Perspectives

Working in L²pdµκq Ă L²pdµ0q we are treating the operator L^κas a small perturbation of the case κ “ 0 with stationary conveyor belt. The natural space to investigate the convergence to F^κin the case κ ą 0 however is L²`F_κ^´1dx dα˘

. In this L²-space the transport operator T ´ Pκis not skew-symmetric and the collision operator Q is not self-adjoint, so the hypocoercivity method [135]

cannot be applied. To get around this, one can split the operator Lκ differently into a transport and a collision part following the approach in [69]. More precisely, we can write Lκ “ ˜Q ´ ˜T

where $

’’

&

’’

%

Qf “ B˜ α

´

DBαf ´^Bα_F^Fκ

κ f¯ ,

Tf “ pτ ` κe˜ 1q ¨ ∇xf ´ Bα“`τ^K¨ ∇xV `<sup>Bα</sup><sub>F κ</sub><sup>Fκ</sup>˘ f‰ . Then in L<sup>2</sup>`F_κ^´1dx dα˘

the operatorQ is symmetric and negative semi-definite, and the operator˜ T is skew-symmetric. Furthermore, the stationary state F˜ ^κlies in the intersection of the kernels of the collision and transport operators, i.e. F^κ P Ker ˜Q X Ker ˜T. The hypocoercivity approach requires microscopic and macroscopic coercivity ofQ and ˜˜ T. To this end, we need to be able to control the behaviour of the stationary state at infinity as in [69], i.e. for large enough |x|,

@ α P S¹, e^{´σ1V pxq}ď Fκpx, αq ď e^{´σ2V pxq}

for some constants σ1, σ2 ą 0. If true, this would be an important physical information on the stationary state, but we still do not know how to prove it. Even with this information at hand,

this approach requires that the existence of the stationary state be known a priori. The rate of convergence one obtains in this case may be different from the rate obtained here, and it is not clear which method yields the better rate as both are most likely not optimal.

There are several ways in which one could seek to improve the results in Chapter5. For exam-ple, one could try to push the convergence result to larger values of κ using bifurcation techniques.

More precisely, for a path p : κ ÞÑ Fkmapping κ to the stationary state Fκof equation (5.33), The-orem (7.2) guarantees that p is defined on a small interval r0, κ0q for some 0 ă κ0! 1. It may be possible to extend this interval by showing that the implicit equation P pκ, Fκq “ 0 defining the stationary state Fκis non-degenerate, i.e. that B2P pκ, Fκq ‰ 0.

Another future avenue would be to apply the techniques developed here to other models where the global equilibrium is not known a priori.

8 Part III: From micro to macro

The 6^thproblem asked by Hilbert³⁰in 1900 is concerned with the axiomatisation of physics. More than 100 years later it is still unresolved, and might never be considered completed as the problem statement is rather broad. Precisely, the original German text Mathematische Probleme states:

Durch die Untersuchungen über die Grundlagen der Geometrie wird uns die Auf-gabe nahegelegt, nach diesem Vorbilde diejenigen physikalischen Disciplinen axioma-tisch zu behandeln, in denen schon heute die Mathematik eine hervorragende Rolle spielt; dies sind in erster Linie die Wahrscheinlichkeitsrechnung und die Mechanik.

Was die Axiome der Wahrscheinlichkeitsrechnung³¹ angeht, so scheint es mir wün-schenswert, daß mit der logischen Untersuchung derselben zugleich eine strenge und befriedigende Entwickelung der Methode der mittleren Werte in der mathematischen Physik, speziell in der kinetischen Gastheorie Hand in Hand gehe.

Ueber die Grundlagen der Mechanik liegen von physikalischer Seite bedeutende Un-tersuchungen vor; ich weise hin auf die Schriften von Mach³², Hertz³³, Boltzmann³⁴ und Volkmann³⁵; es ist daher sehr wünschenswert, wenn auch von den Mathematik-ern die Erörterung der Grundlagen der Mechanik aufgenommen würde. So regt uns beispielsweise das Boltzmannsche Buch über die Principe der Mechanik an, die dort angedeuteten Grenzprocesse, die von der atomistischen Auffassung zu den Gesetzen über die Bewegung der Continua führen, streng mathematisch zu begründen und durchzuführen. Umgekehrt könnte man die Bewegung über die Gesetze starrer Kör-per durch Grenzprocesse aus einem System von Axiomen abzuleiten suchen, die auf der Vorstellung von stetig veränderlichen, durch Parameter zu definirenden Zustän-den eines Zustän-den ganzen Raum stetig erfüllenZustän-den Stoffes beruhen - ist doch die Frage nach der Gleichberechtigung verschiedener Axiomensysteme stets von hohem principiellen Interesse.

