That is, the attributes of each particle should be independent of those attributes of any other particle. The attributes of particles should be able to fill the entire attribute space in a manner which is consistent with the physics. For example, if we are launching particles into a hemispherical space above a surface, then we should be able to approach completely filling the hemisphere with outgoing directions, as we approach an infinite number of particles launched. This means that any subsequence of random numbers should not be correlated with any other subsequence of random numbers. Most especially, n-tuples of random numbers should be independent of one another.
More explanation is provided below.
Uniformity - The sequence of random numbers should be uniform, and unbiased. Often, when there is a lack of uniformity, there are n-tuples of random numbers which are correlated. In this case, the space might be filled in a definite, easily observable pattern. Thus, the properties of uniformity and uncorrelated sequences are loosely related. The generator should be efficient.