2 systems are said to be in thermal equilibrium with each other if there is no change in co- ordinates of either system when they are separated by a diathermal wall.
You are having a confusion between Thermal Eqb. and Steady State.
Thermal Eqb.: Ṫ = 0 = ∇T
Steady State: ∂T/∂t = 0; where T = T(r,t) is the temperature of the system.
@ Meir Achuz
But what about the gauge condition? Isn't it another constraint. So now there r 4 unknowns and 5 equations. Again a question of mutual consistency arises.
@ Phrak
I really don't get all that mathematics. Sorry, but I can't help. I guess I'm not upto ur level in mathematical...
The no. of unknowns in the Maxwell's Equations is 6- the components of E and B. But there are 8 partial differential equations if we separate them out component- wise. So when we find the solution to them how do we ensure their mutual consistency?