  The Jarzynski equality allows the calculation of free-energy differences
using values of work measured in nonequilibrium trajectories. The number of
trajectories required to accurately estimate free-energy differences in this
way grows sharply with the size of work fluctuations, motivating the search for
protocols that perform desired transformations with minimum work. However,
protocols of this nature can involve varying temperature, to which the
Jarzynski equality does not apply. We derive a variant of the Jarzynski
equality that applies to varying-temperature protocols, and show that it can
have better convergence properties than the standard version of the equality.
We derive this modified equality, and the associated fluctuation relation,
within the framework of Markovian stochastic dynamics, complementing related
derivations done within the framework of Hamiltonian dynamics.
