Complementarity relation for irreversible processes near steady states

A relation giving a minimum for the irreversible work in quasi-equilibrium processes was derived by Sekimoto et al. [K. Sekimoto, S. Sasa, J. Phys. Soc. Japan 66 (1997) 3326] in the framework of stochastic energetics. This relation can also be written as a type of “uncertainty principle” in such a way that the precise determination of the Helmholtz free energy through the observation of the work 〈W〉 requires an indefinitely large experimental time Δt. In the present article, we extend this relation to the case of quasi-steady processes by using the concept of non-equilibrium Helmholtz free energy. We give a formulation of the second law for these processes that extends that presented by Sekimoto [K. Sekimoto, Prog. Theoret. Phys. Suppl. No. 130 (1998) 17] by a term of the first order in the inverse of the experimental time. As an application of our results, two possible experimental situations are considered: stretching of a RNA molecule and the drag of a dipolar particle in the presence of a gradient of electric force.