When a system’s particles interact under a repulsive inverse power law, it means that particles will repel each other and the force with which one particle experiences from another can be stated generally as:

**F = k / r**^{n}

^{n}

where **F** is the force exerted on the particle, **k** is a constant, **r** is the distance between the two two particles and **n** is the order of the interaction.

Particles that behave like this can be seen to exist suspended in a fluid when that fluid is near the point where both the liquid and solid forms of it can coexist. This is called a inverse power law melt. Now all liquids occasionally demonstrate collective motion, where a large number of particles move together. This behaviour is quite hard to directly understand in liquids but a gateway can be made through the fact that collective motion in liquids has many parallels with elastic waves in disordered solid materials. Many of the expressions created for the dispersion relation (description of how the wave spread out when moving through a material) can then be reapplied. One of the most successful methods of analysing elastic waves in an amorphous solid is called the quasicrystalline approximation, a simple theory that accurately recreates experimental data.

However there are problems which have been known about this theory since its inception. For instance the thermal interaction which acts to move the particles of the liquid is not considered (although near the phase boundary it is not a major fault) and effects that result in energy loss are also not accounted for. The real question is “when do these effects make the quasicrystalline approximation invalid?” Recently work was done by comparing the molecular dynamic simulations (which are difficult and time consuming) with the much more simple answers calculated by the quasicrystalline approximation. By varying the exponent, the **n** in the equation, between values of 10 and 100 in the calculations it was found that the approximation becomes invalid once **n** ≥ 20 about. Features such as the similarity to the elastic motion of solids also fails to apply once a hard sphere particle interaction begins to take hold.

Paper links: Collective modes in simple melts: Transition from soft spheres to the hard sphere limit