  An ideal equilibrium Bose--Einstein condensate (BEC) is usually considered in
the grand canonical ($\mu V T$) ensemble, which implies the presence of the
chemical equilibrium with the environment. However, in most experimental
scenarios, the total amount of particles in BEC is determined either by the
initial conditions or by the balance between dissipation and pumping. As a
result, BEC may possess the thermal equilibrium but almost never the chemical
equilibrium. In addition, many experimentally achievable BECs are non-ideal due
to interaction between particles. In the recent work
[10.1103/PhysRevLett.128.065301], it has been shown that invariant subspaces in
the system Hilbert space appear in non-equilibrium BEC in the fast
thermalization limit. In each of these subspaces, Gibbs distribution is
established with a certain number of particles that makes it possible to
investigate properties of non-ideal non-equilibrium BEC independently in each
invariant subspace. In this work, we analyze the BEC stability due to change in
dispersion curve caused by non-ideal interactions in BEC. Generally, non-ideal
interactions lead to the redshift or blueshift of the dispertion curve and to
the change in the effective mass of the particles. We show that the redshift of
the dispersion curve can lead to the negative compressibility of BEC, whereas
the change in the effective mass always makes BEC more stable. We find the
explicit condition for the particle density in BEC, at which the negative
compressibility appears.
