  We investigate the relaxation dynamics of open non-integrable quantum
many-body systems in the thermodynamic limit by using a tensor-network
formalism. We simulate the Lindblad quantum master equation (LQME) of infinite
systems by making use of the uniform matrix product operators (MPO) as the
ansatz of their density matrices. Furthermore, we establish a method to measure
the thermodynamic equivalence between two states described by the uniform MPOs.
We numerically show that when an initial state of the LQME is a thermal Gibbs
state, a time evolved state is always indistinguishable from a Gibbs state with
a time-dependent effective temperature in the weak-dissipation and
thermodynamic limit.
