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Oliver Penrose, with Lars Onsager, generalized Bose-Einstein condensation to interacting particles using the one-particle reduced density matrix. With its trace normalized to particle number $N$, condensation occurs when its largest eigenvalue grows proportionally to $N$, rather than remaining microscopic.^[ref]

This defines macroscopic occupation without first assuming that an interacting liquid behaves like an ideal gas in a particular momentum state. The condensate fraction also need not equal the superfluid fraction: coherence and superfluid response are related physical questions, but not interchangeable numerical definitions.