  In the standard framework of thermodynamics the work produced or consumed in
a process is a random variable whose average value is bounded by the change in
the free energy of the system. This work is calculated without regard for the
size of its fluctuations. We find that in some processes, such as reversible
cooling, the fluctuations of the work can diverge. Small or fragile thermal
machines may be unable to cope with large fluctuations. Hence, with the present
focus on nano- scale thermodynamics, we analyse how thermodynamic efficiency
rates are modified when the size of the fluctuations around the average is
restricted. We quantify the work content and work of formation of any state
when the work fluctuations are bounded by a given amount c. By varying c we
interpolate between the standard free energy c = infinity and the min-free
energy c = 0, defined in the context of single-shot thermodynamics. We derive
fundamental relations between average work and its fluctuations, and explore
the emergence of irreversibility and partial order on state transformations
when bounding fluctuations. We also study the efficiency a single qubit thermal
engine model with constrained fluctuations, and derive the corrections to the
Carnot efficiency.
