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Mathematical Physics Seminar

Webinar: Hans Jürgen  Herrmann - Rolling Matter 

Location:  Zoom
Date & time: Wednesday, 10 July 2024 at 10:45AM - 11:45AM




Hans Jürgen  Herrmann – ETH Zurich

Wednesday, July 10th, 10:45AM EDT 


Rolling Matter      


We will discuss realizations of solids, which can sustain internally rotational degrees of freedom. They are discrete versions of Cosserat media. We will consider bearings, which are systems of touching spheres or disks that roll on each other without slip. In these systems every loop must be even. Depending on the size of these loops and on the spatial dimension a bearing can present a large number of possible configurations of angular velocities. Bearings states can be perceived as physical realizations of networks of oscillators with asymmetrically weighted couplings. These networks can exhibit optimal synchronization properties through tuning of the local interaction strength as a function of node degree or the inertia of their constituting rotor disks through a power-law mass-radius relation. We frustrate a system of touching spheres by imposing two different bearing states on opposite sides and search for the configurations of lowest energy dissipation. For Coulomb friction (with random friction coefficients) in two dimensions, a sharp line separates the two bearing states and we prove that this line corresponds to the minimum cut. Astonishingly however, in three dimensions, intermediate bearing domains, that are not synchronized with either side, are energetically more favourable than the minimum-cut surface. This novel state of minimum dissipation is characterized by a spanning network of slip-less contacts that reaches every sphere. Such a situation becomes possible because in three dimensions bearings of loops of size four have four degrees of freedom. Since Apollonius of Perga we know that it is possible to fill space with circles of very different sizes. The self-similarity of this construction generates a power-law size distribution defining a fractal dimension. It is possible to generate also space-filling packings of disks in such a way that they can all roll without slip on each other. They can be constructed using conformal transformations. In two dimensions, discrete families of different topologies can be classified algebraically. One finds that they synchronize fastest, when they are hollow. One can also construct random variants, study the mechanical properties and the energy spectrum and observe anomalous diffusion of tracer particles. Applications of these packings are models for turbulence exhibiting Kolmogorov scaling and anomalous heat conduction and models for tectonic gouge in seismic gaps. Space-filling bearing states can be viewed as a realization of solid turbulence.