  The Carnot theorem, one expression of the second law of thermodynamics,
places a fundamental upper bound on the efficiency of heat engines operating
between two heat baths. The Carnot theorem can be stated in a more generalized
form for heat engines operating with multiple heat baths, where the maximum
efficiency is achieved for reversible heat engines operating quasistatically
between two heat baths. In this study, we determine the irreversible efficiency
of heat engines operating with multiple heat baths in a linear response regime,
i.e., under small temperature differences and a slow variation of the control
parameters, by quantifying the impact of the dissipation by irreversible
operations. The Carnot theorem is derived as a natural consequence of it.
Because the result obtained is based on the linear response relation and
fluctuation-dissipation theorem in the universal framework of linear response
theory, it has wide applicability to irreversible heat engines operating in the
linear response regime.
