We have used the Gibbs ensemble Monte Carlo method to investigate fluids in
the N dimensional space. We consider the practical way to perform these si
mulations and give expressions for the long-range corrections. Results for
the four-dimensional Lennard-Jones fluid are presented. The calculated phas
e diagram and several thermodynamic properties for this fluid are given. Es
timates of the critical point and the critical exponent are obtained from t
he phase diagram. The simulations were performed with two different simulat
ion techniques: random choice and a fixed sequence of Monte Carlo steps. (C
) 1999 American Institute of Physics. [S0021-9606(99)52041-2].