  Thermodynamics describes large-scale, slowly evolving systems. Two modern
approaches generalize thermodynamics: fluctuation theorems, which concern
finite-time nonequilibrium processes, and one-shot statistical mechanics, which
concerns small scales and finite numbers of trials. Combining these approaches,
we calculate a one-shot analog of the average dissipated work defined in
fluctuation contexts: the cost of performing a protocol in finite time instead
of quasistatically. The average dissipated work has been shown to be
proportional to a relative entropy between phase-space densities, to a relative
entropy between quantum states, and to a relative entropy between probability
distributions over possible values of work. We derive one-shot analogs of all
three equations, demonstrating that the order-infinity Renyi divergence is
proportional to the maximum possible dissipated work in each case. These
one-shot analogs of fluctuation-theorem results contribute to the unification
of these two toolkits for small-scale, nonequilibrium statistical physics.
