Abstract
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
Highlights
Many body systems exhibit a myriad of phases, which may be stable or metastable
When speaking about frozen systems a naïve picture springs to mind in which the position of each particle is fixed relatively to the other particles, in contrast to particles in a fluid which keep changing their relative positions
Though, that in a frozen system, each particle sees for long times, more or less, the same neighbors
Summary
Mixing in many body systems is intuitively understood as the change in time of the set of neighbors surrounding each particle. We introduce a parameter which describes the mixing rate in many body systems in terms of changes of a properly chosen adjacency matrix. Like ordered solids or glasses, the relative positions of particles keep changing all the time, due to finite temperature. (Even at very low temperature the particles cannot be seen as fixed due to zero-point motion, but in that case the system is not in the classical regime.) the naïve criterion described above, when taken strictly is not really helpful It is true, though, that in a frozen system, each particle sees for long times, more or less, the same neighbors. If the cells are larger than l, the adjacency matrix will change all the time and the system which we would describe as frozen will have a non-vanishing mixing rate f. An alternative but related approach is to employ instead of the adjacency matrix at a given time, t, a cumulative adjacency matrix that gives who are the neighbors at a given time and the neighbors at earlier and later times
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