  We discuss the physical meaning and significance of statistical forces on
quasi-static probes in first order around detailed balance for driven media.
Exploiting the quasi-static energetics and the structure of (McLennan) steady
nonequilibrium ensembles, we find that the statistical force obtains a
nonequilibrium correction deriving from the excess work of driving forces on
the medium in its relaxation after probe displacement. This reformulates,
within a more general context, the recent result by N. Nakagawa (Phys. Rev. E
90, 022108 (2014)) on thermodynamic aspects of weakly nonequilibrium adiabatic
pumping. It also proposes a possible operational tool for accessing some excess
quantities in steady state thermodynamics. Furthermore, we show that the point
attractors of a (macroscopic) probe coupled to a weakly driven medium realize
the predictions of the minimum entropy production principle. Finally, we
suggest a method to measure the relative dynamical activity through different
transition channels, via the measurement of the statistical force induced by a
suitable driving.
