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 , condensation occurs when its largest eigenvalue grows proportionally to , rather than remaining microscopic.[ref]
Bose-Einstein Condensation and Liquid Helium by Oliver Penrose and Lars Onsager. Abstract and definition of condensation for interacting particles
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.
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Oliver Penrose is a British physicist and mathematician, known for his contributions to theoretical physics, particularly in the fields of quantum mechanics and statistical mechanics. He is notably recognized for his work on the foundation of quantum theory and for developing the concept of Penrose diagrams, which are used in general relativity to represent the causal structure of spacetime.