  Thermalization of a system when interacting with a thermal bath is an
interesting problem. If a system eventually reaches a thermal state in the long
time limit, it's expected that its density matrix would resemble the mean-force
Gibbs state. Moreover, the correlation function must satisfy the
Kubo-Martin-Schwinger (KMS) condition or equivalently the
Fluctuation-Dissipation Relation (FDR). In this paper, we derive a formal
expression for the non-Markovian two-point function within the context of the
weak coupling limit. Using this expression, we explicitly compute the two-point
function for specific models, demonstrating their adherence to the KMS. In
addition, we have formulated a non-perturbative approach in the form of a
self-consistent approximation that includes a partial resummation of
perturbation theory. This approach can capture strong coupling phenomena while
still relying on simple equations. Notably, we verify that the two-point
function obtained through this method also satisfies the KMS condition.
