Dynamic dependence of nonequilibrium work limits the validity of the Jarzynski Equality

The Jarzynski equality (JE) is analyzed in regard to its validity for both quasi-static transformations in the thermodynamic limit and Hamiltonian evolutions of the work protocol. In the first case, we show that the JE holds for isothermal transformations only; in the second case, we show that the work done (linked to the final state Hamiltonian) depends on the temporal dynamics of the work protocol (including its speed), thus precluding the possibility of identifying it with the free energy or any other thermodynamic state function. Even in the case of thermodynamic limit and infinitesimally slow transformations of the Hamiltonian (adiabatic invariance) following states of thermodynamic equilibrium, the resulting work expression does not default to the JE, but to the work relation for adiabatic thermodynamic transformations, W=Delta U, where W is work and U is internal energy.