Bose-Einstein condensation, the macroscopic occupation of a single quantum
state, appears in equilibrium quantum statistical mechanics and persists also
in the hydrodynamic regime close to equilibrium. Here we show that even when a
degenerate Bose gas is driven into a steady state far from equilibrium, where
the notion of a single-particle ground state becomes meaningless, Bose-Einstein
condensation survives in a generalized form: the unambiguous selection of an
odd number of states acquiring large occupations. Within mean-field theory we
derive a criterion for when a single and when multiple states are Bose selected
in a non-interacting gas. We study the effect in several driven-dissipative
model systems, and propose a quantum switch for heat conductivity based on
shifting between one and three selected states.Comment: 5+3 pages, 2+2 figure
