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For Q$_{int}$I 2 $_{dc}$Im β > 1, the steady state rate equation in Fig. 5a can only be satisfied with ˙ W$_{e}$ + ˙ U$_{i}$ > 0. There- fore external work W$_{e}$ is generated and the engine beam operates as a heat engine. The distinction between the work and heat flows in Fig. 5 is subtle. For T$_{eff}$ → ∞ it follows that Q$_{eff}$ tends to infinity such that the motion of the resonator is perfectly sinusoidal, like in Fig. 2c. This energy can theoretically be stored with 100% efficiency and should be called work. At any finite value of T$_{eff}$ the resonator performs a thermally driven random motion, which cannot be fully converted into work according to Carnot’s theorem 19 and should be called heat. 

<img src='content_image/423764.jpg'>

For the miniaturization of engines a high power den- sity p$_{dens}$ is essential, since it determines the minimum size the engine can have in order to perform its task within the specified time. The maximum power ˙ W$_{he}$ that the demonstrated heat engine can generate is given by 1 $_{2}$k < x 2 > ω$_{0}$I 2 $_{dc}$Im β . In order to achieve the maximum power density, a current I$_{dc}$ =5 mA is applied for which the amplitude of the centre of mass is limited by colli- sions with electrode T3 to x =100 nm. For an optimum value of Im β =100 A − 2 (see Fig. 4c) it is found that ˙ W$_{he}$ =12.7 nW, about 4% of which is dissipated by the intrinsic losses ( ˙ Q$_{fd}$ ). It thus follows that the piezoresis- tive heat engine, with an engine displacement volume of 

## Performance 

