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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.

The one-body density matrixuses a creation operator and an annihilation operator to describe first-order spatial quantum coherence. Its diagonal is particle-number density and its trace is the expectation value of the particle number operator, $\mathrm{Tr}{G}^{(1)}=⟨N⟩$. Dividing by $⟨N⟩$ instead gives a unit-trace convention, so the normalization matters when interpreting its eigenvalues.

For bosons, an eigenvalue that remains proportional to particle number in the thermodynamic limit identifies Bose-Einstein condensation. This criterion extends occupation of a single-particle mode to interacting systems.^[ref]

In a homogeneous system of volume $V$, an occupied plane-wave mode contributesA constant long-distance limit therefore corresponds to a zero-momentum condensate; a moving condensate can retain a spatial phase. Finite local density or short-distance quantum coherence alone does not establish off-diagonal long-range order.