  The electrostatic screening properties of ionic fluids are of paramount
importance in countless physical processes. Yet, the behavior of ionic
conductors out of thermal equilibrium has to date mainly been studied in the
context of thermodiffusion phenomena by virtue of direct extensions of
Debye-H\"uckel theories. We investigate how the static response of a symmetric
ionic fluid is influenced by the presence of a thermal gradient by introducing
a theory of electrostatic screening under a stationary temperature profile. By
borrowing mathematical methods commonly used in the semiclassical approximation
of quantum particles, we find analytical solutions to the asymptotic decay of
the charge density which can be used to describe the non-equilibrium response
of the system to external charge perturbations and for arbitrary ionic
concentrations. Notably, a transition between monotonic and oscillatory
screening regimes is observed as an effect of the temperature variation which
generalizes known results of thermal equilibrium to out of equilibrium
conditions. A final quantitative example on the screening of charged surfaces
in aqueous electrolytes shows that the deviation from thermal equilibrium
predicted by our solutions is generally larger than thermodiffusion effects,
and should therefore be taken into account for a comprehensive description of
the electrical double layer. Our findings pave the way to the rigorous
treatment of non-equilibrium steady states in ionic systems with potential
applications to the study of energy materials, nanostructured systems and
waste-heat-recovery technologies.
