Horizontal segregation of mono-layer granules coordinated by vertical motion

We experimentally investigate the segregation of a binary mixture of spherical beads confined between two horizontal vertically vibrating plates. The two kinds of beads are of equal diameter and mass but have different restitution coefficients. Segregation occurs in particular ranges of vibration amplitude and frequency. We find that the collisions between beads at an angle to the horizontal plane induce an effective horizontal repulsive force. When one or both bead types bounce up and down in synchronization, the effective repulsive force between the two types of beads is likely to be larger than that found within a single bead type, resulting in the mixture segregating. Non-horizontal collisions also play a role in stabilizing the segregation state by transferring the horizontal kinetic energy back into vertical motion

Brazil nut effect in a rectangular plate under horizontal vibration

An intruder to a group of identical small beads enclosed in a rectangular plate will gradually migrate to either the center or one side of the plate when the plate is subjected to a horizontal vibration. By considering probabilities for a bead to move into and off the space between the intruder and the near side of the plate, we predict that the size ratio and the mass ratio of the intruder to small bead have equal but opposite effects in determining the direction of migration. The prediction is confirmed by a molecular dynamics simulation

Collective motion of inelastic particles between two oscillating walls

This study theoretically considers the motion of N identical inelastic particles between two oscillating walls. The particles' average energy increases abruptly at certain critical filling fractions, wherein the system changes into a solid-like phase with particles clustered in their compact form. Molecular dynamics simulations of the system show that the critical filling fraction is a decreasing function of vibration amplitude independent of vibration frequency, which is consistent with previous experimental results. This study considers the entire group of particles as a giant pseudo-particle with an effective size and an effective coefficient of restitution. The N-particles system is then analytically treated as a one-particle problem. The critical filling fraction's dependence on vibration amplitude can be explained as a necessary condition for a stable resonant solution. The fluctuation to the system's mean flow energy is also studied to show the relation between the granular temperature and the system phase

Energy and phase transition in a horizontally vibrating granular system

The study focuses on the average energy of a monolayer of granular particles confined in a rectangular container. The container is shaken sinusoidally in a horizontal plane. The motion of every particle is recorded by a CCD camera so that the kinetic energy of the system can be analyzed by tracking the trajectory of each particle. It is found that the average energy changes abruptly at a certain critical filling fraction while the configuration of the particles makes a transition from a disordered to a solid-like state. We determine the critical value of the filling fraction and the energy of the solid-like state using a resonant condition

Brazil nut effect in annular containers

This paper investigates the motion of particles between two co-axial cylinders which are subjected to a sinusoidal vertical vibration. We measure the rising time of a large intruder from the bottom of the container to the free surface of the bed particles and find that the rising time as a function of intruder density decreases to a minimum and then increases monotonically. The result is qualitatively opposite to the previous findings in experiments using cylindrical containers where a maximal instead of minimal rising time in the small-density regime was found. The experimental results suggest that the topology of the container plays an important role in the Brazil nut effect

Frequency distributions of complex systems

With a modi. cation to the standard minority game we built a simple model exhibiting the property of self-organized criticality. We found the frequency distribution of the model can be of the form of either a power law, a power law with an exponential cutoff or a log-normal, depending on the value of the information parameter relative to the size of the system. Power laws appear in complex physical systems such as the sandpile and earthquake systems whose information parameters are small compared with their system sizes

Find the Mandelbrot-like sets in any mapping

The Mandelbrot-like sets appear in the parameter spaces of many one-parameter complex mappings. We find that the type of these sets depends on the multiplicity of the critical points of the mappings. We give the effective map for parameters in these sets, and accordingly calculate the positions, the sizes, and the orientations of these Mandelbrot-like sets for any one-parameter complex mappings

Structure of the cubic mappings

The parameter and dynamic spaces of the cubic mappings consist of many small copies of the Mandelbrot-like sets and Julia sets, respectively of the standard quadratic mapping z- > z * z + c. We find the effective quadratic mappings for the kth iteration of the cubic mappings in the neighborhood of the centers of these small copies. Thus we are able to calculate the positions, sizes and orientations of these small copies. The structure of the cubic mappings is therefore understood

The parameter spaces of the cubic polynomials

We plot the two-dimensional projections of the parameter spaces of the cubic mappings. The projection of the parameter points that have non-totally disconnected Julia sets can be seen as a combination of Mandelbrot-like sets. The regularities of these projections with respect to parameters are explained using elementary analysis

Phyllotaxis: Its geometry and dynamics

We have found a relation between the irrational divergence angles and the number of spirals based on the properties of the generalized Fibonacci numbers. Our numerical simulation shows that the patterns of the spiral phyllotaxis depend mainly on the initial growing speed of the primordia