Collective Dynamics of Disc Particles II Stability Analysis
Imagine a completely smooth, mirror-like surface of a quiet lake. A pebble falling in the water produces a small wave — in scientific terms, a perturbation — which rapidly dies out. However, if at that moment the wind is blowing, the wave will not be damped, but be amplified, to become — depending on the strength of the wind — a small ripple or a breaker wave. If we look at this situation with the eyes, not of a poet, but of a scientist we can say that if there is no wind the equilibrium state of the system (the water surface) is stable (the perturbations die out) while the wind causes an instability (growth of the initial perturbations). The instability in turn produces a new structural state of the system — a regular wave sequence. If we are dealing with a system which has not yet been studied in detail, then we may ask the question whether a given spatially uniform state of the system is stable. First of all, it would be well to ascertain whether there is an energy source in the system without which it would be impossible to produce any structure. However, the existence of such a source does not necessarily mean an instability or the appearance of a new structural state; hence we must study the dynamics of the system in detail. We may assume that we know the equations which more or less completely describe our system. How can we then examine such a system for stability?
KeywordsEnergy Balance Equation Linear Oscillation Collective Dynamics Energy Instability Diffusion Instability
Unable to display preview. Download preview PDF.