Fri March 6th 2009
Seminar Force network statistics
Brian Tighe


In disordered athermal media like grains, foams, and emulsions, stresses display strong spatial inhomogeneities, especially near the transition to mechanical rigidity, or jamming. This inhomogeneity can be characterized by the contact force probability distribution function (``force distribution''), which gives the probability of finding a contact force of a particular magnitude. In giving a microscopic statistical measure of the stresses in a disordered system, it plays a similar and complementary role to the more familiar pair correlation function.

A prinicple motivation for studying the force distribution is to determine how its shape evolves with the distance to the jamming point, a nonequilibrium critical point. In particular, one expects fluctuations to increase as the jamming point is approached, and the force fluctuations are strongly influenced by the form of the tail of the distribution. A number of simple models predict that the tail should be robustly exponential in the jammed phase. This prediction generally follows from an analogy to the microcanonical ensemble, in which a conserved stress replaces the role of energy. Thus the tail of the force distribution also speaks to the minimal ingredients in an athermal ensemble-based approach to jammed matter.

I will report on numerical and analytical results in the force network ensemble, a simple model that emphasizes the role of vector force balance in static disordered media. In particular, we find that the force distribution is not robustly exponential; though broad, and hence admitting large fluctuations, it generally decays as a Gaussian in two dimensions. I will demonstrate that the Gaussian tail follows from a previously hidden conserved quantity, the area of a reciprocal tiling, that exists as a consequence of local force balance. I will discuss the role of the conserved quantity in both the ensemble and ``real'' numerical systems, as well as implications for the interpretation of experiments.
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The 10th Complex Motion in Fluids 2021
Max Planck Gesellschaft
Centre for Scientific Computing