Mixing rate in Classical Many Body Systems

Mixing in many body systems is intuitively understood as the change in time of the set of neighbors surrounding each particle. Its rate and its development over time hold important clues to the behavior of many body systems. For example, gas particles constantly change their position and surrounding particles, while in solids one expects the motion of the atoms to be limited by a fixed set of neighboring atoms. In other systems the situation is less clear. For example, agitated granular systems may behave like a fluid, a solid or glass, depending on various parameter such as density and friction. Thus, we introduce a parameter which describes the mixing rate in many body systems in terms of changes of a properly chosen adjacency matrix. The parameter is easily measurable in simulations but not in experiment. To demonstrate an application of the concept, we simulate a many body system, with particles interacting via a two-body potential and calculate the mixing rate as a function of time and volume fraction. The time dependence of the mixing rate clearly indicates the onset of crystallization.