Van der Waals type equation of state for Lennard-Jones fluid and the fluctuation of the potential energy by molecular dynamics simulations

Molecular dynamics (MD) simulations of a Lennard-Jones system are performed to obtain the pVT and UVT relations. An extended van der Waals equation of state (EOS) is derived by statistical mechanics on the perturbation approximation. A hard sphere system is used as the reference system. The Ree-Hoover EOS is assumed for the hard sphere system. The attraction energy term in the canonical ensemble partition function is extended by a cluster expansion. The new EOS includes three parameters, two of which are the interaction parameters in the Lennard-Jones interaction. The last parameter is the effective volume of the hard sphere system as the reference. The extended van der Waals EOS reproduces the pVT and UVT relations, at least qualitatively, whereas the original van der Waals EOS can explain only the pVT relation. The present EOS can also explain the fluctuations of the potential energy as calculated by MD simulations with 1000 molecules in the unit cell. In this sense, the present EOS gives better heat capacities at constant volume C-v, even at lower temperatures than the original van der Waals EOS, which includes the attraction part of the equation, with no temperature dependence in the internal energy U.