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Lars Onsager formulated reciprocal relations connecting cross-coupled transport coefficients near thermodynamic equilibrium. The Onsager reciprocal relations link microscopic reversibility with symmetry properties of macroscopic linear response, subject to the relevant time-reversal and field conventions.^[ref]

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.

An Onsager reciprocal relation connects cross-couplings between near-equilibrium transport processes. Choose thermodynamic forces $X_j$ and fluxes $J_i$ so that their products give the entropy production, and write $J_i={\sum }_j{L}_{ij}{X}_j$. With microscopic reversibility, no imposed time-reversal-breaking field and variables with the appropriate matching time-reversal parity, the coefficients obey $L_{ij}=L_{ji}$. The choice of conjugate force and flux matters: arbitrary rescaling of just one coefficient need not preserve its numerical symmetry.

For coupled thermal and electrical transport, a thermal driving force can induce charge flow while an electrical driving force induces heat flow. Reciprocity relates those cross effects, reducing the number of independent transport measurements. Lars Onsager's 1931 paper develops this reasoning from microscopic reversibility and distinguishes it from thermodynamic positivity alone.^[ref] External magnetic fields and odd time-reversal variables require a generalized parity-sensitive relation rather than an unconditional symmetric matrix.

By Lars Onsager. Physical Review 37, 405–426. Published 15 February 1931. The abstract and §§1–2 explain coupled irreversible processes and the derivation from microscopic reversibility; §7 states limitations with imposed magnetic fields and rotation.