Experiments demonstrate that the null space of the rigidity matrix determines grain motion during vibration-induced compaction

Using a previously developed experimental method to reduce friction in mechanically stable packings of disks, we find that frictional packings form tree-like structures of geometrical families that lie on reduced dimensional manifolds in configuration space. Each branch of the tree begins at a point in configuration space with an isostatic number of contacts and spreads out to sequentially higher dimensional manifolds as the number of contacts are reduced. We find that gravitational deposition of disks produces an initially under-coordinated packing stabilized by friction on a high-dimensional manifold. Using short vibration bursts to reduce friction, we compact the system through many stable configurations with increasing contact number and decreasing dimensionality until the system reaches an isostatic frictionless state. We find that this progression can be understood as the system moving through the null-space of the rigidity matrix defined by the interparticle contact network in the direction of the gravitational force. We suggest that this formalism can also be used to explain the evolution of frictional packings under other forcing conditions.