Entropy linear response theory with non-Markovian bath

We developed a perturbative calculation for entropy dynamics considering sudden coupling between system and bath. The theory we can work in general environment without Markovian approximation. A formula is given bosonic fermionic environment, respectively. find the Renyi response only related to spectral functions of together with specific statistical kernel distribution function. t^2 growth/decay short time limit t linear longer scale second entropy. non-monotonic behavior systems found be quite when environment's temperature lower. Fourier's law heat transport obtained two systems' are close each other. consistency check made Sachdev-Ye-Kitaev model free fermions, Page curve alike process dual black hole evaporation. An oscillation entanglement gapped spectrum environment.