  In an ideal Bose gas that is driven into a steady state far from thermal
equilibrium, a generalized form of Bose condensation can occur. Namely, the
single-particle states unambiguously separate into two groups: the group of
Bose-selected states, whose occupations increase linearly with the total
particle number, and the group of all other states whose occupations saturate
[Phys. Rev. Lett. 111, 240405 (2013)]. However, so far very little is known
about how the number of Bose-selected states depends on the properties of the
system and its coupling to the environment. The answer to this question is
crucial since systems hosting a single, a few, or an extensive number of
Bose-selected states will show rather different behavior. While in the former
two scenarios each selected mode acquires a macroscopic occupation,
corresponding to (fragmented) Bose condensation, the latter case rather bears
resemblance to a high-temperature state of matter. In this paper, we
systematically investigate the number of Bose-selected states, considering
different classes of the rate matrices that characterize the driven-dissipative
ideal Bose gases in the limit of weak system-bath coupling. These include rate
matrices with continuum limit, rate matrices of chaotic driven systems, random
rate matrices, and rate matrices resulting from thermal baths that couple to a
few observables only.
