  In this paper, we derive a generalized second fluctuation-dissipation theorem
(FDT) for stochastic dynamical systems in the steady state. The established
theory is built upon the Mori-type generalized Langevin equation for stochastic
dynamical systems and only uses the properties of the Kolmogorov operator. The
new second FDT expresses the memory kernel of the generalized Langevin equation
as the correlation function of the fluctuation force plus an additional term.
In particular, we show that for nonequilibrium states such as heat transport
between two thermostats with different temperatures, the classical second FDT
is valid even when the exact form of the steady state distribution is unknown.
The obtained theoretical results enable us to construct a data-driven nanoscale
fluctuating heat conduction model based on the second FDT. We numerically
verify that the new model of heat transfer yields better predictions than the
Green-Kubo formula for systems far from the equilibrium.
