  In existing linear response theories for adiabatically driven cyclic heat
engines, Onsager symmetry is identified only phenomenologically, and a relation
between global and local Onsager coefficients, defined over one cycle and at
any instant of a cycle, respectively, is not derived. To address this
limitation, we develop a linear response theory for the speed of adiabatically
changing parameters and temperature differences in generic Gaussian heat
engines obeying Fokker--Planck dynamics. We establish a hierarchical
relationship between the global linear response relations, defined over one
cycle of the heat engines, and the local ones, defined at any instant of the
cycle. This yields a detailed expression for the global Onsager coefficients in
terms of the local Onsager coefficients. Moreover, we derive an efficiency
bound, which is tighter than the Carnot bound, for adiabatically driven linear
irreversible heat engines based on the detailed global Onsager coefficients.
Finally, we demonstrate the application of the theory using the simplest
stochastic Brownian heat engine model.