The problem, suggested by Boltzmann’s work on the principles of mechanics [45], is therefore to develop “mathematically the limiting processes [...] which lead from the atomistic view to the laws of motion of continua”, namely to obtain a unified description of gases, including all levels

30David Hilbert (1862-1943) was a German mathematician and is recognised as one of the most influential and uni-versal mathematicians of the 19^thand early 20^thcenturies. He was invited to address the 2^ndInternational Congress of Mathematicians in Paris in 1900, where he proposed 23 problems that are known today as Hilbert’s problems.

31Vgl. Bohlmann, Ueber Versicherungsmathematik 2te Vorlesung aus Klein und Riecke, Ueber angewandte Mathe-matik und Physik, Leipzig und Berlin 1900

32Die Mechanik in ihrer Entwickelung, Leipzig, zweite Auflage. Leipzig 1889

33Die Principien der Mechanik, Leipzig 1894

34Vorlesungen über die Principien der Mechanik, Leipzig 1897

35Einführung in das Studium der theoretischen Physik, Leipzig 1900

of description. In other words, the challenging question is whether macroscopic concepts can be understood microscopically.

The set of methods for making the connection between microscopic and macroscopic models are called multiscale analysis or scaling process or limiting process. The idea of multiscale analysis is to mathematically derive one particular model describing macroscopic phenomena in the observable physical world, from a microscopic model that is based on interactions between atoms, particles, or agents. Typically, the microscopic model (depending on space, time and velocity) contains more information than the macroscopic one (depending only on space and time). One can make the connection between these two regimes by averaging over the velocities and rescaling the time and space variables. Mathematically, this corresponds to ’zooming out’, and so we are exchanging the loss of information on the kinetics with the ability to capture emerging dynamics of the bulk of particles that were only implicit in the kinetic equation. The choice of rescaling influences which phenomena we are able to observe on the macroscopic scale and has to be chosen in a sensible way to match the physical context: if we speed up time too much with respect to the scaling in space, the particles may escape to infinity and we see nothing; if we do not speed up time fast enough, no change will occur on the macroscopic level and so no interesting phenomena arise. Since certain information are lost in the scaling process, it is possible that different kinetic models lead to the same macroscopic equation. Examples of limiting processes for kinetic equations can be found in the classical references [263,103,294]. Let us mention that the terms ’microscopic’, ’macroscopic’

and ’mesoscopic’ are sometimes used ambiguously in the literature. In this thesis, we will use ’mi-croscopic’ in the sense of ’kinetic’ as opposed to a regime describing individual particle dynamics.

Building on the ideas of Maxwell in [232], in 1872, Boltzmann published his famous work [46]

on what can be considered the master equation of kinetic theory

Btf ` v ¨ ∇xf “ Qpf, f q , (8.49)

where x P R^N represents position and v P R^N velocity, for the probability density f pt, x, vq. The bilinear collision operator Q may differ depending on the type of microscopic interactions at play.

Equation (8.49) is known as the Boltzmann equation and was derived for a monoatomic rarefied gas by merging mechanical concepts and statistical considerations [232,46]. It describes gas particles undergoing free transport and collisions. In the modern literature, the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of some quantity such as energy, charge or particle number in a thermodynamic system.

Chapter6is centred around the idea of understanding the relationship between different ki-netic and macroscopic models using multiscale analysis. Diffusion approximations to kiki-netic

equations have been studied in various works, see for example [293,13,26,121,210,258] and the references therein. In this section, we discuss two particular scaling approaches that play a role in Chapter6, grazing collision limits and parabolic diffusion limits, exemplified by the Boltz-mann equation (8.49) for different choices of collision kernels Q. The latter shows how a limiting process can be used to derive the classical Keller–Segel model (1.1) from a kinetic description for bacterial motion.