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Patterns such as periodic subharmonic waves and kink-pairs are observed experimentally in a two-dimensional vertically vibrated granular layer located in an annular channel which has a sawtooth-shaped base. Using the thickness of the granular layer and external excitation as control parameters, we find the transitions between different patterns in the layer. The dependence of the kink-pair's velocity on the external excitation is revealed. The local velocity fields of the granular layer indicate that, in addition to horizontal transport in the layer, there is convection in the node area of a kink.

The fact that trapezoid clusters exist in 2D vertically vibrated granular systems leads us to construct a cluster model, in which wave-like motions are explained as the result of cluster-plate and cluster-cluster collisions. By analyzing the collision of one cluster with the plate in detail, we deduce a basic equation from velocity relationship, which could be separated into two correlative equations: one relates wave-like motion with exciting acceleration, and we call it the excitation condition; the other relates wavelength with exciting frequency, viz., the dispersion relation. The theoretical results are in agreement with the experimental ones, which supports the idea of the cluster model. Moreover, from the cluster model, we also predict a possibility of abnormal dispersion relation of a 2D granular system.

The heat conduction in a one-dimensional (1D) hard-point model with mass gradient is studied. Using numerical simulation, we find an asymmetric heat conduction in this model with greater heat current in the direction of mass increase. The increase of temperature gradient, mass gradient and system size are found to enhance the asymmetric heat conduction. Based on the collision dynamic of a hard-point particle, we give a qualitative explanation for the underlying mechanism of asymmetric effect.