What is the fundamental difference between equilibrium and non-equilibrium thermodynamics? Can we use the formalism of equilibrium thermodynamics for a system that is in non-equilibrium steady state?
 A: Thermodynamic non-equilbrium is a negative characterization (the system is  not at thermodynamic equilibrium). It leaves plenty of room for different behaviors which may allow or not to use the formalism of equilibrium thermodynamics.
The basic requirements for thermodynamic equilibrium are:


*

*time independence of the macroscopic physical quantities for the system;

*the state of the system is characterized by a few variables not related to any kind of macroscopic motion;

*the state of the system is independent on the past history.


Only condition n.1 is not enough, since it does not exclude steady fluxes in the system.
Condition n.2 excludes the possibility of treating the whole system as an equilibrium  thermodynamic system if steady state fluxes are present. However, there is the possibility that a system of this class could be treated as a system in Local Thermodynamic Equilibrium (LTE). The condition is that  each subvolume of the system containing a macroscopic number of degrees of freedom could be considered at equilibrium over the typical scale of time connected to the macroscopic motion.
For instance, a steady jet of gas coming out from a high temperature source, could be in LTE conditions if the velocity distribution of small regions of the jet, in the center of mass frame, would be well approximated by a Maxwellian. In that case, one can speak about  local density, temperature, internal energy, etc.
A: Equilibrium impies that the thermodynamic coordinates and hence their functions do not change in time. In a non equilibrium situation these quantities will change with time.
