Background

Granular Matter

As mentioned in the introduction, the term Granular Matter is used for systems consisting of many solid particles (grains) of macroscopic size.

A common example of such a system is sand. In some respects it resembles other many-particle-systems like fluids (e.g. in a hourglass) or solids (e.g. a sandcastle). But there are also significant differences. In granular matter microscopic effects like Brownian Motion or quantum effects are not an issue but instead interactions between the particles are characterized almost completely by inelastic collisions and friction.

This gives rise to the dissipative nature of granular matter which is of huge importance for its behaviour. During the collisions energy is transferred from the macroscopic degrees of freedom into the microscopic ones and hence lost for the macroscopic motion. The same holds for the entropy which yields that the equilibrium state has no longer to be a spatially uniform one. Instead there could other be distributions with smaller macroscopic entropy but higher total entropy because of the inner entropy.

Theoretical Modelling and Stationary Solutions

To analyse the properties of granular matte rmore precisely in our experiment, we introduce a parameter $\mathfrak{a}$ as a measure of asymmetry for a set of particles that is distributed on two different boxes: $$\mathfrak{a} := \frac{N_l -\frac{N}{2}}{N}$$ where $N_l$ denotes the number of particles on the left-hand side and $N$ the total number of particles. One can derive that - for a two dimensional system - the time evolution of $\mathfrak{a}$ can be described by the differential equation $$\frac{\partial \mathfrak{a}}{\partial t} = F_0 N \left[ \left(\mathfrak{a} -\frac{1}{2}\right)^2 e^{-\mu \left(\mathfrak{a} -\frac{1}{2}\right)^2} - \left(\mathfrak{a} +\frac{1}{2}\right)^2 e^{-\mu \left(\mathfrak{a} +\frac{1}{2}\right)^2} \right] + \xi$$ where $\xi$ is a noise term and $$\mu = 4 \pi \cdot \frac{g h}{v_b^2} \cdot (r\bar{N})^2 \cdot (1-\varepsilon)^2$$ is a parameter that contains all relevant information about the system (particle density $\bar{N}$ and radius $r$, velocity of the bottom plate $v_b$, gravity $g$, height of the mid-wall $h$ and coefficient of restitution $\varepsilon$) (see J. Eggers. Phys. Rev. Lett., 83:5322, 1999).

The stationary solutions of this equation correspond to equilibrium states. For large values of $\mu$ asymmetric solutions are possible.