  Graham has shown in Z. Physik B 26, 397-405 (1977) that a
fluctuation-dissipation relation can be imposed on a class of non-equilibrium
Markovian Langevin equations that admit a stationary solution of the
corresponding Fokker-Planck equation. The resulting equilibrium form of the
Langevin equation is associated with a nonequilibrium Hamiltonian. Here we
provide some explicit insight into how this Hamiltonian may loose its time
reversal invariance and how the "reactive" and "dissipative" fluxes loose their
distinct time reversal symmetries. The antisymmetric coupling matrix between
forces and fluxes no longer originates from Poisson brackets and the "reactive"
fluxes contribute to the ("housekeeping") entropy production, in the steady
state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian
contribute in qualitatively different but physically instructive ways to the
entropy. We find instances where fluctuations due to noise are solely
responsible for the dissipation. Finally, this structure gives rise to a new,
physically pertinent instance of frenesy.
