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Below you can find a list of my publications, together with links to the journal where appropriate:
[17] On Invariant Measures, Cylindrical Algebraic Decomposition, and Second Boundary-Value Problems for Boltzmann Hard Sphere Flows
Wilkinson, M. On Invariant Measures, Cylindrical Algebraic Decomposition, and Second Boundary-Value Problems for Boltzmann Hard Sphere Flows (2024). arXiv:
Abstract: Motivated by a remark in R. K. Alexander's 1975 Berkeley PhD thesis and associated article (Time evolution for infinitely many hard spheres, Communications in Mathematical Physics, 49(3), pp.217-232, (1976)), by using a cylindrical algebraic decomposition (CAD) of fibres of the tangent bundle of the hard sphere table modelling the dynamics of 2 hard spheres on Euclidean space $\mathbb{R}^{3}$, we derive a characterisation of regular linear momentum- and angular momentum-conserving billiard flows which admit the Liouville measure as an invariant measure. More precisely, we prove that the scattering map associated to any such billiard flow is a classical solution of a second boundary-value problem for a Jacobian PDE posed on the half-space of pre-collisional velocities. In turn, this allows us to characterise all invariant measures on the tangent bundle which are absolutely continuous with respect to the Liouville measure. The CAD we construct yields a new explicit representation formula for flows on the tangent bundle of the table which we term the invading wave representation of the flow. We conclude by drawing some connections between the Boltzmann scattering of hard spheres and Inverse Optimal Transport.
[16] Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line
Wilkinson, M. Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line. J Stat Phys 191, 93 (2024). https://doi.org/10.1007/s10955-024-03310-y
Abstract: For any $N\geq 3$, we study invariant measures of the dynamics of $N$ hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold $\mathcal{M}$ of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all $N$ hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable $N$-body scattering maps which generate a billiard dynamics on $\mathcal{M}$ admitting a canonical weighted Hausdorff measure on $\mathcal{M}$ (that we term the Liouville measure on $\mathcal{M}$) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linear $N$-body scattering map yields a billiard dynamics which admits the Liouville measure on $\mathcal{M}$ as an invariant measure.
[15] On the Initial Boundary-Value Problem for Newton’s Equations of Motion for Hard Particles: Non-existence
M. Wilkinson (2024). On the Initial Boundary-Value Problem for Newton’s Equations of Motion for Hard Particles: Non-existence (revised version submitted). arXiv:1805.04611
Abstract: In the first of two papers, we study the initial boundary-value problem that underlies the theory of the Boltzmann equation for general non-spherical hard particles. In this work, for two congruent ellipses and for a large class of associated boundary conditions, we identify initial conditions for which there do not exist local-in-time weak solutions of Newton’s equations of motion. To our knowledge, this is the first time the necessity of rolling in the energy-conserving dynamics of strictly-convex rigid bodies has been demonstrated. This study was, in part, motivated by a recent observation of Palffy-Muhoray, Virga, Wilkinson and Zheng on the interpenetration of strictly-convex rigid bodies.
[14] Weyl Group Representation of Billiard Trajectories for One-dimensional Hard Sphere Dynamics
Abstract: We present an exact formula for the dynamics of $N$ hard spheres of radius $r>0$ on an infinite line which evolve under the assumption that total linear momentum and kinetic energy of the system is conserved for all times. This model is commonly known as the one-dimensional Tonks gas or the hard rod gas model. Our exact formula is expressed as a sum over the Weyl group associated to the root system $A_{N-1}$ and is valid for all initial data in a full-measure subset of the tangent bundle of the hard sphere table. As an application of our explicit formula, we produce a simple proof that the associated billiard flow admits the Liouville measure on the tangent bundle of the hard sphere table as an invariant measure.
[13] A Lie Algebra-theoretic Approach to Characterisation of Collision Invariants of the Boltzmann Equation for General Convex Particles
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
Abstract: By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that the collision invariant is of class $\mathscr{C}^{1}$. This work extends that of Saint-Raymond and Wilkinson (Communications on Pure and Applied Mathematics (2018), 71(8), pp. 1494-1534), in which the authors characterise collision invariants only in the case of the so-called canonical physical scattering family. Indeed, our method extends to the case of non-canonical physical scattering, whose existence was reported in Wilkinson (Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055-2083). Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson as we place no symmetry hypotheses on the underlying non-spherical particles which make up the gas under consideration. The techniques established in this paper also yield a new proof of the result of Boltzmann for collision invariants of class $\mathscr{C}^{1}$ in the classical case of hard spheres.
[12] Semi-discrete Optimal Transport Methods for the Semi-geostrophic Equations
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
Abstract: We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.
[11] A new Implementation of the Geometric Method for Solving the Eady Slice Equations
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
Abstract: We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations, which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero.
[10] Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on R^3
S. Lisai and M. Wilkinson (2021). Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on R^3. Journal of Mathematical Fluid Mechanics, vol. 23, no. 3, 54. https://doi.org/10.1007/s00021-021-00574-2
Abstract: We consider a class of steady solutions of the semi-geostrophic equations on
R^3 and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L^2(R^3, R^3) introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R^3. We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.
