Irreversible effects of memory

The steady state of a Langevin equation with short-ranged memory and coloured noise is analyzed. When the fluctuation-dissipation theorem of second kind is not satisfied, the dynamics is irreversible, i.e. detailed balance is violated. We show that the entropy production rate for this system should include the power injected by "memory forces". With this additional contribution, the fluctuation relation is fairly verified in simulations. Both dynamics with inertia and overdamped dynamics yield the same expression for this additional power. The role of "memory forces" within the fluctuation-dissipation relation of first kind is also discussed.