  We consider finite-range, many-body fermionic lattice models and we study the
evolution of their thermal equilibrium state after introducing a weak and
slowly varying time-dependent perturbation. Under suitable assumptions on the
external driving, we derive a representation for the average of the evolution
of local observables via a convergent expansion in the perturbation, for small
enough temperatures. Convergence holds for a range of parameters that is
uniform in the size of the system. Under a spectral gap assumption on the
unperturbed Hamiltonian, convergence is also uniform in temperature. As an
application, our expansion allows us to prove closeness of the time-evolved
state to the instantaneous Gibbs state of the perturbed system, in the sense of
expectation of local observables, at zero and at small temperatures. As a
corollary, we also establish the validity of linear response. Our strategy is
based on a rigorous version of the Wick rotation, which allows us to represent
the Duhamel expansion for the real-time dynamics in terms of Euclidean
correlation functions, for which precise decay estimates are proved using
fermionic cluster expansion.
