Abstract
In the standard framework of thermodynamics, work is a random variable whose average is bounded by the change in free energy of the system. This average work is calculated without regard for the size of its fluctuations. Here we show that for some processes, such as reversible cooling, the fluctuations in work diverge. Realistic thermal machines may be unable to cope with arbitrarily large fluctuations. Hence, it is important to understand how thermodynamic efficiency rates are modified by bounding fluctuations. We quantify the work content and work of formation of arbitrary finite dimensional quantum states when the fluctuations in work are bounded by a given amount c. By varying c we interpolate between the standard and minimum free energies. We derive fundamental trade-offs between the magnitude of work and its fluctuations. As one application of these results, we derive the corrected Carnot efficiency of a qubit heat engine with bounded fluctuations.
Highlights
In the standard framework of thermodynamics, work is a random variable whose average is bounded by the change in free energy of the system
We find that in many protocols, for example thermodynamically reversible cooling, the work must have fluctuations that diverge in size
In this article we have derived tight bounds on the minimal fluctuations in work associated with thermodynamically reversible protocols, for which the average work is given by difference in free energy between initial and final states
Summary
In the standard framework of thermodynamics, work is a random variable whose average is bounded by the change in free energy of the system. The work cost of forming a state from the Gibbs state is given by the max-free energy[1,4], which is generally significantly larger than the deterministic work that can be extracted from the state This discrepancy between the work cost and work content of states in the single-shot regime results in thermodynamic irreversibility when transforming between states. We find that in many protocols, for example thermodynamically reversible cooling, the work must have fluctuations that diverge in size This makes realising these protocols practically infeasible, especially for small or fragile machines. We derive expressions for the c-bounded work that interpolate between these two regimes of deterministic and freely fluctuating work We apply these results to the study of a single qubit thermal engine, and derive a corrected Carnot efficiency when fluctuations in the work produced by the engine are constrained
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