Behringer, RP, *The scientist in the sandbox: Time-dependence, fractals and waves*,
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 7 no. 5
(1997),
pp. 963-978 .
**Abstract:**

*Granular materials exhibit a rich variety of dynamical behavior, much of which is poorly understood. Fractal-like stress chains, convection, a variety of wave dynamics, including waves which resemble capillary waves, 1/f noise, and fractional Brownian motion provide examples. Although granular materials consist of collections of interacting particles, there are important differences between the dynamics of a collection of grains and the dynamics of a collection of molecules. In particular, the ergodic hypothesis is generally invalid for granular materials, so that ordinary statistical physics does not apply. Fluctuations on laboratory scales in such quantities as the stress can be very large - as much as an order of magnitude greater than the mean. Below is a brief review of some of the theoretical approaches to granular flow followed by a discussion of several recent experiments. These experiments focus on complex structures and fluctuations in the flow of granular materials in a hopper or in simple shear flow. The experimental work at Duke has been carried out in collaboration with a number of investigators, including G. W. Baxter, R. Leone, H. K. Pak, E. Van Doom, C. O'Hern, and B. Miller. Elsewhere in this issue, Pak et al. discuss experiments to characterize and better understand the convective flows which occur when granular materials are shaken with accelerations exceeding that of gravity.*