<
strong>MATHEMATICAL PHYSICS WEBINAR
RUTGERS UNIVERSITY
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Wednesday, July 10th, 10:45AM EDT
Rolling Matter ;
We will discuss realizations of solids, which can sustain internally rotational degrees of freedom. They a re discrete versions of Cosserat media. We will consider bearings, which ar e 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 perce ived as physical realizations of networks of oscillators with asymmetricall y weighted couplings. These networks can exhibit optimal synchronization pr operties through tuning of the local interaction strength as a function of node degree or the inertia of their constituting rotor disks through a powe r-law mass-radius relation. We frustrate a system of touching spheres by im posing two different bearing states on opposite sides and search for the con figurations of lowest energy dissipation. For Coulomb friction (with random friction coefficients) in two dimensions, a sharp line separates the two bear ing states and we prove that this line corresponds to the minimum cut. Asto nishingly however, in three dimensions, intermediate bearing domains, that are not synchronized with either side, are energetically more favourable th an the minimum-cut surface. This novel state of minimum dissipation is char acterized by a spanning network of slip-less contacts that reaches every sp here. Such a situation becomes possible because in three dimensions bearing s of loops of size four have four degrees of freedom. Since Apollonius of P erga we know that it is possible to fill space with circles of very differe nt sizes. The self-similarity of this construction generates a power-law si ze distribution defining a fractal dimension. It is possible to generate al so space-filling packings of disks in such a way that they can all roll wit hout slip on each other. They can be constructed using conformal transforma tions. In two dimensions, discrete families of different topologies can be classified algebraically. One finds that they synchronize fastest, when the y are hollow. One can also construct random variants, study the mechanical properties and the energy spectrum and observe anomalous diffusion of trace r particles. Applications of these packings are models for turbulence exhib iting Kolmogorov scaling and anomalous heat conduction and models for tecto nic gouge in seismic gaps. Space-filling bearing states can be viewed as a realization of solid turbulence.
DTSTAMP:20260828T015309 DTSTART;TZID=America/New_York:20240710T104500 DTEND;TZID=America/New_York:20240710T114500 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR