Validity condition of the Jarzynski relation for a classical mechanical system

Recently, Jarzynski suggested a striking thermodynamic equation that relates free energy change of a system and work done on the system during arbitrary nonequilibrium processes, which has been believed to hold irrespective of detailed nature of the nonequilibrium process. However, we show here that the Jarzynski equation does not hold for an adiabatic process unless the phase-space extension of the system on completion of the adiabatic process coincides with that of the final equilibrium state of the system. This condition can be satisfied only when the adiabatic process does not change parameters on which the equilibrium phase-space extension of our system is dependent.