  We propose and analyze a new candidate Lyapunov function for relaxation
towards general nonequilibrium steady states. The proposed functional is
obtained from the large time asymptotics of time-symmetric fluctuations. For
driven Markov jump or diffusion processes it measures an excess in dynamical
activity rates. We present numerical evidence and we report on a rigorous
argument for its monotonous time-dependence close to the steady nonequilibrium
or in general after a long enough time. This is in contrast with the behavior
of approximate Lyapunov functions based on entropy production that when driven
far from equilibrium often keep exhibiting temporal oscillations even close to
stationarity.