[9] On the Non-uniqueness of Physical Scattering for Hard Non-spherical Particles
M. Wilkinson (2020). On the Non-uniqueness of Physical Scattering for Hard Non-spherical Particles. Arch. Rational Mech. Anal. 235, 2055–2083. https://doi.org/10.1007/s00205-019-01460-y
Abstract: We prove the existence of uncountably-many physical scattering maps for non-spherical hard particles which, when used to construct global-in-time weak solutions of Newton’s equations of motion, conserve the total linear momentum, angular momentum and kinetic energy of the particle system for all time. We prove this result by first exhibiting the non-uniqueness of a classical solution to a constrained Monge–Ampère equation posed on Euclidean space, and then applying the deep existence theory of Ballard for hard particle dynamics. In the final section of the article, we briefly discuss the relevance of our observations to the statistical mechanics of hard particle systems.
[8] The Stability Principle and Global Weak Solutions of the Free Surface Semi-geostrophic Equations in Geostrophic Coordinates
M. J. P. Cullen, T. Kuna, B. Pelloni and M Wilkinson (2019). The Stability Principle and Global Weak Solutions of the Free Surface Semi-geostrophic Equations in Geostrophic Coordinates. Proc. R. Soc. A., 47520180787. https://doi.org/10.1098/rspa.2018.0787
Abstract: The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimization argument originally inspired by the Stability Principle as studied by Cullen, Purser and others, uses optimal transport techniques as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of the Stability Principle in a rigorous mathematical framework.
[7] Smooth Solutions of the Surface Semi-geostrophic Equations
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
Abstract: The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip R^2×(0,1), that models a semi-geostrophic flow in regime of constant potential vorticity. This system is an expression of an Eulerian semi-geostrophic flow in a coordinate system originally due to Hoskins, to which we shall refer as Hoskins’ coordinates. We obtain results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in Hölder spaces.
[6] A Derivation of the Liouville Equation for Hard Particle Dynamics with non-conservative Interactions
B. D. Goddard, T. D. Hurst and M. Wilkinson (2020). A Derivation of the Liouville Equation for Hard Particle Dynamics with non-conservative Interactions. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 151(3):1040-1074. doi:10.1017/prm.2020.49
Abstract: The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using two examples.
[5] On a Paradox in the Impact Dynamics of Smooth Rigid Bodies
P. Palffy-Muhoray, E. Virga, M. Wilkinson, and X. Zheng (2018). On a Paradox in the Impact Dynamics of Smooth Rigid Bodies. Mathematics and Mechanics of Solids, Volume 24, Issue 3. doi:10.1177/1081286517751262
Abstract: Paradoxes in the impact dynamics of rigid bodies are known to arise in the presence of friction. We show here that, on specific occasions, in the absence of friction, the conservation laws of classical mechanics are also incompatible with the collisions of smooth, strictly convex rigid bodies. Under the assumption that the impact impulse is along the normal direction to the surface at the contact point, two convex rigid bodies that are well separated can come into contact, and then interpenetrate each other. This paradox can be demonstrated in both 2D and 3D when the collisions are tangential, in which case no momentum or energy transfer between the two bodies is possible. The post-collisional interpenetration can be realized through the contact points or through neighboring points only. The penetration distance is shown to be O(t^3). The conclusion is that rigid-body dynamics is not compatible with the conservation laws of classical mechanics.
[4] On Collision Invariants for Linear Scattering
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
Abstract: In this article, we extend the result of Boltzmann on the characterization of collision invariants from the case of hard disks to a class of two-dimensional compact, strictly convex particles.
[3] On the Convergence of Soft Potential Dynamics to Hard Sphere Dynamics
M.Wilkinson (2018). On the Convergence of Soft Potential Dynamics to Hard Sphere Dynamics. Asymptotic Analysis, vol. 107, no. 1-2, pp. 1-32. doi:10.3233/ASY-171448
Abstract: We address a question raised in the work of Gallagher, Saint-Raymond and Texier [MR3157048] that concerns the convergence of soft-potential dynamics to hard sphere dynamics. In the case of two particles, we establish that hard sphere dynamics is the limit of soft sphere dynamics in the weak-star topology of BV. We view our result as establishing a topological method by which to construct weak solutions to the ODE of hard sphere motion.
[2] Strict Physicality of Global Weak Solutions of a Navier-Stokes Q-tensor System with Singular Potential
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
Abstract: We study the existence, regularity and so-called ‘strict physicality’ of global weak solutions of a Beris–Edwards system which is proposed as a model for the incompressible flow of nematic liquid crystal materials. An important contribution to the dynamics comes from a singular potential introduced by John Ball and Apala Majumdar which replaces the commonly employed Landau-de Gennes bulk potential. This is built into our model to ensure that a natural physical constraint on the eigenvalues of the Q-tensor order parameter is respected by the dynamics of this system. Moreover, by a maximum principle argument, we are able to construct global strong solutions in dimension two.
[1] Dynamic Statistical Scaling in the Landau-de Gennes Theory of Nematic Liquid Crystals
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
Abstract: In this article, we investigate the long time behaviour of a correlation function which is associated with a nematic liquid crystal system that is undergoing an isotropic-nematic phase transition. Within the setting of Landau–de Gennes theory, we confirm a hypothesis in the condensed matter physics literature on the average self-similar behaviour of this correlation function in the asymptotic regime at time infinity. In the final sections, we also pass comment on other scaling regimes of the correlation function.
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