  In this paper we study computationally feasible bounds for relative free
energies between two many-particle systems. Specifically, we consider systems
out of equilibrium that do not necessarily satisfy a fluctuation-dissipation
relation, but that nevertheless admit a nonequilibrium steady state that is
reached asymptotically in the long-time limit. The bounds that we suggest are
based on the well-known Bogoliubov inequality and variants of Gibbs' and
Donsker-Varadhan variational principles. As a general paradigm, we consider
systems of oscillators coupled to heat baths at different temperatures. For
such systems, we define the free energy of the system relative to any given
reference system (that may or may not be in thermal equilibrium) in terms of
the Kullback-Leibler divergence between steady states. By employing a two-sided
Bogoliubov inequality and a mean-variance approximation of the free energy (or
cumulant generating function, we can efficiently estimate the free energy cost
needed in passing from the reference system to the system out of equilibrium
(characterised by a temperature gradient). A specific test case to validate our
bounds are harmonic oscillator chains with ends that are coupled to Langevin
thermostats at different temperatures; such a system is simple enough to allow
for analytic calculations and general enough to be used as a prototype to
estimate, e.g., heat fluxes or interface effects in a larger class of
nonequilibrium particle systems.
