Principle of Minimal Work Fluctuations

Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, in considering the Jarzynski equality $\langle e^{-\beta W} \rangle=e^{-\beta \Delta F}$, a change in the fluctuations of $e^{-\beta W}$ may impact on how fast the statistical average of $e^{-\beta W}$ converges towards the theoretical value $e^{-\beta \Delta F}$, where $W$ is the work, $\beta$ is the inverse temperature, and $\Delta F$ is free energy difference between two equilibrium states. Motivated by our previous study aiming at the suppression of work fluctuations, here we obtain a principle of minimal work fluctuations. In brief, adiabatic processes as treated in quantum and classical adiabatic theorems yield the minimal fluctuations in $e^{-\beta W}$. In the quantum domain, if a system initially prepared at thermal equilibrium is subject to a work protocol but isolated from a bath during the time evolution, then a quantum adiabatic process without energy level crossing (or an assisted adiabatic process reaching the same final states as in a conventional adiabatic process) yields the minimal fluctuations in $e^{-\beta W}$, where $W$ is the quantum work defined by two energy measurements in the beginning and at the end of the process. In the classical domain where the classical work protocol is realizable by an adiabatic process, then the classical adiabatic process also yields the minimal fluctuations in $e^{-\beta W}$. Numerical experiments based on a Landau-Zener process confirm our theory in the quantum domain, and our theory in the classical domain explains our previous numerical findings regarding the suppression of classical work fluctuations [G.~Y.~Xiao and J.~B.~Gong, Phys. Rev. E {\bf 90}, 052132 (2014)].