Research

We know that matter around us is made up of atoms and molecules, but the equations with which many material scientists and physicists work to model the dynamics of matter are so rarely derived under the atomic hypothesis. More commonly, scientists derive the PDE governing matter (like the Navier-Stokes equations, for example) informally using a variant of Newton’s Second Law applied to parcels of the continuum. Following David Hilbert’s programme as laid out by his Sixth Problem, mathematicians seek to derive weak solutions of the PDE governing the motion of matter rigorously as limits of the dynamics of particle systems, as the number N of particles tends to infinity. 

Fundamental to this programme is the mathematical understanding of the dynamics of N particles, be they “spherical” like helium atoms, or “non-spherical” like diatomic oxygen molecules. This is a significant mathematical challenge unto itself. This amounts to the construction of a billiard dynamics on the tangent bundle of the hard particle table, which in general admits the structure of a manifold with corners:

Even if we focus on the intuitively-simple system of N hard spheres (where, for instance, N is of the order of Avogadro's constant for applications in the real world), the following mathematical questions are open:

• How do N hard spheres in simultaneous contact with one another scatter so that total momentum and energy is conserved?

• Can you unfold the hard sphere table in a way that produces a smooth or real-analytic manifold? Does that manifold admit the structure of an algebraic variety?

• What is the maximum number of collisions that N hard spheres can endure on their time domain?

In the case of non-spherical particles like ellipsoids, there are similarly challenging open problems:

• Assuming Boltzmann scattering, what is the maximum number of times two isolated ellipsoids can collide with one another before complete separation?

• How do you resolve a collision between two ellipsoids for at collision points for which the scattering problem is ill-posed?

I find problems on hard particle dynamics beautiful, as they model the real world yet involve use of techniques from a wide variety of fields in mathematics – analysis, geometry, and algebra. Indeed, in a recent paper of mine, I make use of techniques from real algebraic geometry to understand the structure of invariant measures of hard sphere dynamical systems.

Here are some articles of mine on particle dynamics:

• M. Wilkinson (2024). Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line, J Stat Phys 191, 93. https://doi.org/10.1007/s10955-024-03310-y 

• M. Wilkinson (2022). A Lie Algebra-theoretic Approach to Characterisation of Collision Invariants of the Boltzmann Equation for General Convex Particles. Kinetic and Related Models, 15(2): 283-315. doi:10.3934/krm.2022008

• L. Saint-Raymond and M. Wilkinson (2018). On Collision Invariants for Linear Scattering. Comm. Pure Appl. Math., 71:1494-1534. doi:10.1002/cpa.21761

Overall, Hilbert's Sixth Problem in the case of gases (one of nature's more straightforward phases of matter) can be summarised by the following diagram:

For an introduction to the kinds of mathematical problem that exist in this field, you might like to consider this monograph of I. Gallagher, L. Saint-Raymond, and B. Texier.

The semi-geostrophic equations (SG) are a well-known model for the dynamics of the layer of the Earth’s atmosphere known as the troposphere. Cite textbook Atmospheric Fluid Dynamics here. As a PDE, the semi-geostrophic equations belong to the class of spatially-inhomogeneous semi-linear active transport equations, with the additional constraint that its solution be a conservative vector field. SG is given explicitly by:

The gradient of the geopotential P models the evolution of the so-called geostrophic velocity field of the atmosphere, as well as its temperature. When written in this way (in Eulerian coordinates), it is particularly difficult to prove the existence of solutions in any reasonable sense of the associated initial-value problem. In this way, they are even more complicated than the well-known Navier-Stokes equations (associated to which there is a Clay Millenium Prize)! To make SG easier to analyse, following Cullen and Benamou-Brenier, the equations are rewritten in so-called geostrophic coordinates. In this coordinate system, SG takes the form of the following active scalar transport equation:

where the activity is determined by the solution operator of a second boundary-value problem for the well-known Monge-Ampere (MA) equation, namely:

As a standalone problem, construction of solutions to the second boundary-value problem for the MA equation are achieved using the theory of Optimal Transport. For an excellent introduction to Optimal Transport, you might like to consider the monographs of Villani, or that of Santambroggio. You might also like to see the following Simons Foundation video in which the achievements of the Fields medal-winning mathematician, Alessio Figalli, in the field of SG are mentioned.

Major open problems in this field include the following:

• Establishing the global-in-time existence of (distributional) solutions of SG in Eulerian coordinates for initial data whose associated temperature profile is bounded in space;

• Derivation of SG from an underlying elementary fluid model (such as the Primitive equations or the compressible Navier-Stokes equations) in an appropriate small Rossby-number limit.

I have worked on various aspects of SG, including the analysis of the so-called surface  semi-geostrophic equations, the analysis of Hoskins’ channel flow, as well as the stability of linear solutions of SG in Eulerian coordinates. Together with my former group in Heriot-Watt, we also analysed SG and the associated Eady model using semi-discrete Optimal Transport. Here is a selection of my publications in the area:

• C. P. Egan, D. P. Bourne, C. J. Cotter, M. J. P. Cullen, B. Pelloni, S. M. Roper, M. Wilkinson (2022). A New Implementation of the Geometric Method for Solving the Eady Slice Equations. Journal of Computational Physics, Volume 469. https://doi.org/10.1016/j.jcp.2022.111542  

• D. P. Bourne, C. P. Egan, B. Pelloni, and M. Wilkinson (2022). Semi-discrete Optimal Transport Methods for the Semi-geostrophic Equations. Calc. Var. 61, 39. https://doi.org/10.1007/s00526-021-02133-z  

• S. Lisai and M. Wilkinson (2020). Smooth Solutions of the Surface Semi-geostrophic Equations. Calc. Var. 59, 16. https://doi.org/10.1007/s00526-019-1664-3  

Nematic liquid crystals are a classical example of a condensed matter system in which the constituent molecules are long and rod-like:

Nematics have been very important to the technology industry, forming the backbone of the display industry for many decades. The mathematical study of nematic liquid crystals also has a long history. When using continuum mechanics to model the physical properties of nematics, one must first decide upon an appropriate order parameter which governs the local ordering and structure of the molecules which make up the material. In the case of uniaxial nematic molecules, one popular choice due to the Nobel Prize-winning physicist Pierre-Gilles de Gennes is the Q-tensor order parameter given by

where rho is an even probability density function on the sphere. From this definition, a Q-tensor is a traceless and symmetric 3 by 3 matrix.

In my PhD, I modelled the dynamics of nematic liquid crystals – both including and excluding hydrodynamic effects – using Q-tensor theory. Here are some articles I have published in the area of Q-tensor theory:

• M. Wilkinson (2015). Strict Physicality of Global Weak Solutions of a Navier-Stokes Q-tensor System with Singular Potential. Arch. Rational Mech. Anal., 218: 487. https://doi.org/10.1007/s00205-015-0864-z  

• E. Kirr, M. Wilkinson and A. D. Zarnescu (2014). Dynamic Statistical Scaling in the Landau-de Gennes Theory of Nematic Liquid Crystals. J. Stat. Phys., 155: 625. https://doi.org/10.1007/s10955-014-0970-6  

For a nice introduction to Q-tensor theory, you might like to consider the lecture notes of Zarnescu.