is large (mean-field limit) and the well-posedness of the limiting equation, see [6] and [4]. The study of rates of convergence to equilibrium has been studied for BGK-type equations using a methodology based on hypocoercivity [1, 2, 3, 7, 8]. The reference [5] includes a study of convergence to equilibrium for the Vicsek model when the spatial variable is ignored.

[1] F. Achleitner, A. Arnold, and E. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic and Related Models, 11(4):953–1009, 2018. (link)
[2] F. Achleitner, A. Arnold, and E. A. Carlen. On linear hypocoercive BGK models. In , pages 1–37. Springer, 2016. (link)
[3] A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. 2001. (link)
[4] M. Briant and S. Merino-Aceituno. Cauchy theory and mean-field limit for general Vicsek models in collective dynamics. arXiv preprint arXiv:2004.00883, 2020.
[5] P. Degond, A. Frouvelle, and J. Liu. Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics. Arch. Ration. Mech. Anal., 216(1):63–115, 2015. (link)
[6] A. Diez. Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles. arXiv preprint arXiv:1908.00293, 2019.
[7] J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus Mathematique, 347(9-10):511–516, 2009. (link)
[8] J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Transactions of the American Mathematical Society, 367(6):3807–3828, 2015. (link)

Involved PhD advisers

Dr. Marc Briant (Paris 5, France), Dr. Sara Merino-Aceituno (main supervisor, University of Vienna) and Prof. Christian Schmeiser (University of Vienna).

Involved collaborators for this project

Prof. Anton Arnold (TU Wien) and Antoine Diez (Imperial College London, UK).

Preferred start date

September 2021 (this is a flexible date, the starting date could be a couple of months before or after).

Contact

For queries on this advertisement, please contact Dr. Sara Merino-Aceituno at sara.merino@univie.ac.at.

Requested documents

• a motivation letter;
• a CV, including contacts of 2 possible references (no letters required at this stage), and a list of publications (if available);
• transcript of the marks of Bachelor and Master (in case that the Master has not been completed yet, please upload whichever marks are available or the list of courses being undertaken).

Online application

link