When would an open system reach the steady state calculated from master equation?

From the master equation for density matrix, it seems that one can have steady state solution requiring the derivative of density matrix equals to zero, but I want to know whether a real open system would reach this steady state after long evolution time?

Or is there any requirement for the open system to reach the steady state? And does the steady state (for example, the number of particles for the open system in the steady state) still decay or not?

• How do you define "reach a steady state?" The most strict reading of steady state is a limit as t approaches infinity, which is never reached by any smoothly varying system. However, there are many slightly varied definitions of "reach a steady state" which are used for practical purposes (such as being "steady state" when your measurement errors are larger than the difference between steady state and not) – Cort Ammon May 30 '19 at 15:53
• thank you for your comment @Cort Ammon, I think here we can restrict the discussion without worrying too much about experimental measurements. for example, if I solve the time dependent master equation, I get a time dependent solution, which is possible to show that the system would decay to its ground state after infinite long time, but this seems differ with the steady state which the system is supposed to become after infinite long time, is this a problem ? – guangcun May 31 '19 at 1:57
• @guangcun A system which is in the ground state is also in a steady state, as its state is not changing with time. – probably_someone May 31 '19 at 11:08
• @probabaly_someone, Yes, I agree with you, but the ground state is different with the steady state which directly obtained from taking the master equation (derivative of density matrix) equals zero. am I right ? – guangcun May 31 '19 at 11:14