  The efficiency at maximum power output of linear irreversible Carnot-like
heat engines is investigated based on the assumption that the rate of
irreversible entropy production of working substance in each "isothermal"
process is a quadratic form of heat exchange rate between the working substance
and the reservoir. It is found that the maximum power output corresponds to
minimizing the irreversible entropy production in two "isothermal" processes of
the Carnot-like cycle, and that the efficiency at maximum power output has the
form as $\eta_{mP}={\eta_C}/(2-\gamma\eta_C)$ where $\eta_C$ is the Carnot
efficiency while $\gamma$ depends on the heat transfer coefficients between the
working substance and two reservoirs. The value of $\eta_{mP}$ is bounded
between $\eta_{-}\equiv \eta_C/2$ and $\eta_{+}\equiv\eta_C/(2-\eta_C)$. These
results are consistent with those obtained by Chen and Yan [J. Chem. Phys.
\textbf{90}, 3740 (1989)] based on the endoreversible assumption, those
obtained by Esposito \textit{et al.} [Phys. Rev. Lett. \textbf{105}, 150603
(2010)] based on the low-dissipation assumption, and those obtained by Schmiedl
and Seifert [EPL \textbf{81}, 20003 (2008)] for stochastic heat engines which
in fact also satisfy the low-dissipation assumption. Additionally, we find that
the endoreversible assumption happens to hold for Carnot-like heat engines
operating at the maximum power output based on our fundamental assumption, and
that the Carnot-like heat engines that we focused does not strictly satisfy the
low-dissipation assumption, which implies that the low-dissipation assumption
or our fundamental assumption is a sufficient but non-necessary condition for
the validity of $\eta_{mP}={\eta_C}/(2-\gamma\eta_C)$ as well as the existence
of two bounds $\eta_{-}\equiv \eta_C/2$ and $\eta_{+}\equiv\eta_C/(2-\eta_C)$.

Statistical Mechanics