  Studying the structure of systems in nonequilibrium steady states
necessitates tools that quantify population shifts and associated deformations
of equilibrium free energy landscapes under persistent currents. Within the
framework of stochastic thermodynamics, we establish a variant of the
Kawasaki--Crooks equality that relates nonequilibrium free energy corrections
in driven systems to heat dissipation statistics along time-reversed relaxation
trajectories computable with molecular simulation. Using stochastic control
theory, we arrive at a variational approach to evaluate the Kawasaki--Crooks
equality and use it to estimate distribution functions of order parameters in
overdamped Langevin models of active matter, attaining substantial improvement
in accuracy over simple perturbative methods.
