  The stochastic efficiency [G. Verley et al., Nat. Commun. 5, 4721 (2014)] was
introduced to evaluate the performance of energy-conversion machines in
micro-scale. However, such an efficiency generally diverges when no heat is
absorbed while work is produced in a thermodynamic cycle. As a result, any
statistical moments of the efficiency do not exist. In this study, we come up
with a different version of the definition for the stochastic efficiency which
is always finite. Its mean value is equal to the conventional efficiency, and
higher moments characterize the fluctuations of the cycle. In addition, the
fluctuation theorems are re-expressed via the efficiency. For working substance
satisfying the equipartition theorem, we clarify that the thermodynamic
uncertainty relation for efficiency is valid in an Otto engine. To demonstrate
our general discussions, the efficiency statistics of a quantum
harmonic-oscillator Otto engine is systematically investigated. The probability
that the stochastic efficiency surpasses the Carnot efficiency is explicitly
obtained. This work may shed new insight for optimizing micro-machines with
fluctuations.
