Quantum thermodynamics with constrained fluctuations in work

In the standard framework of thermodynamics the work produced or consumed in a process is a random variable whose average is bounded by the change in the free energy of the system. This average 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 given by the change in free energy can diverge. Small or fragile thermal machines may be unable to cope with arbitrarily large fluctuations. Hence, with the present focus on nano-scale thermodynamics, we analyse how thermodynamic efficiency rates are modified when the size of these fluctuations are restricted. We quantify the work content and work of formation of arbitrary finite dimensional quantum states when the work fluctuations are bounded by a given amount c. By varying c we interpolate between the work given by the change in the standard free energy c=∞ and the min-free energy c=0, defined in the context of single-shot thermodynamics. We derive fundamental relations between average and deterministic work, 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.