Robustness of wave functions of interacting many bosons in a leaky Box

We study the robustness, against the leakage of bosons, of wave functions of interacting many bosons confined in a finite box by deriving and analyzing a general equation of motion for the reduced density operator. We identify a robust wave function that remains a pure state, whereas other wave functions, such as the Bogoliubov's ground state and the ground state with a fixed number of bosons, evolve into mixed states. Although these states all have the off-diagonal long-range order, and the same energy, we argue that only the robust state is realized as a macroscopic quantum state.