  The Jarzynski equality (JE) provides a nonequilibrium method to measure and
calculate the free energy difference (FED). Note that if two systems share the
same Hamiltonian at two equilibrium states, respectively, they share the same
FED between these two equilibrium states as well. Therefore the calculation of
the FED of a system may be facilitated by considering instead another virtual
system designed to this end. Taking advantage of this flexibility and the JE,
we show that by introducing an integrable virtual system, the evolution problem
involved in the JE can be solved. As a consequence, FED is expressed in the
form of an equilibrium equality, in contrast with the nonequilibrium JE it is
based on. Numerically, this result allows FED to be computed by sampling the
canonical ensemble directly and the computational cost can be significantly
reduced. The effectiveness and efficiency of this scheme are illustrated with
numerical studies of several representative model systems.
