Michael Wilkinson


Research summary

I have contributed to several areas of low energy physics. The emphasis is on achieving a thorough  understanding of physically realistic model problems using analytical methods, numerical investigations and simple experiments. I work on topics where there are fundamental points of principle which remain to be understood and have identified significant problems which had not previously been considered.

Areas where I made sustained contributions are listed below, in roughly chronological order. The numbers refer to my list of publications, and the contribution of my collaborators is acknowledged by listing the authors there. Publication numbers [2,34,47,50,53,77] do not fit into the categories below.

Magnetic field effects in solids

My Ph.D. thesis concerns Bloch electrons in a magnetic field. In numerical experiments Hofstadter found that the spectrum is a non self-similar fractal set; its explanation was a long standing problem. I showed how the spectrum can be understood using a chaotic renormalisation-group (RG) transformation. The fractal spectrum is associated with unstable fixed points forced by a fourfold symmetry of the lattice. Initially I approximated the RG transformation using WKB methods [1]. Later I developed an exact form [10], which has connections with Chern numbers describing the quantised Hall effect. This led to new definitions of magnetic Wannier functions [32,46] and new representations of metaplectic transformations [46,51]. For lattices with threefold symmetry, the spectrum has an even richer fractal structure, explained in [15]. An early application of the 'Berry phase' was discovered in the WKB theory of this problem [3]. With Y. Last, I found that the spectrum satisfies remarkable sum rules [24]. Other works in this area are [4,7,31,38,39,42].

Semiclassical methods

I showed that matrix elements obey sum rules related to periodic classical orbits, analogous to Gutzwiller's celebrated relation between periodic orbits and the density of states [8,11,55]. This is relevant to periodic orbit effects in electrical conductance of mesoscopic systems. Analysis of the contributions for long orbits provides insight into connections between semiclassical results and random matrix theory [11].

I also analysed quasi-degeneracies where multiplets are associated with phase space tori which transform into each other under a symmetry operation. The energies are split due to a tunnelling effect [5,6,9,37].

Energy diffusion theory of dissipation

Most discussions of the response of macroscopic systems are based upon linear response theory. This is problematic, because the 'Kubo formula' is based upon quantum mechanical perturbation theory, which is questionable when the perturbation is larger than the separation of energy levels. In applications to macroscopic bodies, this is a serious concern. It was addressed by developing a new approach, the 'energy diffusion theory' of dissipation, which is both simpler and more general than the Kubo formula [17,16]. Energy diffusion theory agrees with the Kubo formula in many regimes, but in some cases it gives dramatically different results [33,49,57,58]. I discovered a regime with enhancement of dissipation due to Landau-Zener transitions [12,13], and other regimes with a suppression of dissipation due to Anderson localisation of Floquet operator eigenstates [18]. More recently, I discovered (with B. Mehlig and D. Cohen) a dynamical behaviour termed 'semilinear response' [66].

Universal dynamics of complex systems

The spectra of complex systems have universal statistical properties, exemplified by random matrix models. I extended this idea to universality of dynamical properties [22,33,49], was the first to calculate statistics of the parametric dependence of energy levels [14,23,26,27]. These were used to derive results on non-adiabatic transitions [12], and the statistics of Chern integers (topological invariants describing quantised Hall conductance) [35,40]. This led to general results on the 'screening' of 'charged' singularities [48,60]. Other work on random matrices has included theories for the parametric dependence of wavefunctions [36], spectral correlations [54] and properties of banded random matrices [19,20,33].

Other topics in condensed matter theory

I explained a surprising universality in exciton spectra of heterostructures (noted by Yang Fang), using statistical topographic ideas [25,28,43]. I have also discovered and explained universality of an exponent describing anomalous phtoconductive response of semiconductors [87].

I made a detailed analysis of the absorption of electromagnetic radiation by small metal particles, showing how semiclassical ideas can be adapted to deal with screening of the electric field in a satisfactory way [29,30,41,44,45,52,56].  

Other work on solid state physics includes a novel mechanism for hopping conduction termed 'adiabatic transport' [21], a novel 'stick-breaking' model for Mott's variable-range hopping process [75,69,73] and anomalous diffusion in solids [91].

Particles in turbulent flows

With Bernhard Mehlig I have made extensive investigations on the motion of particles in turbulent flows. A new method for determining the particle Lyapunov exponents was developed [61,62,63]. We obtained an exact solution of a 'path coalecsence transition' [59,82,90]. By developing the Lyapunov exponents as power series and applying Borel summation, we have shown how the fractal dimension of particle clustering depends upon the 'Stokes number', a dimensionless measure of particle inertia [64,71,82,83,86]. We identified new dynamical regimes, including a generalised Ornstein-Uhlenbeck process which is analysed using novel annihilation/creation operators [65,67].  

I have also considered advection of rod-like particles in turbulence, showing that their textures exhibit apparent singularities similar to those of fingerprints [79,81,85,95]. With Alain Pumir I studied tumbling in turbulent flows [89], including a surprising `irrational quantisation' phenomenon [88].

This work also led to the definition of a new type of dimension, the spectal dimension [94].

 Planet formation and gravitational collapse

The standard model of planet formation hypothesises that planets form by the aggregation of dust particles in a circumstellar accretion disc. Using estimates of relative velocities of particles in turbulent flows [70,78], we argued that the collision are too energetic to allow dust grains to aggregate [74]. We proposed an alternative hypothesis for planet formation, 'concurrent collapse' [91], where planets form by gravitational collapse at the same time as their star. The planets may subsequently become entrained to the accretion disc with circular orbits, and the lighter elements ablated away to leave a rocky core. This model explains way many extra-solar planets have highly eccentric orbits.

This investigation also led to new theory of gravitational collapse which describes the role of shocks in fragmentation of a molecular cloud [84].


With B. Mehlig I introduced the concept of `caustics' into studies of inertial particles [62], and showed that this is essential to understanding the collision rate of inertial particles. Our results help to explain the rapid onset of rainfall from cumulus clouds [68] (see also [72,76]).