Functional Inequalities in Kinetic Theory

One of the most important many body models one encounter in Mathematics, Physics and real life is that which models the behaviour of gases, and in particular rarefied gases. Kacs model is a simple model describing such gas that still possesses many interesting features and problems that appear in more complicated many body models. One of the most interesting problems related to Kacs model is the study of how fast the gas reaches a state of equilibrium (balance). The work I propose to undertake is to advance this study using the help of the notion of the Entropy - a quantity that somehow measures the disorder in the system. Moreover, one would hope to do so in a way that is independent in the number of particles, allowing us to gain information about the behaviour of an average particle. There are several possibilities to attack this problem: 1. Using concepts from the field of Optimal Transportation, one can find new inequalities related to the entropy on the appropriate energy sphere. 2. The state where a few particles have very high energy while the rest are almost stationary seems to cause a slow decay to equilibrium. Quantifying this mathematically, and identifying how to exclude this option is another approach to try. 3. Kacs notion of chaoticity, crucial to his work, hasn`t been fully used in the above study. I propose to try and reintroduce it to the problem, and find appropriate quantitative versions of it that might be of use. The problem of using the entropy to investigate the rate of convergence to equilibrium, known as Cercignanis conjecture, is not a new idea yet one that seems to grasp the physical problem of going from many particles to an average one adequately. Moreover, Kacs model seems to capture a lot of the geometric differences of considering the energy sphere versus the normal space with the statistical exponential distribution (the so-called problem of the equivalence of ensembles). Any advance in this field should have ripples beyond the immediate scope of the problem. The methods suggested above include new venues of research that harken to the original model and the geometry of the process, ones that have not been considered to this day, and ones I believe will have great impact in Kinetic Theory. This problem, as intuitive to explain and understand as it is, has received more attention recently by many prominent mathematician. New tools and techniques were discovered, and more connection between different fields of mathematics emerged. This is the perfect time to continue and investigate the conjecture, and see its effect on an entire field.

The greatest success of the project is in uncovering many connections between different fields inmathematics, and showing how intimately related they are. The original goal of the project was to investigate a question of the trend to equilibrium in an important many particle model by investigating tools from the field of optimal transportation, new notions of asymptotic correlations between the particles and how to identify bad states that are the cause of most of the issues in this problem. It has become apparent relatively quickly that all these points are connected, and one must investigate them all simultaneously. Partial results have been obtained, pushing us in the right direction, yet there is still more to be done. Appropriate quantities, and their dependency in the number of particles in the system, as well as their control over bad situations have been identified, yet it is still unclear if they respect the dynamics of the process that governs the evolution of the model. The realisation of the complexity of this issue led me to attempt and investigate more connections between the realms of functional inequalities, long time behaviour of solutions to evolution equations, and probability. During the fellowship I have successfully resolved five distinct problems, most of which are connected to the above. Together with Anton Arnold and Tobias Wöhrer I have explored degenerate and effective Fokker- Planck equations, equations that are very relevant when one considers standard white noise, and in particular have shown what can be done when the normal approach to study convergence to equilibrium, the same approach as in the many particle model, fails. With Jonathan Ben Artzi I have explored connections between geometric functional inequalities that lie in the heart of an evolution flow, such as those that appear in the many particle model, and the spectral properties of the operator that governs the flow. In particular, we have focused on the case where the operator has no invariant directions (eigenvectors) but does, in some sense, have a smear of directions that are almost invariant (the spectrum is continuous, and we have a density of states). With Michael Cwikel I have investigated in between spaces (interpolation spaces) of weighted Sobolev spaces, trying to understand when these spaces are again weighted Sobolev spaces. As most dierential equations require an understanding of the behaviour of the derivatives of the solution, most likely in a certain space that has a weight associated to it, the problem we explored is very natural and important. We are hoping our result will help study a greater variety of equations in the near future. Lastly, together with José Cañizo and Bertrand Lods I have explored convergence to equilibrium in the Linear Boltzmann equation, an equation that can describe neutron scattering, radiative transfer, and cometary flows, using the same technique used to study the many particle model mentioned above. An explicit and qualitative rate of convergence has been obtained for the so- called soft potentials cases, where the collisions may involve singularities. In addition, we have investigated the behaviour of the moments of solutions to an important coagulations and fragmentation system (Becker-Döring). The study of such moments is usually fundamental in understanding solvability and long time behaviour for many such equations, including the aforementioned many particle system. We have managed to show that under very mild conditions such moments remain bounded uniformly.