  Work fluctuation and total entropy production play crucial roles in small
thermodynamic systems subject to large thermal fluctuations. We investigate a
trade-off relation between them in a nonequilibrium situation in which a system
starts from an arbitrary nonequilibrium state. We apply the variational method
to study this problem and find a stationary solution against variations over
protocols that describe the time dependence of the Hamiltonian of the system.
Using the stationary solution, we find the minimum of the total entropy
production for a given amount of work fluctuation. An explicit protocol that
achieves this is constructed from an adiabatic process followed by a
quasi-static process. The obtained results suggest how one can control the
nonequilibrium dynamics of the system while suppressing its work fluctuation
and total entropy production.
