Approximating the entire spectrum of nonequilibrium steady-state distributions using relative entropy: An application to thermal conduction.

Distribution functions for systems in nonequilibrium steady states are usually determined through detailed experiments, both in numerical and real-life settings in the laboratory. However, for a protocol-driven distribution function, it is usually prohibitive to perform such detailed experiments for the entire range of the protocol. In this article 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) subject to constraints supplied by the problem definition and new measurements. MaxRent is found to be superior to the principle of maximum entropy (MaxEnt), which maximizes Shannon's informational entropy for estimating distributions but lacks the ability to incorporate additional prior information. The MaxRent principle is illustrated using a toy model of ϕ4 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 four states are known to us in detail, one of which is used as input into MaxRent at a time. We find that MaxRent approximates the phase-space density functions for every value of the protocol, even when they are far from the known distribution. 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, making it unnecessary to perform detailed experiments for all values of the protocol when one wishes to obtain approximate distributions.