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We address the question of how a non-equilibrium steady state (NESS) isreached in the Linbdladian dynamics of an open quantum system. We develop anexpansion of the density matrix in terms of the NESS-excitations, each of whichhas its own (exponential) decay rate. However, when the decay rates tend tozero for many NESS-excitations (the spectral gap of the Liouvillian is closedin the thermodynamic limit), the long-time dynamics of the system can exhibit apower-law behaviour. This relaxation to NESS expectation values is determinedby the density of states close to zero spectral gap and the value of theoperator in these states. We illustrate this main idea on the example of thelattice of non-interacting fermions coupled to Markovian leads at infinite biasvoltage. The current comes towards its NESS value starting from a typicalinitial state as $\tau^{-3/2}$. This behaviour is universal and independent ofthe space dimension.

Power-law approach to steady state in open lattices of non-interacting electrons