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 simulations and give expressions for the long range corrections. Results for the four-dimensional Lennard-Jones fluid are presented. The calculated phase diagram and several thermodynamic properties for this fluid are given. Estimates of the critical point and the critical exponent are obtained from the phase diagram. The simulations were performed with two different simulation techniques: random choice and a fixed sequence of Monte Carlo steps.

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