Efficiency at maximum power is studied for two simple engines (three- and five-state systems). This quantity is found to be sensitive to the variable with respect to which the maximization is implemented. It can be wildly different from the well-known Curzon-Ahlborn bound (one minus the square root of the temperature ratio), or can be even closer than previously realized. It is shown that when the power is optimized with respect to a maximum number of variables the Curzon-Ahlborn bound is a lower bound, accurate at high temperatures, but a rather poor estimate when the cold reservoir temperature approaches zero (at which point the Carnot limit is achieved).
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