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In the standard framework of thermodynamics the work produced or consumed ina process is a random variable whose average value is bounded by the change inthe free energy of the system. This work is calculated without regard for thesize of its fluctuations. We find that in some processes, such as reversiblecooling, the fluctuations of the work can diverge. Small or fragile thermalmachines may be unable to cope with large fluctuations. Hence, with the presentfocus on nano- scale thermodynamics, we analyse how thermodynamic efficiencyrates are modified when the size of the fluctuations around the average isrestricted. We quantify the work content and work of formation of any statewhen the work fluctuations are bounded by a given amount c. By varying c weinterpolate between the standard free energy c = infinity and the min-freeenergy c = 0, defined in the context of single-shot thermodynamics. We derivefundamental relations between average work and its fluctuations, and explorethe emergence of irreversibility and partial order on state transformationswhen bounding fluctuations. We also study the efficiency a single qubit thermalengine model with constrained fluctuations, and derive the corrections to theCarnot efficiency.

From single-shot to general work extraction with bounded fluctuations in work