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14.2.1 Problem Description

Although we live in a three-dimensional world, many important processes involve interactions on surfaces, which are effectively two-dimensional. While experimental studies of two-dimensional systems have been successful in probing some aspects of such systems, computer simulation is another powerful tool that can be used to measure their properties. We have used a computer simulation to study the melting  transition of a two-dimensional system of interacting particles [Johnson:86a;86b]. One purpose of the study is to investigate whether melting in two dimensions occurs through a qualitatively different process than it does it three dimensions. In three dimensions, the melting transition is a first-order transition which displays a characteristic latent heat. Halperin and Nelson [Halperin:78a], [Nelson:79a] and Young [Young:79a] have raised the possibility that melting in two dimensions could occur through a qualitatively different process. They have suggested that melting could consist of a pair of higher-order phase transitions,  which lack a latent heat, that are driven by topological defects in the two-dimensional crystal lattice.

We studied a two-dimensional system of particles interacting through a truncated Lennard-Jones potential.  The Lennard-Jones potential is

where is the energy parameter, is the length parameter, and r is the distance between two particles. The potential is attractive at distances larger than and repulsive smaller distances. The potential energy of the whole system is the sum of the potential energies of each pair of interacting particles. In order to ease the computational requirements of the simulation, we have truncated the potential at a particle separation of .

Mark A. Johnson wrote the Monte Carlo  simulation of melting in two dimensions for his Ph.D. research at Caltech.

Guy Robinson
Wed Mar 1 10:19:35 EST 1995