Classical limit of Jarzynski Equality

The study of mesoscopic systems out of balance is actually one of the most active areas of physics. Several approaches to the treatment of these systems have been developed in the last decade, highlighting the results of Jarzynski as well its experimental verification in many systems. The discovery dates the nineties of last century, where an exact relationship between non equilibrium work and variation of the Helmholtz free energy of the system prepared interest in two states of thermal equilibrium at temperature T. It is conjectured that, in the coming years, that quantum devices can be built using this rule, which will work in your state nonequilibrium. Theoretically, it has been shown that quantum systems 'ratchet' are candidates for a new type of mesoscopic device working well in contact with a thermal reservoir and subjected to external driving. In this work, we investigated the classical and quantum equilibrium relationships for the harmonic oscillator and analyze the classical limit equal Jarzynski for the system. The model for the study consists of a harmonic oscillator with a linear disturbance. We used Statistical Mechanics methods based on the canonical ensemble and perturbation theory to obtain the quantum analog, also we use quantum open system technics. We calculated the partition function, the Helmholtz free energy, stochastic work and showed classical Jarzynski equality for the harmonic oscillator with external drive. The same calculation were obtained for its quantum analogue, considering the expected value of work in two areas: the work as the variation of energy and work as an operator in the Heisenberg representation. The two definitions for quantum work and its corresponding averages correctly describe classical result for $\beta\ll 1$, otherwise we are in the quantum regime and Jarzynski equality is not satisfied.\\

Acknowledgments: Capes, CNPq and Fapemig.