  In the present paper, a global Lindbladian ansatz is constructed which leads
to thermalization at temperature $T$ to the Gibs state of the investigated
system. This ansatz connects every two eigenstates of the Hamiltonian and leads
to a simple master equation known in the literature as the relaxation time
approximation (RTA). The main message of this paper is that RTA, being a
Lindbladian approach itself, can be used as Lindbladian securing thermalization
when modeling physical processes, and can be consequently combined with other
types of Lindbladians which would drive the system of the equilibrium state. I
demonstrate it with two applications. The first application is the slow cooling
(or heating) of quantum systems by varying the environment temperature to a
critical point. With this RTA-Lindblad ansatz, one can directly relate to the
equilibrium behavior of the system, and if an order parameter has the exponent
$\Psi$, the remaining value at the phase transition will decrease with
$1/\tau^{\Psi}$, where $\tau$ is the overall time of the slow process. In the
second application, I investigate the change in the expectation value of a
conserved quantity (an operator commuting with the Hamiltonian) due to an extra
Lindbladian term which would drive the system out from equilibrium, while the
thermalizing RTA-Lindbladian term is also present. I give a closed perturbative
expression in the first order for the expectation value in the new steady state
using only expectation values calculated in the original thermal equilibrium.
