On the work–free energy relations in a driven isothermal system: A historical and stochastic perspective

Equality relationships, connecting the work performed in a non-equilibrium driven process of an isothermal system with the difference of equilibrium thermodynamic potentials, have been established by various authors since Jarzynski’s work of 1997, mostly based on a classical phase–space description. Quantum considerations have been limited and stochastic derivations have been mainly ad hoc. However, such processes are better described by an ensemble in the a-space for fluctuating state (or mesoscopic) variables. Going back to the probabilistic results of Boltzmann and Einstein and employing a few basic properties of (generally non-stationary) Markov processes, these relationships follow almost naturally, providing the initial and final states are in contact with the heat bath. The essential point is that in the exponentiated work exp(–βW), the non-equilibrium entropy functions of both initial and final states cancel out. The results obtained are valid for any particle number N. We note that the connection with the microscopic quantum statistical evolution must be considered in order to proceed. In addition, some aspects of irreversibility are briefly being discussed.