We report experimental measurements of particle dynamics on slowly sheared granular matter in a three-dimensional Couette cell. A closely packed ensemble of transparent spherical beads is confined by an external pressure and filled with fluid to match both the density and refractive index of the beads. This allows us to track tracer particles embedded in the system and obtain three-dimensional trajectories $\left[r\left(t\right),\theta \left(t\right),z\left(t\right)\right]$ as a function of time. We study the probability distribution function of the vertical and radial displacements, finding Gaussian and exponential distributions, respectively. For slow shear rates, the mean-square fluctuations in all three directions are found to be dependent only on the angular displacement of the Couette cell, $\Delta {\theta }_e$, $⟨\Delta z^2⟩\sim \Delta {\theta }_e$, $⟨\Delta r^2⟩\sim \Delta {\theta }_e^{\alpha }$, $⟨\Delta {\theta }^2⟩\sim \Delta {\theta }_e^{\beta }$, where $\alpha$ and $\beta$ are constants. With $\Delta {\theta }_e$ proportional to the time between measurements, the values of the constants, $\alpha$ and $\beta$, are found to be subdiffusive and superdiffusive, respectively. ThFe linear relation between $⟨\Delta z^2⟩$ and angular displacement implies a diffusive process, from which we can calculate an “effective temperature,” $T_{\text{eff}}$, in the vertical direction, through a fluctuation-dissipation relation. It is of interest to determine whether these systems can be described by analogous equilibrium statistical mechanics concepts such as “effective temperature” and “compactivity.” By studying the dynamics of tracer particles, we find the effective temperature defined by the Stokes-Einstein relation to be independent of the tracer particle characteristic features, such as density and size, and dependent only on the packing density of the system. For slow shear rate, both the diffusivity and mobility of tracer particles are proportional to the shear rate, giving rise to a constant effective temperature, characteristic of the jammed system. We finally discuss the significance of the existence of $T_{\text{eff}}$ for a statistical mechanics formulation of granular matter.

• Received 19 February 2008

DOI:https://doi.org/10.1103/PhysRevE.77.061309