1.1   Lennard-Jones potential

The physics and emergent behaviour of interacting particles has long standing history: classical computational problems include the simulation of the behaviour of many atoms in liquids, solids, proteins etc, and common simulation techniques are Molecular Dynamics and (Metropolis) Monte Carlo methods.

For both, we need to know the pairwise interaction potential between two particles (we ignore here systems that require 3 and more body interactions to be considered).

Figure A: (Left) The Lennard Jones potential: the distance A0 is the optimum distance as it has the lowest potential energy. (Right) A purely repulsive potential: particles exposed to this will try to separate from each other as far as possible. We have used arbitrary units on both axes for these schematic plots.

For example, the well known Lennard Jones potential (shown in Figure A (left) above) for two particles such as inert atoms, has a repulsive term that for short distances R increases the energy (of the type \(1/R^{12}\), and an attractive term (of the type \(1/R^6\)). The first term originates in strong repulsion of electron orbitals that start to overlap, the second in weak attraction from induced electrical polarisation. The two terms combined result in a potential as shown in figure 1a), which has a minimum at a distance A0 (approximately 3 for the schematic sketch): each pair of particles has the lowest possible energy if they can be separated by this distance A0.

1.2   Repulsive potential

Another area of complex system research that uses particle-based simulation techniques such as Molecular Dynamics and Monte Carlo methods is that of the the dynamics of vortex lines in (Type II) superconductors. In these systems, the vortex lines always repel each other, and a corresponding potential is shown in figure A (right): for all distances, the energy decreases if we increase the distance between the two interacting objects, separated by a distance R.

In these systems, the vortex lines cannot escape the sample, so that they will arrange in a way to minimise they energy, which is - in the absence of any other disordering effects and absence of geometrical constraints of the sample - a hexagonal lattice.

Figure B: Delaunay triangulation of (nearly) hexagonal system of vortices that repel each other with a potential as in Figure A (right). They are also exposed to a small disordering effect that causes the deviation from the perfect hexagonal lattice. [Taken from Figure 5 in Vortex Dynamics in two-dimensional systems at high driving forces (postprint pdf).]