A "mean-field approximation" on the phase transitions of three-dimensional Lennard-Jones model

It is difficult to derive the solid-fluid transition theoretically from microscopic models, although this phenomenon itself has been investigated for a long time. We previously constructed an exactly-solvable model with the solid-fluid transition. This model resembles the infinite-range (or mean-field) model in spin systems in some points, hence it can be called a "mean-field model" of the solid-fluid transition. In the present paper, we construct a "mean-field approximation" of the solid-fluid transition by using the "mean-field model" introduced in our previous study, and tries to describe the phase transitions of the three-dimensional Lennard-Jones model as an example. This approximation succeeds in describing the phase diagram which contains three (gas, liquid, and fcc-solid) phase, at least qualitatively.