If there is only one species, then we are done. But, if there are, say, two species, then there will be additional potentials such as U [ μ μ 1 , μ μ 2 ] = U − − μ μ 1 N 1 − − μ μ 2 N 2 and so on. If there are D dimensions to the thermodynamic space, then there are 2 unique thermodynamic potentials. For the most simple case, a single phase ideal gas, there will be three dimensions, yielding eight thermodynamic potentials.