  We show that distribution functions of nonequilibrium steady states (NESS)
evolving under a slowly varying protocol can be accurately obtained from
limited data and the closest known detailed state of the system. In this
manner, one needs to perform only a few detailed experiments to obtain the
nonequilibrium distribution function for the entire gamut of nonlinearity. We
achieve this by maximizing the relative entropy functional (MaxRent), which is
proportional to the Kullback-Leibler distance from a known density function,
subject to constraints supplied by the problem definition and new measurements.
MaxRent is thus superior to the principle of maximum entropy (MaxEnt), which
maximizes Shannon's informational entropy for estimating distributions but
lacks the ability of incorporating additional prior information. The MaxRent
principle is illustrated using a toy model of $\phi^4$ thermal conduction
consisting of a single lattice point. An external protocol controlled
position-dependent temperature field drives the system out of equilibrium. Two
different thermostatting schemes are employed: the Hoover-Holian deterministic
thermostat (which produces multifractal dynamics under strong nonlinearity) and
the Langevin stochastic thermostat (which produces phase space-filling
dynamics). Out of the 80 possible states produced by the protocol, we assume
that 4 states are known to us in detail, one of which is used as input into
MaxRent at a time. We find that MaxRent accurately approximates the phase space
density functions at all values of the protocol even when the known
distribution is far away. MaxEnt, however, is unable to capture the fine
details of the phase space distribution functions. We expect this method to be
useful in other external protocol driven nonequilibrium cases as well.
